The Mysterious Crop Circles

George and Iris Owen
55 Charles St. West, #1201
Toronto, Ontario, Canada
M5S 2W9
THE MYSTERIOUS CROP CIRCLES.
A New Horizons Note.
Copyright:
New H o r i z o n s Research F o u n d a t i o n .
December 1991.
INTRODUCTION.
About t e n y e a r s ago a f a s c i n a t i n g , s t r a n g e , and u n u s u a l
phenomenon made i t s appearance.
I t seemed to have
l i t t l e r e l a t i o n s h i p to any o t h e r known phenomena, and
i t d i d n ' t seem t o b e l o n g under any c a t e g o r y .
However,
a number o f c l a i m s have been made o v e r the y e a r s w h i c h
a t t e m p t to c a t e g o r i s e the phenomena as b e l o n g i n g t o
some one p a r t i c u l a r b e l i e f o r c a t e g o r y .
We r e f e r to the
phenomenon known as "crop c i r c l e s " .
The d i s c i p l i n e
c o n c e r n i n g the study o f these c i r c l e s i s known as
"cerealogy".
T h i s p a p e r i s an attempt t o draw t o g e t h e r some o f the
i n f o r m a t i o n about crop c i r c l e s , and a r r i v e a t an
u n d e r s t a n d i n g o f what i s known, o r b e l i e v e d , about them,
today.
The phenomenon has become w i d e l y known d u r i n g the l a s t
t e n y e a r s , and i n f a c t , i n i t s p r e s e n t form, i t has
a p p a r e n t l y o n l y been o c c u r r i n g d u r i n g the l a s t t e n y e a r s .
I t t a k e s the form o f l a r g e p a t t e r n s o f p e r f e c t l y b e n t
c o r n b e i n g formed i n f i e l d s o f g r o w i n g c o r n .
Initially,
these were l a r g e b e a u t i f u l l y formed c i r c l e s , hence
the name; b u t , more r e c e n t l y , the p a t t e r n s formed have
changed, i n c r e a s i n g i n c o m p l e x i t y from y e a r to y e a r ;
the
p a t t e r n s t h a t appeared l a s t y e a r , f o r i n s t a n c e , were
n o t o n l y e x t r e m e l y complex, b u t t h e y were i n d e e d b e a u t i f u l .
Most o f these p a t t e r n s i n the c o r n f i e l d s have appeared
i n a c e r t a i n p a r t o f England, namely the south w e s t e r n
a r e a , i n the v i c i n i t y o f Stonehenge, and n e a r the o t h e r
v a r i o u s a n c i e n t monuments o f t h a t r e g i o n .
However,
t h e y have a l s o been r e p o r t e d i n o t h e r p a r t s o f England,
i n the M i d l a n d s and e a s t e r n r e g i o n s , and i n o t h e r
c o u n t r i e s , n o t a b l y Canada, A u s t r a l i a , Germany, New Z e a l a n d ,
F r a n c e , The U n i t e d S t a t e s o f A m e r i c a , and one has been
r e p o r t e d i n Sweden.
A l t h o u g h the phenomenon has caused so much e x c i t e m e n t
i n recent years, s t o r i e s of simple c i r c l e s appearing i n
f i e l d s o f g r o w i n g c o r n go back many c e n t u r i e s .
There a r e
legends d a t i n g from the Middle Ages t h a t t a l k o f
c i r c l e s b e i n g formed i n the f i e l d s o v e r n i g h t , and these
were a t t r i b u t e d to f a i r i e s dancing through the corn,
o r i t was s a i d mowing d e v i l s came i n the n i g h t and cut
the corn i n ' r i n g s .
One o f us (IMO) grew up on an a r a b l e farm i n e a s t e r n
England and remembers such simple c i r c l e s b e i n g found
i n the c o r n f i e l d s i n the 1920's and 1930s when h a r v e s t
time came around.
L o c a l farmers a t t r i b u t e d them to
weather d i s t u r b a n c e s , the g e n e r a l t h e o r y b e i n g t h a t they
were caused by m i n i a t u r e w h i r l w i n d s .
T h i s was b e f o r e
there were many a i r c r a f t i n the sky, and as the c i r c l e s
were u s u a l l y s i t u a t e d r i g h t i n the middle o f the c o r n f i e l d s ,
they were not u s u a l l y d i s c o v e r e d u n t i l the machines
c u t t i n g the corn reached them.
They were i n v i s i b l e
from the s i d e s o f the f i e l d s , and so i t was i m p o s s i b l e
to know e x a c t l y when they were formed.
I t goes without
s a y i n g t h a t nobody walks through a f i e l d o f growing
corn, i t i s so easy to damage i t .
I n these e a r l y
c i r c l e s i t was sometimes obvious t h a t the c i r c l e had
been formed e a r l i e r i n the growing season, as the c o m
was l y i n g f l a t , but the e a r s had continued to grow,
and were p u t t i n g out green shoots.
The p r e s e n t r a s h o f phenomena s t a r t e d w i t h j u s t such
c i r c l e s , a simple round, some twenty o r more f e e t i n
diameter, with the corn l a y i n g down, a l l i n one d i r e c t i o n .
I t was l a i d p e r f e c t l y smoothly, as i f i t had been swept
downwards by some strange f o r c e .
T y p i c a l l y the r o o t s
were n o t damaged, and the s t a l k s were n o t broken, o n l y
bent.
The corn continued to grow i n t h i s p o s i t i o n .
The c i r c l e s u s u a l l y appeared o v e r n i g h t , and were q u i c k l y
n o t i c e d , not o n l y by p i l o t s o f a i r c r a f t o v e r f l y i n g the
f i e l d s , but a l s o because i n many cases they appeared i n
f i e l d s overlooked by h i l l y country.
However, t h i s simple p a t t e r n was n o t to remain f o r l o n g .
D u r i n g succeeding years the p a t t e r n s grew more and more
e l a b o r a t e , f i r s t t a k i n g the form o f double and t r i p l e
r i n g s around the o r i g i n a l c i r c l e , and then a c q u i r i n g
f u r t h e r e x t e n s i o n s t h a t i n many cases looked l i k e
h i e r o g l y p h i c s , o r some p r i m i t i v e attempt a t communication.
Recent c i r c l e s were a l s o c o n s i d e r a b l y l a r g e r , some
b e i n g as many as hundreds o f f e e t i n diameter, and
occupying i n t h e i r d e s i g n some thousands o f f e e t o f c o r n f i e l d .
F o r many farmers these formations proved more than a
minor problem.
Not o n l y were l a r g e t r a c t s o f growing
corn b a d l y damaged, and i n some cases r u i n e d , but the
r e s t o f the f a r m e r ' s crops were u s u a l l y trampled by the
rush o f the s i g h t s e e r s and i n v e s t i g a t o r s who came i n
the wake o f such s i g h t i n g s .
Many farmers had t h e i r
whole crops d e s t r o y e d .
I f , as has been suggested, some
o f these c i r c l e s are caused by hoaxers and p r a n k s t e r s ,
i t i s a v e r y i r r e s p o n s i b l e course o f a c t i o n indeed,
and one would t h i n k t h a t the farmers who s u f f e r would
have a c l a i m f o r compensation.
THE PHENOMENA.
T h i s c u r r e n t r a s h o f phenomena s t a r t e d i n 1980, when
a l a r g e number o f c i r c l e s were found i n the south western
p a r t o f England.
They appeared to be cut w i t h almost
s u r g i c a l p r e c i s i o n i n f i e l d s o f growing wheat and
barley.
The e a r l i e r c i r c l e s were always r e p o r t e d
as b e i n g i n c o r n f i e l d s , b u t o f r e c e n t years the p a t t e r n s
have been found i n f i e l d s o f rape, among growing r o o t
c r o p s , such as sugar b e e t , i n hayfmelds, i n sugar cane
i n A u s t r a l i a , and there are even a few r e p o r t s o f the
f o r m a t i o n s o c c u r r i n g on the s i d e s o f dusty roads, and
one i s r e p o r t e d as happening i n a f i e l d o f p r i c k l y
thistles.
An i n t e r e s t i n g f e a t u r e o f many o f these p a t t e r n s i s
t h a t even a f t e r the crop has been h a r v e s t e d , the p l a c e
where the c i r c l e has been i s s t i l l obvious, as i f the
ground
s u r f a c e were i n some way c h a n g e d .
I t seems,
i n these cases, as i f something more than j u s t the bending
o f the corn took p l a c e .
However, a l l t r a c e s o f the
p a t t e r n s d i s a p p e a r s when w i n t e r comes and the s o i l
i s p r e p a r e d f o r next y e a r ' s c r o p .
A l s o , there are no
r e p o r t s , as f a r as we are aware, o f a c i r c l e , o r p a t t e r n ,
r e a p p e a r i n g on the same spot the f o l l o w i n g y e a r , o r
o f the crop b e i n g i n any way a f f e c t e d the next y e a r .
I f the s o i l had been a f f e c t e d by e i t h e r chemicals,
or b u r n i n g , one would expect subsequent crops to show
evidence o f t h i s .
As we have s a i d i n the I n t r o d u c t i o n , o r i g i n a l l y these
p a t t e r n s took the form o f simple c i r c l e s , some s m a l l , and
o t h e r s huge i n diameter, but d u r i n g the succeeding
y e a r s the p a t t e r n s took on the forms o f oblongs, squares,
r e c t a n g l e s , and t r i a n g l e s , and o t h e r g e o m e t r i c a l p a t t e r n s .
C i r c l e s would be connected, o r there would be a l a r g e
c i r c l e with s e v e r a l small ones grouped around, or a
s e r i e s o f r i n g s would surround an i n n e r c i r c l e .
L a t e r s t i l l the p a t t e r n s became even more c o m p l i c a t e d ,
and some took on i n s e c t - l i k e forms, o t h e r s produced
s q u i g g l y p a t t e r n s , l i k e n e d to b r a i n p a t t e r n s , and
o t h e r s resembled a c h i l d ' s s c r i b b l i n g .
We have
a t t a c h e d a number o f examples to t h i s p a p e r .
During
the t e n years from 1980 - 1990,
almost every c o n c e i v a b l e
k i n d o f p a t t e r n emerged.
I n a l l the r e p o r t e d cases the corn i s bent i n one
d i r e c t i o n o n l y , i t i s not trampled h a p h a z a r d l y , i t
i s not crumpled, i t l a y s s t r a i g h t .
The s t a l k s , although
bent, are not broken, and the corn continues to grow
i n the ground.
Corn continues to grow and r i p e n
when the s t a l k s are bent, i f the r o o t s are not p u l l e d
out and the s t a l k s not broken.
The corn w i l l continue
to r i p e n , and when r i p e , because o f the c o n t a c t w i t h
the ground,the wheat i n the ears w i l l sprout green
shoots.
The l o s s to the farmer l i e s i n the f a c t
t h a t he i s unable to reap the h a r v e s t .
A f u r t h e r word o f e x p l a n a t i o n .
I n a l l the p u b l i s h e d
p i c t u r e s there are s t r a i g h t l i n e s r u n n i n g the l e n g t h o f
the f i e l d s .
These s t r a i g h t l i n e s are furrows made
by the wheels o f the t r a c t o r s t h a t sowed the seed i n
the s p r i n g t i m e .
The seed does not f a l l i n these furrows,
and so these l i n e s are l e f t when the corn grows.
Because modern machinery i s o f t e n v e r y l a r g e , and the
v e h i c l e s have wide wheels these furrows can be wide
enough f o r persons to walk i n w i t h o u t damaging the
corn.
I n the days when h o r s e s : p u l l e d the seeding
machinery these furrows d i d not e x i s t .
I f one d e v i a t e s
from the t r a c k s , however, the corn w i l l be damaged.
Growing corn i s v e r y heavy i n the e a r s , and i t i s easy
to damage i t , and the damage i s o b v i o u s .
This i s a
f a c t o r to take i n t o c o n s i d e r a t i o n i f one i s l o o k i n g
a t the p o s s i b i l i t y o f these p a t t e r n s b e i n g the r e s u l t
o f hoaxes o r p r a c t i c a l j o k e s .
The p a t t e r n s u s u a l l y have o c c u r r e d d u r i n g the months
of J u l y and August, when the crops are r i p e n i n g .
However,
i n 1991, the weather i n England i n the springtime was
c o l d and wet, and so the h a r v e s t was l a t e r than u s u a l .
However, i n many cases, the c i r c l e s s t a r t e d forming a t
the u s u a l time, appearing i n the young, s t i l l green,
corn.
T h i s would cause one to wonder i f the time o f
y e a r was a f a c t o r , and not n e c e s s a r i l y the r i p e n e s s o f
the c o r n .
T h i s l a s t y e a r , 1991. saw some v e r y v a r i e d and complicated
p a t t e r n formations indeed.
The e a r l i e r ones were
l a b e l l e d 'insectograms' by the i n v e s t i g a t o r s , as they
resembled some weird k i n d o f i n s e c t .
About a dozen
' i n s e c t o g r a m s ' appeared i n the c o u n t i e s o f W i l t s h i r e
and Hampshire d u r i n g the e a r l y 1991 ' s e a s o n ' .
As
the summer went on f u r t h e r strange shapes and p a t t e r n s
appeared.
A r e p o r t i n the F o r t e a n Times g i v i n g the
types o f formations d u r i n g that summer l i s t s ' d u m b - b e l l s ' ,
( c o n s i s t i n g o f two or sometimes three p l a i n o r r i n g e d
c i r c l e s j o i n e d by a narrow c o r r i d o r ) }
h a l o e s were
sometimes added to these shapes, o r perhaps r e c t a n g u l a r
' c o f f i n s ' or o t h e r appendages.
A l s o l i s t e d are ' c u r l y *
p a t t e r n s resembling a c h i l d ' s s c r i b b l i n g s , a 'wiggly
p i p e c l e a n e r man', an "Irminsul" ( d e s c r i b e d as the
a n c i e n t long-suppressed symbol o f German paganism),
and an e l a b o r a t e m a z e - l i k e formation which was a p t l y
named "The B r a i n " .
Another type o f formation was i n
the shape o f a whale, and some o f these had' f l i p p e r s .
One o f the most s t r i k i n g o f t h i s y e a r ' s ' c r o p ' appeared
a t Barbary C a s t l e , i n a w h e a t f i e l d j u s t below the
ruined castle.
I t was f i r s t spotted by the p i l o t o f
a l i g h t a i r c r a f t f l y i n g i n the neighbourhood,
A copy
o f the p i c t u r e o f t h i s geometric p a t t e r n i s found i n
the Appendix.
Before i t was damaged by storms and
v i s i t o r s i t was surveyed a c c u r a t e l y , and an a n a l y s i s
made o f the meanings conveyed by i t s geometry, a h i c h
are s a i d to be s t a r t l i n g and r e v e l a t o r y .
We quote from an a r t i c l e e n t i t l e d "Geometry & Symbolism
a t Barbefry C a s t l e " which appeared i n The C e r e a l o g i s t
(No. 4, Summer 1991).
"The Barbary C a s t l e f o r m a t i o n i s a r e g u l a r but p r e v i o u s l y
unknown form o f g e o m a t r i c a l diagram
We are
p r e s e n t e d w i t h a n o v e l a n d / i n t e r e s t i n g l e s s o n i n geometry,
but the i m p l i c a t i o n s o f t h i s f i g u r e are not merely academic,
f o r the n u m e r i c a l s t r u c t u r e behind i t s dimensions are
f a m i l i a r to students o f the a n c i e n t n u m e r i c a l s c i e n c e ,
and have deep c o s m o l o g i c a l s i g n i f i c a n c e .
There are many m y s t e r i e s about t h i s f o r m a t i o n , i n c l u d i n g
the method and sequence o f i t s c r e a t i o n , and the
strange l i g h t s and other phenomena which c o i n c i d e d
w i t h i t s appearance.
A l s o enigmatic i s the impressive
r a t c h e t s p i r a l a t the s o u t h - e a s t c o r n e r .
Yet the
b a s i c g e o m e t r i c a l idea behind the diagram i s c l e a r
enough.
I t demonstrates the p r i n c i p l e o f Three i n One
by means o f a c e n t r a l c i r c l e which e x a c t l y c o n t a i n s
the combined a r e a s o f the three c i r c l e s around i t .
Moreover, the sum o f a l l the f o u r c i r c u l a r a r e a s i n
the diagram i s 31680 square f e e t .
The s i g n i f i c a n c e
o f t h i s number i n a r i t h m e t i c , cosmology, a n c i e n t theology,
and temple a r c h i t e c t u r e , was f i r s t explored i n C i t y o f
R e v e l a t i o n (1972), and i s summed up i n a s e c t i o n o f
The Dimensions o f P a r a d i s e (Thames and Hudson, 1988).
In t r a d i t i o n a l cosmology, 31680 m i l e s was taken to
be the measure around the s u b - l u n a r y w o r l d , and the
e a r l y C h r i s t i a n s c h o l a r s c a l c u l a t e d the number 3168
as emblematic o f Lord Jesus C h r i s t .
The same number
was p r e v i o u s l y a p p l i e d to the name o f a l e a d i n g
p r i n c i p l e i n the pagan r e l i g i o n . "
The w r i t e r o f the a r t i c l e goes on to say "Neither
p h y s i c a l l y n o r i n t e l l e c t u a l l y , does t h i s f i g u r e g i v e
s i g n s o f b e i n g a human c r e a t i o n .
To i d e n t i f y i t s author
seems t h e r e f o r e to be a problem f o r t h e o l o g y .
One's
r a t i o n a l mind s h r i n k s away from the i m p l i c a t i o n t h a t
t h i s diagram c o n s t i t u t e s a d i v i n e r e v e l a t i o n " .
The B a r b e r y C a s t l e f i g u r e was spotted on the 17th J u l y .
D u r i n g the second week o f August, however, an even more
s t a r t l i n g geometric f i g u r e appeared i n a c o r n f i e l d i n
Cambridgeshire, a t a v i l l a g e c a l l e d I c k l e t o n , and
some c o n s i d e r a b l e d i s t a n c e away from Barbary C a s t l e .
A c c o r d i n g to a r e p o r t i n F o r t e a n Times, " t h i s f i g u r e
shocked the minds o f every mathematician who saw i t .
George W i n g f i e l d , an i n i t i a t e o f f r a c t a l geometry and
chaos theory, f l e w over i t , and r e c o g n i s e d the p e r f e c t ,
unmistakeable form o f the Mandelbrot Set, the b u g - l i k e
f i g u r e from the world beyond the surface o f apparent
forms, which D r . Mandelbrot d i s c o v e r e d about 15 y e a r s ago.
I t s image i n the w h e a t f i e l d was b e a u t i f u l l y executed,
the wheatstalks "being l a i d i n m u l t i p l e s w i r l e d l a y e r s
to create the c h a r a c t e r i s t i c p l a i t e d e f f e c t which no
human i m i t a t o r s o f crop c i r c l e s have been a b l e to
reproduce".
(A note on the Mandelbrot Set i s found
i n the A p p e n d i x ) .
One accompaniment to the phenomena we have not as
y e t mentioned i s that o f l i g h t s and sound.
While
the m a j o r i t y o f the p a t t e r n s appear d u r i n g the n i g h t t i m e ,
and a p p a r e n t l y without any s i g n o f t h e i r happening,
i n some cases people have r e p o r t e d h e a r i n g n o i s e s , such
as bangs, and seeing l i g h t s o f v a r i o u s k i n d s .
I n many
o f these i n s t a n c e s people nearby have o f t e n jumped to
the c o n c l u s i o n that v i s i t i n g UFO's have caused the
p a t t e r n s to o c c u r .
There have, o f course, a l s o been
a few r e p o r t s o f close encounters with a l i e n s , and .
strange people, a l l e g e d l y from o u t e r space.
These
same people b e l i e v e t h a t the p a t t e r n s are some form o f
attempted communication with people on e a r t h .
N e v e r t h e l e s s , i n most o f the cases the phenomena
have appeared o v e r n i g h t , and completely s i l e n t l y .
I n some s i t u a t i o n s i n v e s t i g a t o r s have a c t u a l l y been
camped out o v e r n i g h t i n the f i e l d s , watching h o p e f u l l y
f o r the c i r c l e s to appear, o n l y to d i s c o v e r , to t h e i r
s u r p r i s e and c h a g r i n when d a y l i g h t a r r i v e s , t h a t the
c i r c l e s and p a t t e r n s have formed d u r i n g the n i g h t i n
the next f i e l d , or j u s t behind them, and they have seen
o r heard n o t h i n g .
Although we have s a i d e a r l i e r i n t h i s a r t i c l e t h a t
the c i r c l e phenomenon has mainly developed d u r i n g the
l a s t t e n years or so, there were many r e p o r t s o f c i r c l e
f o r m a t i o n d u r i n g the 1970's, and these came from many
other countries.
Crop c i r c l e s were r e p o r t e d i n
Langenberg, Sask, Canada, f o r i n s t a n c e , i n 1974, and
they continue to occur i n both Saskatchewan and A l b e r t a .
They were u s u a l l y a t t h a t time a t t r i b u t e d to UFO's
and were f r e q u e n t l y designated as ' l a n d i n g r i n g s ' .
T h e i r p h y s i c a l appearance was e x a c t l y l i k e the p a t t e r n s
o f r e c e n t times - the corn was bent, not broken, and
the s t a l k s a l l l a y p e r f e c t l y s w i r l e d i n one d i r e c t i o n .
They a l s o o c c u r r e d i n A u s t r a l i a , where they were
commonly r e f e r r e d to as "UFO n e s t s " .
They appeared
there from the b e g i n n i n g o f the 1 9 7 0 ' s ,
These
"UFO nests" n o t o n l y appeared i n growing corn, but
they appeared i n other c r o p s , such as h a y f i e l d s ,
and f i e l d s o f sugar cane.
In one p a r t i c u l a r i n s t a n c e
the ' n e s t ' was found deep w i t h i n a f i e l d o f p r i c k l y
saffron t h i s t l e s .
I t i s worth q u o t i n g the account
g i v e n by the i n v e s t i g a t o r from the l o c a l UFO s o c i e t y
who v i s i t e d the ' n e s t ' .
"It was
a sense
and the
mirages
a day o f d r y heat, o f bush f i r e s , there was
o f n o t h i n g l i v i n g i n the p a l e y e l l o w d r i e d g r a s s e s ,
e n d l e s s a s p h a l t road s h i f t e d and shimmered w i t h
o f the M y t h i c a l I n l a n d Sea.
At f i r s t s i g h t the UFO n e s t seemed from a d i s t a n c e to
have been scorched i n t o the f i e l d by the g i g a n t i c r e d
sun hanging low i n a h e a t - e r a z e d sky.
The c i r c l e
c o u l d n o t have been p l a c e d i n a more i n h o s p i t a b l e
environment, gouged as i t was deep w i t h i n a f i e l d o f
p r i c k l y saffron t h i s t l e s .
F o r t u n a t e l y f o r our l e g s ,
but r a t h e r i n c o n v e n i e n t l y f o r the r e c o r d i n g o f the
c i r c l e f o r p o s t e r i t y , M r . V i v H u c k e l , the proud r e c i p i e n t
o f t h i s n e s t had ploughed up to the c i r c l e a t the request
o f the ABC (The A u s t r a l i a n B r o a d c a s t i n g Commission)
the day b e f o r e .
And f o r a very down-to-earth A u s s i e
farmer he had p l e n t y to say on the s u b j e c t .
What h i t me when I f i r s t saw i t was t h a t i t was v e r y
s i m i l a r to n e s t s found i n the sugar cane beds up i n
Queensland" s a i d V i v ; "I drove around, f i n i s h e d
p l o u g h i n g , and came back and had another s q u i z z a t i t .
What i n t r i g u e d me was t h a t the centre p o r t i o n (about
f o u r f e e t a c r o s s ) was almost completely b a r e , and a f t e r
t h a t you s t a r t e d g e t t i n g l i t t l e stumps o f s a f f r o n
t h i s t l e s - t h e y ' d been shredded.
The f u r t h e r you got
to the p e r i m e t e r , p i e c e s o f s t a l k were j u s t broken up;
b u t the l a s t couple o f f e e t o f t h i s t l e s - and these are
two f e e t h i g h minimum - were completely knocked down i n
an a n t i c l o c k w i s e d i r e c t i o n , t w i s t e d up, and some had
been completely t o r n out by the; roots'.
It's pretty
hard to p u l l a green s a f f r o n t h i s t l e by the r o o t s , m a t e . "
The farmer went on to say t h a t the damage c o u l d not
have been caused by c a t t l e .
He had had o n l y 5 head o f
cows i n t h a t paddock d u r i n g the p r e v i o u s s i x months,
and he s t a t e d t h a t n e i t h e r cows n o r sheep would ' b u l l '
t h e i r way i n t o s a f f r o n t h i s t l e s .
N e i t h e r was i t a
' h a r e s p l a y g r o u n d ' he s a i d .
He a l s o d i s c o u n t e d the idea
t h a t i t c o u l d have "been a w h i r l w i n d - "It d o e s n ' t
s t a y i n the one spot - i t sweeps s t r a i g h t through a
paddock and l e a v e s a dread s t r a i g h t s c a r o f knocked
down t h i s t l e s , say, a l l heading i n one d i r e c t i o n .
Anyhow, i t h a s n ' t been hot enough f o r a w h i r l w i n d " .
This i s a very i n t e r e s t i n g account.
The t h i s t l e s
e v i d e n t l y were more broken than corn i s u s u a l l y under
such circumstances, but t h i s t l e s t a l k s are s t i f f e r ,
and one would expect them to break, wheras c o r n w i l l
bend.
However, we have h e r e , as i n a l l the o t h e r
c i r c l e phenomena the crop a l l swept down i n the one
d i r e c t i o n , and not trampled around as one might expect
were c a t t l e r e s p o n s i b l e ,
T h i s s i t e was two hundred m i l e s west o f Sidney, i n
the c o u n t r y s i d e .
There can be no doubt t h a t hoaxers
were not a t work i n t h i s remote c o r c n e r o f A u s t r a l i a ,
i n a f i e l d of t h i s t l e s .
The above d e s c r i p t i o n s w i l l , we hope, g i v e the r e a d e r
an idea o f what has been happening with r e g a r d to
the ' c r o p c i r c l e phenomenon' d u r i n g r e c e n t y e a r s .
At the l a s t r e c k o n i n g , r e s e a r c h e r s who have been
s t u d y i n g these strange p a t t e r n s e s t i m a t e there have
been over two thousand s i t i n g s r e p o r t e d ;
most o f
these have o c c u r r e d i n England, m a i n l y i n the south
west c o r c n e r , but there have a l s o been r e p o r t s from
E a s t A n g l i a and the M i d l a n d s .
The p a t t e r n s have a l s o
been r e p o r t e d from America, Canada, Germany, A u s t r a l i a ,
New Zealand and F r a n c e .
There are many Japanese
r e s e a r c h e r s i n t e r e s t e d i n c e r e a l o g y , although we
have not a c t u a l l y seen r e p o r t s o f the phenomena ...
o c c u r i n g i n t h a t country.
One would presume t h a t
i t may be t h a t r i c e crops, which are the main s t a p l e
i n e a s t e r n c o u n t r i e s , b e i n g s h o r t e r , and l i g h t e r , might
not respond i n the same way as the h e a v i e r crops o f
the west to whatever i t i s t h a t i s i n f l u e n c i n g these
plants.
However, g i v e n the weight o f evidence
we t h i n k i t may q u i t e p r o p e r l y be d e s c r i b e d as a w o r l d wide phenomenon.
Dozens o f magazine a r t i c l e s have been w r i t t e n , many
r a d i o and t e l e v i s i o n programmes a i r e d , a number o f
v e r y b e a u t i f u l books have been p u b l i s h e d , and many
o r g a n i s a t i o n s have been set up to undertake r e s e a r c h .
These l a t t e r p u b l i s h t h e i r own j o u r n a l s , and the s u b j e c t
o f the r e s e a r c h has a c q u i r e d the t i t l e o f "Cerealogy".
Study o f the phenomenon has become w e l l e s t a b l i s h e d .
POSSIBLE CAUSES.
When the f i r s t c i r c l e s appeared they were v e r y simple,
as we have s a i d .
They were g e n e r a l l y j u s t round areas
o f c o r n t h a t had a p p a r e n t l y been f l a t t e n e d o v e r n i g h t ,
and they u s u a l l y occurred on c l e a r calm n i g h t s .
In o t h e r
words unusual weather p a t t e r n s , wind, r a i n , or storms
d i d not appear to be a f a c t o r .
However most people
f e l t there was p r o b a b l y a simple e x p l a n a t i o n .
This
type o f c i r l e had been known f o r many y e a r s , as we
have mentioned there were r e f e r e n c e s to these c i r c l e s
i n the M i d d l e Ages.
Many farmers a t t r i b u t e d the
f o r m a t i o n s to s m a l l w h i r l w i n d s , s i m i l a r to the 'dust
d e r v i s h e s ' one can sometimes see forming on dusty
roadsides.
B u t , as we have j u s t s a i d , they o f t e n
appeared i n c o n d i t i o n s , and i n p l a c e s , where such a
wind was u n l i k e l y ,
There was s p e c u l a t i o n t h a t a t
the s i t e s i n western England, where there are many h i l l s ,
perhaps there were some unusual wind c u r r e n t s , but they
appear a l s o i n the f l a t areas o f E a s t A n g l i a , where
there are no h i l l s .
They are not ' f a i r y r i n g s ' .
Such r i n g s are caused by the outward growth of a fungus
i n the e a r t h , and t h e i r mechanism i s w e l l understood.
They a l s o appear i n the same p l a c e y e a r a f t e r y e a r ,
and the c i r c l e p a t t e r n s do not u s u a l l y r e c u r i n the
same s p o t .
The simple e x p l a n a t i o n seemed l e s s l i k e l y as the numbers
o f such p a t t e r n s i n c r e a s e d .
D u r i n g the 1970's there
was a g r e a t d e a l o f i n t e r e s t i n UFOs, and the p o s s i b i l i t y
t h a t e a r t h was b e i n g v i s i t e d by a l i e n s from o t h e r
u n i v e r s e s or s t e l l a r systems.
A l o n g with the d e s c r i p t i o n s
o f s i g h t i n g s o f UFO's there were many r e p o r t s o f what
came to be c a l l e d ' l a n d i n g r i n g s ' .
UFO r e s e a r c h e r s
b e l i e v e d these so c a l l e d ' r i n g s ' denoted a p l a c e where
an a l i e n ship had landed, u s u a l l y d u r i n g the n i g h t i m e ,
and then had taken o f f a g a i n a t d a y l i g h t .
During
the 1970's there were many r e p o r t s o f such phenomena.
These ' l a n d i n g r i n g s ' had many o f the c h a r a c t e r i s t i c s
o f the c i r c l e s t h a t appeared i n the e a r l y 1980's, and
so i t was i n e v i t a b l e t h a t sc-me r e s e a r c h e r s b e l i e v e
t h a t a l l the c u r r e n t p a t t e r n s are caused by v i s i t o r s
from another w o r l d .
I t i s t r u e t h a t i n some cases,
l i g h t s and n o i s e s were r e p o r t e d as h a v i n g o c c u r r e d
d u r i n g the n i g h t when the p a t t e r n s were formed, j u s t
as i s r e p o r t e d i n a l l e g e d UFO l a n d i n g s cases.
However,
i n UFO l a n d i n g r e p o r t s i t i s o f t e n s a i d t h a t the e a r t h
i t s e l f i s a c t u a l l y scorched, o r b u r n t , and t h i s has
not been so i n r e c e n t p a t t e r n s .
However, one common
f e a t u r e i s the f a c t t h a t the corn i s bent, not broken,
and i s l a i d s y m e t r i c a l l y i n one d i r e c t i o n .
Many
UFO i n v e s t i g a t o r s b e l i e v e t h a t the i n t r i c a t e p a t t e r n s
o f the more r e c e n t phenomena are some attempt a t
communication on the p a r t o f the a l i e n v i s i t o r s .
The w h i r l w i n d theory c o l l a p s e d when the more e l a b o r a t e
f o r m a t i o n s s t a r t e d to happen. Whirlwinds c e r t a i n l y
do not create the k i n d o f p a t t e r n s t h a t were a p p e a r i n g .
Some s c i e n t i s t s , however, are s t i l l l o o k i n g f o r a
' n a t u r a l ' cause, and have put forward the t h e o r y that
the h o l e t h a t i s p r e s e n t l y i n the e a r t h ' s ozone l a y e r
i s r e s p o n s i b l e f o r c r e a t i n g these p a t t e r n s .
They c l a i m
t h a t the unprecedented atmospheric c o n d i t i o n s caused
by t h i s hole have l e d to v i o l e n t d i s t u r b a n c e s i n the
e a r t h ' s magnetic f i e l d , and i t has been claimed t h a t these
d i s t u r b a n c e s have been d e t e c t e d i n the areas where
the phenomena have appeared.
Other s c i e n t i s t s have claimed t h a t a m y s t e r i o u s , and
as y e t not understood, v o r t e x system i s r e s p o n s i b l e f o r
the phenomena.
S c i e n t i s t s from a l l over the world
appear to have put forward t h e i r p e t t h e o r i e s as to
the cause o f the f i g u r e s , but none have succeeded i n
providing a convincing explanation.
I t has been suggested, f o r i n s t a n c e , t h a t the e l e v e n
y e a r c y c l e o f sunspot a c t i v i t y has some b e a r i n g on the
formation of c i r c l e s .
A g a i n , i t was suggested t h a t the
frequency o f power breakdowns i n the v i c i n i t y o f some o f
the phenomena has caused ' i o n i s a t i o n ' o f the c r o p s ,
due to i n c r e a s e d i o n i s a t i o n o f the atmosphere under
these c o n d i t i o n s .
I n some m y s t e r i o u s way, i t i s
a l l e g e d , t h i s ' i o n i s a t i o n ' causes the corn to bend
over i n these strange p a t t e r n s .
As the p a t t e r n s continued to become more complex,
and e s p e c i a l l y a f t e r the f o r m a t i o n o f the geometric and
h i e r o g l y p h i c p a t t e r n s , many people have begun to b e l i e v e
t h a t they c o n s t i t u t e messages.
These messages could
be coming from people on o t h e r p l a n e t s o r s t a r s , o r
they c o u l d be ' r e l i g i o u s ' messages, as has been suggested
i n r e g a r d to the Barbary C a s t l e f i g u r e o f 1991.
A
comparison has been drawn between some o f the f i g u r e s
and the famous l i n e s drawn on the p l a i n s o f Nazca i n
South America.
These Nazca l i n e s have b a f f l e d e x p e r t s
and i n v e s t i g a t o r s f o r many y e a r s .
They cover l a r g e
a r e a s o f c o u n t r y s i d e , and c o n s i s t o f l o n g s t r a i g h t
l i n e s , drawn i n s p e c i a l c o n f i g u r a t i o n s .
They can
o n l y be p r o p e r l y seen and a p p r e c i a t e d from the a i r .
I n a more l i g h t h e a r t e d v e i n , because the f i g u r e s are
so p l e a s i n g , and set, u s u a l l y i n b e a u t i f u l country,
i t has been suggested t h a t they are meant as p i c t u r e s ,
o r a r t forms, c r e a t e d by these beings from o u t e r
space.
They are u s i n g the background o f e a r t h ' s
c o u n t r y s i d e as a canvas to p a i n t p i c t u r e s .
Certainly
many o f the books f u l l o f p i c t u r e s o f these p a t t e r n s
make l o v e l y coffee t a b l e books.
The f a c t s o f the matter are that nobody up u n t i l the
p r e s e n t time has been a b l e to come up with a reasonable
e x p l a n a t i o n as to how these p a t t e r n s are formed, o r
the reason f o r t h e i r e x i s t e n c e .
A HOAX?
To many people the obvious answer as to t h e i r f o r m a t i o n
i s t h a t they are a l l a g i g a n t i c hoax.
T h i s may, indeed,
t u r n out to be the case.
However, i n f a i r n e s s , one should p u t the p o i n t s a g a i n s t
the p o s s i b i l i t y o f the whole a f f a i r b e i n g a hoax.
I n the f i r s t p l a c e , the simple c i r c l e s , which s t a r t e d
up the c y c l e o f p a t t e r n s , have been i n e x i s t e n c e f o r
many y e a r s .
These e a r l y c i r c l e s were almost c e r t a i n l y
not hoaxes, and even then there was no s a t i s f a c t o r y
explanation f o r t h e i r formation.
As we have s a i d , i t
was assumed t h a t they were caused by w h i r l w i n d s , but
then, as now, there was no evidence f o r the presence
o f w h i r l w i n d s a t the times and p l a c e s they o c c u r r e d .
However, r e c e n t l y i t has "been claimed by two landscape
a r t i s t s i n England t h a t they had been r e s p o n s i b l e
f o r c r e a t i n g the v a r i o u s p a t t e r n s and c i r c l e s .
They
claimed t h a t they went out a t n i g h t , on many o c c a s i o n s ,
and p e r s o n a l l y c r e a t e d the p a t t e r n s .
Many people
b e l i e v e d t h e i r d e s c r i p t i o n o f what they had done, and
assumed t h a t the matter was c l o s e d .
But one i s
l e f t to wonder.
In the f i r s t p l a c e , even i n d a y l i g h t ,
i t would not be easy to create these p a t t e r n s i n the
middle o f f i e l d s o f growing corn without l e a v i n g
evidence o f h a v i n g been t h e r e .
I t i s easy to see
where people have trodden i n f i e l d s o f r i p e c o r n , a
s t a l k once bent cannot be stood up a g a i n , and even
a n i m a l s f o o t p r i n t s are u s u a l l y v e r y e a s i l y seen i f
they have entered a f i e l d .
To perform a l l these
c o m p l i c a t e d manoevres and create a l l these complex
p a t t e r n s i n the dead o f n i g h t , and i n u t t e r s i l e n c e ,
assumes a r e a l d e d i c a t i o n to a p r a c t i c a l joke.
But to c a r r y out t h i s hoax so many many times, i n
such v e r y d i f f e r e n t areas (even i f one were o n l y t a l k i n g
about E n g l a n d ) , and over such a l o n g p e r i o d o f time,
assumes a tremendous d e d i c a t i o n to a hoax - and f o r
what purpose?
One might assume that many many people were i n v o l v e d
i n the hoax.
That hundreds o f people were c r e e p i n g
about the f i e l d s each n i g h t , drawing c i r c l e s and
p a t t e r n s i n the growing c o r n .
But why would they do
this?
U n t i l the two E n g l i s h a r t i s t s ' c o n f e s s e d '
nobody e l s e claimed to have been r e s p o n s i b l e f o r the
patterns.
Over a p e r i o d o f more than ten y e a r s , one
would have expected o t h e r people to have owned up,
o r one might expect t h e i r f r i e n d s who were aware o f
what they were doing would have ' t o l d ' on them.
One i s c e r t a i n l y l e f t to wonder why such p r a n k s t e r s
were never d i s c o v e r e d , e s p e c i a l l y c o n s i d e r i n g t h a t
on many o c c a s i o n s c i r c l e s and p a t t e r n s appeared o v e r n i g h t
i n f i e l d s r i g h t next to where r e s e a r c h e r s and i n v e s t i g a t o r s
were camped, i n the hope o f b e i n g p r e s e n t a t the time
these p a t t e r n s were c r e a t e d .
And what about a l l
the phenomena t h a t happened i n o t h e r c o u n t r i e s ;
were
these a l s o the r e s u l t o f p r a c t i c a l jokes?
I f the whole t h i n g i s r e a l l y a v a s t p r a c t i c a l joke,
i t has been v e r y i r r e s p o n s i b l e b e h a v i o u r on the p a r t
o f the p r a n k s t e r s .
Many farmers have s u f f e r e d severe
f i n a n c i a l l o s s , due to h a v i n g t h e i r crops d e s t r o y e d ,
and one would imagine t h a t these farmers might have
a c l a i m f o r compensation.
Not o n l y have the f a r m e r ' s
crops been damaged by the f o r m a t i o n s , but i n most
cases the r e s t o f the crop has been trampled and r u i n e d
by the ho gardes o f s i g h t s e e r s , and i n v e s t i g a t o r s .
N e v e r t h e l e s s , i n s p i t e o f what we have j u s t s a i d about
hoaxes, there can be no doubt t h a t there have been
some hoaxes.
I n v e s t i g a t o r s c l a i m t h a t they have
u s u a l l y been a b l e to d e t e c t these hoaxes, and they
are aware o f them.
T h a t ' s as may be.
In every
type o f i n v e s t i g a t i o n o f t h i s k i n d , and as we are
e s p e c i a l l y aware when s t u d y i n g p a r a p s y c h o l o g y , the
hoaxer w i l l appear.
The e f f e c t , g e n e r a l l y , i s to
'muddy the water' and make i t d i f f i c u l t to a r r i v e
a t a p r o p e r s c i e n t i f i c c o n c l u s i o n , and t h a t i s c e r t a i n l y
true i n t h i s case.
I t has been remarked t h a t " i f the whole phenomenon
should t u r n out to be a hoax, i t c o u l d t e l l us something
about the s t a t e o f s o c i e t y a t a time when s c i e n c e and
technology have achieved t h e i r p r e s e n t s o p h i s t i c a t i o n .
S k e p t i c s w i l l c l a i m t h a t i t proves p u b l i c g u l l i b i l i t y
and f a l l i b i l i t y , but the ' f a l l - o u t - from the d e s t r u c t i o n
o f t r u s t c o u l d w e l l b l a n k e t the s c i e n t i f i c community
as w e l l .
T r u s t , on which c i v i l i s e d s o c i e t y and
b e h a v i o u r are founded cannot s u r v i v e such b a t t e r i n g ,
whether i n s c i e n c e , p o l i t i c a l and economic l i f e , o r
i n human r e l a t i o n s h i p s . "
CONCLUSIONS.
To sum up, the phenomenon o f the f o r m a t i o n o f crop
c i r c l e s and p a t t e r n s i s a v e r y i n t e r e s t i n g one indeed.
I t i s e n t i r e l y new i n our e x p e r i e n c e .
We do not a t t h i s time, have an e x p l a n a t i o n as to
how the p a t t e r n s are formed, although i t i s accepted
t h a t a few, a t l e a s t , are the r e s u l t o f hoaxes.
However,
on b a l a n c e , i t must be conceded t h a t i t i s u n l i k e l y
t h a t the g r e a t e r number are hoaxes}
there has to be
some o t h e r e x p l a n a t i o n .
I t i s a v e r y widespread phenomenon,
areas o f the w o r l d .
o c c u r r i n g i n many
D e s c r i b e d as 'the g r e a t e s t o f modern m y s t e r i e s , ' the
study o f the phenomena has been g i v e n the t i t l e
o f ' c e r e a l o g y ' and t h i s study has spawned a h o s t o f
j o u r n a l s , many newspaper and magazine a r t i c l e s , as
w e l l as numerous books.
Many i n v e s t i g a t o r s c u r r e n t l y
term themselves ' c e r e a l o g i s t s ' .
International
conferences have been h e l d , as w e l l as many r e g i o n a l
conferences and symposia.
I t may w e l l rank with ' u f o l o g y ' as one o f the
consuming areas o f i n t e r e s t and s p e c u l a t i o n o f
our modern t i m e s .
I s i t , perhaps, a modern urban legend?
I f i t were
not f o r the f a c t that the evidence i s so v i s i b l e and
t a n g i b l e , one might be tempted to r e g a r d cerealogy
as such.
But the evidence i s there f o r a l l to see't
We j u s t do not know, a t p r e s e n t , who o r what, p u t
i t there.
REFERENCES.
John M i c h e l l
"This Y e a r ' s Crop", F o r t e a n Times,
No. 59. Sept. 1991. p . 28-29
Bob
"The Crop C i r c l e Phenomenon",
F o r t e a n Times. No. 53 • Winter
1989-90 p . 32-37
Skinner
Diane Kearns
"An A u s t r a l i a n UFO Nest".
The C e r e a l o g i s t , N o . 4 .
Summer 1991. p . 13.
ACKNOWLEDGEMENT.
The w r i t e r s would l i k e to express t h e i r
g r a t i t u d e to M r . B a r r y Withers f o r the l o a n o f extremely
u s e f u l c u r r e n t m a t e r i a l on the crop c i r c l e formations*
and a l s o f o r many s t i m u l a t i n g and i n f o r m a t i v e
discussions.
APPENDIX*
Notes on some of the f o r m a t i o n s .
As has been s a i d , r e c e n t y e a r s have seen an i n c r e a s e n o t
o n l y i n the number o f f o r m a t i o n s , but i n t h e i r s i z e and v a r i e t y
of patterns.
The o r i g i n a l g e o m e t r i c a l s i m p l i c i t y
when the p a t t e r n s were r e s t r i c t e d t© c i r c l e s and s t r a i g h t
l i n e s has been l o s t .
We may, l i k e the poet i n the
Prelude to Goethe's F a u s t f e e l uneasy t h a t " u n r e l a t e d
t h i n g s t h a t know no b l e n d i n g parade before us".
In
the b e g i n n i n g were simple c i r c l e s , then a n n u l i and
c o n c e n t r i c r i n g s , then p a t t e r n s composed o f both c i r c l e s
and s t r a i g h t l i n e s , then c i r c l e s with appendages such as
arrowheads and key shaped e x t e n s i o n s .
( F i g u r e s 7,2,3)•
Then some formations appeared which seemed to be s t y l i z e d
r e p r o d u c t i o n s o f a n c i e n t pagan symbols — sungods, and
a l s o the u b i q u i t o u s "Great Goddess" o f whom so much
has been w r i t t e n i n r e c e n t decades.
Worse y e t was a somewhat r e p u l s i v e "squiggle" which some
o b s e r v e r s nicknamed "The B r a i n " , and an i r r i t a t i n g l y
"arch" f o r m a t i o n dubbed " C u r l y " .
Next, b i o l o g i c a l
elements i n t r u d e d themselves — the "Whale", a l s o a
swarm o f "insectograms"
forms r e c o g n i z a b l e as
depictions of insects.
Some o f t h e s e , l i k e the Whale
and C u r l y convey an i m p r e s s i o n o f humour and a degree o f
"cockiness" and "cheekiness"j
c e r t a i n l y they c o u l d be
d e s c r i b e d as "jaunty".
V e r y r e c e n t l y two a n t i q u a r i a n
themes, i n a d d i t i o n to the Sun God and the Goddess
symbols have been r e p r e s e n t e d .
©ne i s an i m i t a t i o n o f
a d e s i g n on the upland p l a i n a t Nazca i n P e r u .
The
o t h e r , r e l a t i n g to a t o t a l l y d i f f e r e n t c u l t u r e , has been
r e c o g n i z e d as the "Irminsul" copied from a monument to
Arminus (Hermann) a German who defeated a Roman Army
in 9 A.D.
L a s t l y , o r almost l a s t l y , the huge and
complex f o r m a t i o n a t Barbury C a s t l e i n W i l t s h i r e i s o f
considerable i n t e r e s t .
T h i s i s n o t o n l y because i t
r e p r e s e n t s a r e t u r n , a t perhaps a h i g h e r l e v e l , to
the a u s t e r e s e v e r i t y o f pure geometry, but a l s o because
a t l e a s t one contemporary s c h o l a r , John M i c h e l l , d i s c e r n s
an a p o c a l y p t i c element i n i t .
F i n a l l y , there came
the "Mandelbrot Set".
No i m p a r t i a l commentator can f a i l to be somewhat
d i s t u r b e d a t the apparent e c l e c t i c nature o f the
t o t a l i t y t d the present day o f the "crop c i r c l e s " .
I f a s c r i b a B l e to a s i n g l e i n t e l l i g e n c e ( t e r r e s t r i a l ,
e x t r a - t e r r e s t r i a l , extra-dimensional, extra-temporal),
the mind i n q u e s t i o n would appear to be a t b e s t d i l e t a n t e ,
l i k e Shapkespeare's A u t o l y c u s — "a snapper up o f
unconsidered t r i f l e s " , and a t worst a pedant.
The
apparent d i l e t a n t e i s m suggests, o f c o u r s e , the work o f
human hoaxers, some o f whom may have a p e d a n t i c streak
as w e l l as b e i n g f u n s t e r s .
But as a g a i n s t t a k i n g the
easy way out we should b e a r i n mind the e x t r a o r d i n a r y
i n c r e a s e i n the i n c i d e n c e o f f o r m a t i o n s , not o n l y
i n Wessex but i n e a s t e r n England a l s o .
The numbers are
such t h a t the a f f a i r could be s a i d to have got beyond
a joke!
( A l s o , as mentioned p r e v i o u s l y , they have
been found i n many o t h e r p a r t s o f the w o r l d , i n c l u d i n g
remote areas o f A u s t r a l i a ) . We t h i n k i t worthwhile to
append some n o t e s g i v i n g background on f o u r o f the
f o r m a t i o n s — the " I r m i n s u l " , the Barbury C a s t l e
p a t t e r n , the v e s i c a p i s c e s , and the Mandelbrot Set a t
Ickleton.
The " I r m i n s u l " .
I n 9 A . D . a Roman e n t e r p r i s e came to a sudden and
c a t a s t r o p h i c end.
Augustus, the f i r s t Roman Emperor,
had s a n c t i o n e d the establishment o f a new P r o v i n c e —
Germania — a c r o s s the Rhine.
I t was a d m i n i s t e r e d by
the m i l i t a r y governor — Varus — who was r a t h e r heavyhanded.
H ermann(= Arminius) was a German h i g h - u p i n the
Roman army i n the P r o v i n c e .
This i s not at a l l s u r p r i s i n g
— the m a j o r i t y o f the Roman l e g i o n s c o n s i s t e d o f
b a r b a r i a n s - Goths, Germans, e t c .
Hermann's
idiosyncrasy^was t h a t he became a r e b e l — a German
p a t r i o t , and r a i s e d a r e v o l t a g a i n s t V a r u s .
Rather
c l e v e r l y he tempted Varus i n t o an ambush i n the Teutoburger
F o r e s t i n which three l e g i o n s were k i l l e d .
The T e u t o n i c
n a t i o n p r i n c i p a l l y i n v o l v e d i n the v i c t o r y were known
to the Romans as the Cheruscans.
Emperor Augustus, who
was not used to f a i l u r e , took the r e v e r s e very b a d l y ;
a l l thought o f Roman expansion beyond the Rhine was
abandoned.
:
A f t e r some c e n t u r i e s (almost m i l l e n i a ) Hermann was
i n s t i t u t e d as a n a t i o n a l German h e r o .
E a r l y i n the,,
n i n e t e e n t h century some w r i t e r s o f the s o - c a l l e d "Sturm
and Drang" movement ( i . e Storm and S t r e s s ) s t i r r e d up
both by the romantic movement o f the e i g h t e e n t h c e n t u r y ,
which i d e a l i z e d e a r l i e r p e r i o d s i n human h i s t o r y as times
when men were n o b l e r because c l o s e r to n a t u r e , and by
the Napoleonic wars which evoked a p r e v i o u s l y h a r d l y
e x i s t e n t German n a t i o n a l sentiment, r e d i s c o v e r e d Hermann
%
as a T e u t o n i c h e r o .
Prominent among these n e o - T e u t o n i c
p a t r i o t s were Johann H . V o s s , who c e l e b r a t e d i n poems
the n o b i l i t y o f the "Volk Thuiskans" ( i . e . the Cheruscans),
the companions o f Hermann, and a l s o C . F . D . Schubart.
He c e l e b r a t e d "freedom" as the f o r c e most revered by the
a n c e s t r a l Germans, and embodied i n Hermann.
He commenced
h i s l i s t o f g r e a t German heros with Hermann, and concluded
w i t h K i n g F r e d e r i c k W i l l i a m , the founder o f the P r u s s i a n
state.
Sometime l a t e r i n the n i n e t e e n t h century o r e a r l y
t w e n t i e t h a memorial i n stone was s e t up to Hermann
somewhere near the l o c a t i o n o f h i s v i c t o r y .
We r e g r e t
n o t to have the p a r t i c u l a r s , but i t may be p l a u s i b l y
surmised t h a t i t was a t the expense o f p r i v a t e donors.
The p e r i o d was v e r y much one f o r s e t t i n g up s t a t u e s
and o t h e r memorials o f p a t r i o t i c c h a r a c t e r s , c . f . the
Washington column i n Washington, D . C . , the statue o f
BoadecSa with c h a r i o t and h o r s e s a t Westminster, and
t h a t o f King A l f r e d a t Wantage.
The a c t u s l p a t t e r n o f the Hermann standard (the"Irminsul")
was p r o b a b l y the product o f some n i n e t e e n t h century a r t i s t ' s
imagination.
Amusingly enough, the n i n e t e e n t h century
German romantic p a t r i o t s when s e e k i n g f o r an e x p l a n a t i o n
why the e a r l y Teutons should have d e c l i n e d from t h e i r
o r i g i n a l moral s t a t u s , blamed i t on the c o r r u p t i n g
i n f l u e n c e o f French c i v i l i s a t i o n * .
However they were
k i n d enough to exempt the Anglo-Saxons o f England from
any c u l p a b i l i t y !
Whether the o c c u r r e n c e o f the "Irminsul"
i n E n g l i s h Wessex has a n y t h i n g to do w i t h t h i s i s a
q u e s t i o n we l e a v e to otherst
The B a r b u r y C a s t l e f o r m a t i o n .
The t o t a l l e n g t h o f the v a r i o u s s e c t i o n s o f the Barbury
C a s t l e f o r m a t i o n i s o f the o r d e r o f a thousand y a r d s ,
i . e . a k i l o m e t e r , and i f done by u n a s s i s t e d manpower
would r e p r e s e n t a very c o n s i d e r a b l e p h y s i c a l e f f o r t , u n l e s s
i t were the work o f a v e r y l a r g e team.
The o n l y i n t e r p r e t a t i o n o f the p a t t e r n y e t g i v e n i s t h a t
i n The C e r e a l o g i s t (No. k, Summer 1991) by John M i c h e l l ,
the a u t h o r o f The View Over A t l a n t i s (eds. 1969 and 1 9 6 2 ) ,
and o t h e r books which i n c l u d e C i t y o f R e v e l a t i o n , and
The Dimensions o f P a r a d i s e .
M i c h e l l ' s i n t e r e s t s are
f a i r l y eclectic;
he a s c r i b e s r e a l i t y to l e y l i n e s ,
and a geomagnetic b a s i s to f u n g - s h u i , as w e l l as to
the g e n e r a l t e n e t s o f the s u b j e c t which i n r e c e n t years
has become known as geomancy.
Michell's interpretation
GEOMETRY AND SYMBOLISM AT
BARBURY CASTLE
Analyzing the geometric and numerical structure of this year's masterpiece in wheat,
John Michell is stunned by its implications.
BARBURY CASTLE CROP FORMATION (17:7:91)
SCALE 1:400
ORDNANCE SURVEY OUT) IH>. SU 152 761 atop TYPE WHEAT
Surveyed (20:7SI) and drawn by J. F. Langrish.
DIRECTION BEARING DISTANCE
XD
0*
105* 10*
XA
120"
106 0*
XB
234*
104* 0*
AB
2M*
17T 0*
BC
24"
91- 0"
CD
30*
93' r
DE
148*
65" 10*
ue
or
HA
ISO*
DQ
0*
38* 0AY
•oa*
*S 6"
BZ "
234*"'' '." 3ff r "
DIAGRAM TO SHOW
DIRECTION OF LAY
-
t
Nouc AB
. *D < - t ir i Tim
p
e
n
i
e
o
f
d)s*B
H>llrt'l'fadi— by™^acfaig
d
o
w
n
b
t
c
s
I
m
d
B
M
p
c
H
I
I
I
B
td
faW
. ooo po
oftrt.tho.a'ttin3*'
-ndM
yC
UA
B
a£
nD
h
tA
> and w«n
c
/
(
h
a
B
O
dud a
i th*
otrai
ttBt
.AHmiJu^
Mm.
U- a
THE BARBURY CASTLE formation is a
regular but previously unknown form of
geometrical diagram. John Langrish's survey
shows that whoever made it lapsed from
perfect symmetry, for the central feature of a
circle and rings has slipped too far to the
north-west and, possibly in connection with
this, the triangle has become distorted. The
dimensions of the various parts are nonetheless consistent, and it is possible from the
data to discover the basic construction of the
figure. We are presented with a novel and
interesting lesson in geometry, but the
implications of this figure are not merely
academic for the numerical structure behind
its dimensions are familiar to students of the
ancient numerical science and have deep
cosmological significance.
There are many mysteries about this
formation, including the method and
sequence of its creation and the strange lights
and other phenomena which coincided with
its appearance. Also enigmatic is the
impressive ratchet spiral at the south-east
comer. Yet the basic geometrical idea behind
the diagram is dear enough. It demonstrates
the principle of Three in One by means of a
TWENTYFOUR • THE CEREALOGIST #4
central circle which exactly contains the
histories revelation is said to have been the
combined areas of the three circles around it source of all cultures and to have inspired
Moreover, the sum of all the four circular
successive renewals of the human spirit It
areas in the diagram is 31680 square feet The occurs, presumably, attimeswhen it is most
significance of this number, in arithmetic,
needed, and its content is always the same,
cosmology, ancient theology and temple
being that cosmic Law, Canon or compilation
architecture, wasfirstexplored in City of
of numerical, musical and geometric
Revelation (1972) and is summed up in a
harmonies which provided the ruling
section of 77i<." Dimensions of Paradise (Thamesstandard of every ancient civilization.
& Hudson, 19S8). In traditional cosmology,
31680 miles was taken to be the measure
Construction
around the sub-lunary world, and the early
Christian scholars calcula ted the number
A figure is said to be 'constructed' when it is
3168 as emblematic of Lord Jesus Christ. The drawn with nothing but a straight-edge and
same number was previously applied to the
compass, the equipment with which,
name of a leading principle in the pagan
according to the geometers' myth, the
religion.
Creator designed the world-plan. The basic
figure at Barbury Castle can be constructed
There is a world of symbolism in this
Barbury Castle wheat impression, some of it by the following stages, beginning with the
most elementary geometric operation, the
already apparent and some still awaiting
recognition. Neither physically nor intellectu- division of a circle into six parts by its radius.
ally does thisfiguregive signs of being a
human creation. To identify its author seems Figure I
therefore to be a problem for theology. One's Draw a circle and, with the compass opening
rational mind shrinks away from the
unchanged, mark off six arcs from the centre
implication that this diagram constitutes a
to the circumference.
divine revelation. Yet in the traditional
Draw a line from the centre of the circle
. .c - > i v between two of the points on its
circumference and, with the same compass
opening, divide the line into six lengths, each
length equal to the radius of the circle.
Draw a circle equal to the first with its
centre on the last of the six division points of
the line.
Form an equilateral triangle on the line
between the centres of the two circles and
ciraw a third equal circle centred upon the
triangle's apex.
Figure II
Find the centre of the triangle by drawing
lines from the centre of each circle to the midpoint of each opposite side of the triangle.
From the triangle's centre draw the circle
contained within the triangle.
The area of the circle within the triangle is
exactly equal to the sum of the areas of the three
;
>c.'es. Takins the radius of each of the
-raller circles to measure • unit makes
ibmed areas of the three circles equal
i«bct4tp geometry at Barbury Castle. Wiltshire
to 3". and 3- Is
s t a t e d ar->a or :hc
central circle
The average radius or the tnree outer
circles is found by inspection of the survey to
be about 41ft. The radius of the central circle
is therefore 41 xV3 = 71ft., and this agrees
with the average of 71ft. shown on the survey
to be the radius of the larger circle. Note that
71/41 is a good approximation toV3. The
calculated area of the larger circle is 15S40
A comparison of the calculated with the
average surveyed lengths of the radii in the
3arbury Castle figure
Outlying circle (N)
Outlying cirde (SW)
Central outer ring
Central inner nr.-.;
Survev
41
40.7
71
49.3
2-1-5
Calculated
41
41
71
50.4
25.2
^:
. and the combined areas of all three
is 31680 sq. ft. The fact that the number
3 i oSO is of special significance within the
ancient scientific tradition is mentioned
above.
Figure III
Draw a second triangle with its angles at the
points where lines from the centre meet the
circumferences of the three outer circles. This
corresponds to the triangle which appears in
the crop formation. Within that triangle
inscribe a circle. Reduce the compass opening
by half and draw the central circle.
Erase the original triangle, drawn in Fig.I,
together with other unneeded construction
lines. Apart from the rims of circles, the
widths of rings and lines and the outlying
ratchet spiral, the Barbury Castle formation is
thus essentially constructed.
With thanks to John Langrish for his survey and to
:
..'.•..-..i::.-v.- :::-,--^,>
d.~"/. :ng •:• diagrams.
o f the Barbury C a s t l e wheat f o r m a t i o n i s however based on a
theme t h a t he developed and a p p l i e d i n The View Over
A t l a n t i s , and which the p r e s e n t w r i t e r s f i n d i t convenient
to e n t i t l e as n u m e r o l o g i c a l cosmography.
N u m e r o l o g i c a l cosmography, o f c o u r s e , i n c o r p o r a t e s many
o f the methods o f o r d i n a r y common o r garden numerology.
Many people f o r instance are f a m i l i a r with the use o f
b i r t h d a t e s to o b t a i n what i s , i n e f f e c t , the most i n e x p e n s i v e ,
though c o r r e s p o n d i n g l y l i m i t e d , form o f horoscope.
If
a p e r s o n i s b o r n on 4 J u l y 1919 = V 7 / 1 9 1 9 o r 04/07/1919,
then a d d i t i o n o f a l l the d i g i t s i n e i t h e r the second o r
t h i r d o f these r e p r e s e n t a t i o n s y i e l d s the sum 4+7+1+9+1+9
= 31. The sum 31 i s now i t s e l f s u b j e c t e d to the same
p r o c e s s o f r e d u c t i o n ( o r , as some would say c a b a l i s t i c
r e d u c t i o n ) , i t i s r e p l a c e d by 3+1 = 4.
The "buck" stops
here;
the number 4 i s used to make a broad c l a s s i f i c a t i o n
o f the p e r o n ' s c h a r a c t e r t y p e .
The number 4 i s i n t e r p r e t e d
i n terms o f t r a d i t i o n a l number m y s t i c i s m , based on a
plenitude of associations - - l i t e r a r y , f o l k l o r i c
and r e l i g i o u s (both c u r r e n t and d e f u n c t r e l i g i o n s b e i n g
of influence).
In our p r e s e n t example 4 i s u s u a l l y
a s s o c i a t e d with "four-square", t h a t i s to say the 4-number
b i r t h d a t e persoxi i s d e p i c t e d as somewhat o f a "square",
i . e . s o l i d and dependable, but with the f a u l t s o f t h e i r
v i r t u e s — perhaps a l i t t l e "stodgy".
The n u m e r o l o g i c a l cosmography o f John M i c h e l l a l s o
i n v o l v e s an a n c i e n t system c a l l e d g e m a t r i a , which
o r i g i n a t e d w i t h R a b b i n i c a l commentators a t a p p r o x i m a t e l y
the dawn o f the C h r i s t i a n e r a .
The s t a r t i n g p o i n t was
t h a t the o r i g i n o f a l l t h i n g s i n the c r e a t e d u n i v e r s e
was the Word o f Sod ( a p r i n c i p l e echoed i n the opening
o f S t . J o h n ' s G o s p e l — "In the B e g i n n i n g was the Word").
I n geomatria the r a b b i s a t t a c h e d to each l e t t e r o f the
Hebrew a l p h a b e t a number.
As Greek was, i n r e a l i t y ,
the spoken and even l i t e r a r y language o f most Jews i n the
Roman Empire, the system was t r a n s f e r r e d to the Greek
a l p h a b e t (see T a b l e ) with m u l t i p l e a p p l i c a t i o n s .
Thus,
i f one was named George, i . e . georg i n Greek, o n e ' s number
was 3+5+70+100+3 + 181 = 1+8+1 = 10*1+0 = 1, which i n
t r a d i t i o n a l a s s o c i a t i o n s was c o n s i d e r e d a r a t h e r good
number, b e i n g equated w i t h God, the F i r s t P r i n c i p l e ,
Wholeness, c r e a t i v e n e s s ,
etc.
A l t h o u g h no human could aspire to d i v i n e v i r t u e , nonet h e - l e s s a man c a l l e d George, b o m J u l y 4,1919, c o u l d
f e e l p r e t t y good about i t ! ( c . f . The S e c r e t s o f Numbers,
L i o n e l Stebbing).
Gematria became a prime element i n
the c a b a l a , a system o f Jewish m y s t i c i s m developed
d u r i n g the C h r i s t i a n era i n d e p e n d e n t l y o f C h r i s t i a n
thought and t h e o l o g y .
Cabala was n o t however p u r e l y
gematriai t i n c l u d e d many o t h e r i d e a s and images.
D u r i n g the Renaissance some s c h o l a r s became aware o f
c a b a l a and developed i t i n C h r i s t i a n terms.
N u m e r o l o g i c a l cosmography had, i t seems o r i g i n s
independent o f gematria o r c a b a l a .
We can o n l y guess, but
we can p o s t u l a t e with f a i r c e r t a i n t y t h a t a powerful
c o n t r i b u t o r y stream o f i n f l u e n c e came from Freemasonry
i n i t s i n i t i a l phase i n the l a t e seventeenth and
e a r l y eighteenth century.
Not b e i n g Freemasons, the
p r e s e n t w r i t e r s have not the advantage o f such e s o t e r i c
h i s t o r i c a l knowledge as may be p o s s e s s e d by e n r o l l e d
Freemasons, and t h e r e f o r e have to take aim i n the dark,
as i t were.
The B r i t i s h founders o f Freemasonry
comprised such d i s t i n g u i s h e d persons as the a n t i q u a r y E l i a s
Ashmele, a f t e r whom the museum a t Oxford i s namel,
the a r c h i t e c t , S i r C h r i s t o p h e r Wren, and p o s s i b l y S i r
I s a a c Newton, the g r e a t mathematician and p h y s i c i s t ,
t o g e t h e r with persons a c t u a l l y connected with the trade
o f stone masons, p a r t i c u a r l y the masons' lodge a t
Kilwinning i n Scotland.
I t seems t h a t from the o u t s e t
the founders e n t e r t a i n e d a j t r a d i t i o n t h a t Solomon's
Temple i n a n c i e n t Jerusalem was designed by g r e a t adepts
o r i l l u m i n a t i , possessed o f a profound knowledge o f the
u n i v e r s e and o f the d i v i n e p l a n on which i t was c o n s t r u c t e d .
S i r I s a a c Newton, who was a d d i c t e d to the study both o f
c h r o n o l o g y and prophecy, worked out the measurememts
f o r a model o f Solomon's Temple which was e x h i b i t e d i n
London some y e a r s b e f o r e the Masonic movement was l a u n c h e d .
I t was somewhat l a t e r t h a t v a r i o u s a n t i q u a r i a n s became
i n t e r e s t e d i n the n u m e r i c a l a s p e c t s o f two o t h e r a n c i e n t
b u t v e r y p r e s t i g i o u s monements.
These werejStonehenge,
Cwhich had been o f g e n e r a l , though n o t n u m e r o l o g i c a l
i n t e r e s t , s i n c e the time o f John Aubrey i n R e s t o r a t i o n
England): and the Great Pyramid o f E g y p t .
I n 18#7,
P r o f e s s o r P i a z z i Smyth, Astronomer R o y a l o f S c o t l a n d ,
p u b l i s h e d a book, Our I n h e r i t a n c e i n the Great Pyramid,
i n which he announced the d i s c o v e r y o f s e v e r a l remarkable
coincidences}
e . g . the P y r a m i d ' s h e i g h t was a thousand
m i l l i o n t h p a r t o f the mean d i s t a n c e o f the e a r t h from
the sun.
A l s o the p e r i m e t e r o f the base was e q u a l to
the c i r c u m f e r e n c e o f a c i r c l e whose r a d i u s e q u a l l e d the
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80
100
200 300
400 500 600 700 800
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Greek name
215/51/55
ek,uia, 'ev
474
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5
T::c -O shape sas discovered from the air by Ron Vtesi in 1990 and -is
mvisii$sied by him and les Stacey. Its site was at Fordham Place, near
CJicki$:er. Essex. The two circles, xuirled clockwise ^om slightly eccr~.:r.c
wots, n-ere identical in size, measuring 10'6" across.
The oZo shape has had great appeal to waggish interpreters.
According to Ralph, it has been taken as an advertisement for a new
washing powder (OZO), while some have seen it as a reference to the
defective OZOne layer. Far more interesting is the result which comes
from analyzing the formation through its inherent geometry.
The plan of the oZo shape, constructed from Ron Wesf $ survey
and photographs, is shown here in-black, and next to it is the
completedfigurewhich naturally develops from it. It is plainly seen
tha t the crop formation indicates a geometrically coherent religious
symbol, only a part of which is visible on the ground. This raises the
possibility that other crop circle patterns represented more elaborate,
but uncompleted geometrical designs. In that case, they can be
compared to masons' marks, traditionally carved by masons onto the
stones they cut. A study of masons' marks shows that they too
"epresen: she cores of more elaborate geometries.
The figure that emerges from theoZo form is the nimbus of glory
'•••'hich surrounds divinities and, in Christian art, is characteristic or
:he \ -.rain Marv. In symbolic geometry it is known as the Vessel,
symbolizing the female source of generation and also being associated with the Grail. Its function in geometry parallels its symbolism,
for it is the first figure by which are generated all the regular geometric shapes. Its construction, by means of two equal circles passing
through each other's centres, demonstrates the fruitful union of
opposites.
The construction lines (broken) in this figure form the square and
the equilateral diamond or rhombus. They define the dimensions o:
the two circles, and they also mark the limits of the swirled areas in
the Z shape.
No more elegant or economical way could be found of convevir.j
unmistakably this most significant of aU symbols. This sureiv is rre
mark of Ceres herself.
Pyramid's height.
I f b i s the l e n g t h o f one s i d e o f the
b a s e , and h i s the h e i g h t , then the l a s t statement i s
e q u i v a l e n t to s a y i n g t h a t *fb = 2 ( P i ) h» i . e . t h a t
2 b / h = P i , where P i i s the w e l l known u n i v e r s a l c o n s t a n t
g i v e n n u m e r i c a l l y (to e i g h t s i g n i f i c a n t f i g u r e s ) by
P i = 3.1-+15927.
The l a b o u r s o f Smyth and l a t e r w r i t e r s
have r e v e a l e d f u r t h e r n u m e r i c a l c o i n c i d e n c e s , which have
encouraged the development o f n u m e r o l o g i c a l cosmography
as a body o f d o c t r i n e .
I n the p r e s e n t century the s u b j e c t
r e c e i v e d a f u r t h e r stimulus from the study o f church
a r c h i t e c t u r e , p a r t i c u l a r l y the e x c a v a t i o n s o f G l a s t o n b u r y
Abbey by B l i g h Bond and h i s n u m e r o l o g i c a l i n t e r p r e t a t i o n s
o f i t s o r i g i n a l i n f e r r e d dimensions.
Besides p u b l i s h i n g
many r e p o r t s and a book on the G l a s t o n b u r y r u i n s , B o n d ,
t o g e t h e r with a D r . T . S . L e a , a u t h o r e d two books, one
on the cabala and one on g e m a t r i a .
H i s work has been
an o b v i o u s source o f i n s p i r a t i o n to John M i c h e l l .
Whet/her o r n o t they proceeded a c c o r d i n g ! t o the r e c o g n i z e d
methods o f numerology, v e r y many C h r i s t i a n e x e g e t i s t s
have t h e o r i z e d over the c e n t u r i e s about the numbers i n
the Book o f R e v e l a t i o n , whose a u t h o r s h i p i s a t t r i b u t e d
t r a d i t i o n a l l y to the A p o s t l e John, who i s supposed a l s o
to have w r i t t e n the Gospel o f S t . J o h n , and two E p i s t l e s .
John was t r a d i t i o n a l l y b e l i e v e d to have composed the R e v e l a t i o n
w h i l e a p r i s o n e r working i n the stone q u a r r i e s on the
i s l a n d o f Patmos d u r i n g the p e r s e c u t i o n o f C h r i s t i a n s under
the Emperor Nero (5^-68).
R e v e l a t i o n has some elements i n common with the two o t h e r
a p o c a l y p s e s i n the B i b l e .
With E z e k i e l , i t shares the
" f o u r l i v i n g c r e a t u r e s " and a l s o the "man w i t h the reed",
i . e . a n g e l w i t h the measuring r o d .
(Indeed, w i t h
a d m i t t e d l y a sense o f s t r e t c h i n g the p a r a l l e l , the Barbury
C a s t l e f o r m a t i o n might be regarded as i n c l u d i n g a r e f e r e n c e
to Ezekielfc
the prophet saw the l i k e n e s s ^ o f a wheel
w i t h i n a wheel".
Biit i t might be unwise to pursue t h i s
p a r t i c u l a r hare.)
With the v i s i o n o f D a n i e l , R e v e l a t i o n
shares the horned b e a s t s .
Of the numbers i n R e v e l a t i o n ,
some o f them, l i k e 7, r e s u l t from the a c t u a l h i s t o r i c a l
f a c t o f there b e i n g then the Seven Churches i n & s i a M i n o r —
Ephesus, S a r d i s , e t c .
The seven a n g e l s , w i t h t h e i r
lamps, e t c . were the g u a r d i a n s p i r i t s a p p o i n t e d by God
f o r those c h u r c h e s .
The f o u r l i v i n g c r e a t u r e s a r e b e n i g n ,
and l a t e r were i d e n t i f i e d w i t h the f o u r e v a n g e l i s t s and
t h e i r gospels.
But there are a l s o f o u r b e a s t s which a r e
n a s t y , was w e l l a s the F o u r Horsemen and two dragons.
Twenty f o u r v i r t u o u s e l d e r s s i t on t h e i r t h r o n e s .
A
number o f e n t i t i t e s o c c u r i n t w e l v e s , b u t t e n i s o n l y
r e p r e s e n t e d by t h e b e a s t w i t h seven heads and t e n h o r n s .
However the i n t e r e s t i n g numbers a r e the b i g ones.
The
b e s t known i s 666 — the 'number o f the b e a s t ' w i t h
w h i c h much has been done by a l m o s t t h a t number o f
commentators o v e r the c e n t u r i e s . The 1260 days o f
p r o p h e c y a r e o f course e q u i v a l e n t t o the 42 months o f
30 d a y s each t h a t "the h o l y c i t y s h a l l t h e y t r e a d under
foot".
The l a r g e s t number i s 144,000, b e i n g the "ransomed
a s t h e f i r s t f r u i t s o f humanity f o r God and the Lamb".
W i t h o u t h a v i n g r e a d Sf. Thomas A c q u i n a s on'the fewness
o f t h e saved ,* i t i s n o n e - t h e - l e s s h a r d t o suppose t h a t
t h e r e a s o n i n g b y which he a r r i v e s a t e x a c t l y t h a t number
was n o t i n f l u e n c e d by i t s o c c u r r e n c e i n R e v e l a t i o n .
1
The number however which p a r t i c u l a r l y i n t e r e s t s us i n
t h e p r e s e n t c o n t e x t i s t h e measure o f the New J e r u s a l e m ,
t h e G o l d e n C i t y , which S t . John i n h i s v i s i o n sees
"coming down o u t o f Heaven from God".
The a n g e l w i t h t h e
g o l d m e a s u r i n g r o d found t h e s i d e o f the C i t y t o be
12,000 f u r l o n g s ; * i t s l e n g t h and i t s b r e a d t h , and i t s
h e i g h t b e i n g equalV
T h a t i s t o say, i t was a cube, w i t h
more t h a n a d e q u a t e accommodation even f o r a p o p u l a t i o n
o f 144,000.
Now, i n h i s books, thfe C i t y o f R e v e l a t i o n
and The D i m e n s i o n s o f P a r a d i s e , M i c h e l l a r r i v e s a t the
number 31680 a s b e i n g b o t h o f cosmographic s i g n i f i c a n c e
and r e l i g i o u s i m p o r t .
U n f o r t u n a t e l y n e i t h e r book i s
to hand.
We can r e a s o n a b l y a c c e p t however, t h a t
3168 i s , by g e m a t r i a , a p p l i e d t o p e r h a p s J e s u s C h r i s t
Our L o r d i n G r e e k , o r some s i m i l a r t i t l e , emblematic o f
Jesus.
&s w i l l be seen by r e f e r e n c e t o h i s a r t i c l e
"Geometry and Symbolism a t B a r b u r y C a s t l e " M i c h e l l says
t h a t , " I n t r a d i t i o n a l cosmology 31680 m i l e s was t a k e n
to be t h e measure around t h e s u b l u n a r y w o r l d " .
This
statement i s a t f i r s t extremely p u z z l i n g .
I n medieval
a s t r o n o m y — t h a t i s the P t o l e m a i c system — the s u b l u n a r y
w o r l d was t h a t p a r t o f the u n i v e r s e b e l o w the moon, i . e .
t h e r e g i o n e n c l o s e d by the sphere o f the moon,
Thus
"the measure a r o u n d the s u b l u n a r y w o r l d " would be, i n any
o r d i n a r y l i n g u i s t i c usage, e q u a l t o 2 ( P i ) ( d i s t a n c e o f
the moon from t h e c e n t r e o f the e a r t h ) .
But i f t h i s
were 31680 m i l e s , t h e n the d i s t a n c e o f the moon from t h e
e a r t h ' s c e n t r e would be o n l y 5042 m i l e s (about) which
would be an a b s u r d l y s m a l l e s t i m a t e even f o r m e d i e v a l
man, who knew t h a t the h e i g h t o f t h e moon was more t h a n
1000 m i l e s .
( I n the P t o l e m a i c system, a s f o r G a l i l e o
and Newton,
and f o r us,
i t was 60 e a r t h r a d i i ) .
Thus John M i c h e l l ' s statement has to be i n t e r p r e t e d
somewhat d i f f e r e n t l y , indeed, l i k e so much i n n u m e r o l o g i c a l
cosmography, i n a s p e c i a l sense.
I l l u m i n a t i o n comes as
soon as we note a f i g u r e which he quotes f o r the e a r t h ' s
d i a m e t e r i n The View Over A t l a n t i s , n a m e l y 7920 m i l e s ,
which i s t o l e r a b l y a c c u r a t e , even i n these days o f
s a t e l l i t e and r a d a r o b s e r v a t i o n s , which have produced
new f i g u r e s f o r the shapes o f the e a r t h .
To t h i s degree
o f n u m e r i c a l a c c u r a c y 31680 m i l e s i s j u s t e i g h t e a r t h
r a d i i o r four earth diameters.
I t i s t h e r e f o r e the
p e r i m e t e r o f a square c i r c u m s c r i b e d to a c i r c l e whose
r a d i u s i s e q u a l to t h a t o f the e a r t h l
Our t r o u b l e s however are not y e t o v e r ;
we have s t i l l
to r e c o n c i l e 3 l 6 8 w i t h o r without an a d d i t i o n a l zero d i g i t
o r so, with the dimensions o f the New Jerusalem which was. i s ,
o r w i l l be 12,000 f u r l o n g s on the s i d e .
Taking " f u r l o n g "
a t i t s face v a l u e equates i t to 10 chains = 220 yards * 660
English feet.
I n yards t h e r e f o r e the cube which i s
the New Jerusalem has an edge l e n g t h o f 22 x 12,000 y a r d s .
We would l i k e , somehow, to r e c o n c i l e t h i s f i g u r e w i t h
31680 = 22 x 144 x 10.
T h i s d i s t a n c e 3168 y a r d s i s
an extremely B r i t i s h , indeed A n g l o - S a x o n , measure.
S q u a l l i n g 144 x 22 yards i t i s "a g r o s s o f c r i c k e t
pitches".
However, except i n the eyes o f an o l d e r
g e n e r a t i o n o f Englishmen, i t can h a r d l y from t h a t p o i n t
o f view be regarded as h a v i n g sacred c o n n o t a t i o n s , u n l e s s
we regarded the Barbury C a s t l e f o r m a t i o n as h a v i n g been
p e r p e t r a t e d by the s o u l s o f g r e a t c r i c k e t e r s such as W.G.
Grace o r Jack Hobbs.
However, g e t t i n g back i n t o J e r u s a l e m
the G o l d e n , can we r e c o n c i l e 3168(0) w i t h 22 x 12,000?
?
As w i l l be seen from v a r i o u s examples o f the type o f
c a l c u l a t i o n p e r m i s s a b l e i n n u m e r o l o g i c a l cosmography which
M i c h e l l g i v e s i n The View Over A t l a n t i s , i t i s l e g i t i m a t e
to drop z e r o s , i . e . to a l t e r the u n i t i n terms o f which
a number i s s p e c i f i e d .
I n f a c t on page 144 (I s y n c h r o n i c ! t y ? )
he says "It now appears t h a t G l a s t o n b u r y was o r i g i n a l l y
c o n c e i v e d on the model o f the New Jerusalem . . . a square
12,000 f u r l o n g s on every s i d e .
I f t h i s i s reduced to
12 f u r l o n g s , the f u r l o n g b e i n g e q u a l to 660 f e e t , the
area c o n t a i n e d . . . i s 1440 a c r e s e x a c t l y " [An a c r e = 10
square c h a i n s = 4840 square y a r d s ] .
He goes on to
develop some n u m e r o l o g i c a l r e a s o n i n g e s p e c i a l l y a p p r o p r i a t e
t o G l a s t o n b u r y which need n o t c o n c e r n us now.
More
g e n e r a l s u b s t i t u t i o n s o f one u n i t f o r a n o t h e r a p p e a r
t o be l e g i t i m a t e i n numerology, which seems as a
d i s c i p l i n e ta.be immune t o any a c c u s a t i o n s o f i n f l e x i b i l i t y
o r dogmatic r e g i d i t y .
Thus, on page 139, a l s o r e
G l a s t o n b u r y , M i c h e l l equates a f u r l o n g (220 y a r d s - 660
f e e t = 7920 i n c h e s ) w i t h 7920 m i l e s , t h e e a r t h ' s d i a m e t e r .
Even i f the more c o n s e r v a t i v e o f us f i n d t h i s a t r i f l e
shock i n g , we must be r e c o n c i l e d t o the n o t i o n t h a t i n
n u m e r o l o g i c a l cosmography perhaps i t i s the case riot t h a t
" a n y t h i n g goes" b u t t h a t a s u r p r i s i n g number o f p r o c e d u r e s
are v a l i d !
W i t h i n t h i s s p e c i a l c o n t e x t i t m i g h t be
l e g i t i m a t e t o argue as f o l l o w s .
The f l o o r a r e a o f the
C i t y i n m i l l i o n s o f square f u r l o n g s i s 144 = 144 x 22 y a r d ,
f u r l o n g s = 3168 y a r d , f u r l o n g s = 3168 s p e c i a l u n i t s .
T h i s may seem a t r i c k — i n d e e d i t i s h a r d n o t t o d e s p i s e
i t , b u t i t may be a d m i s s i b l e i n the p r e s e n t c o n t e x t .
There a r e o f course (as must be the case i n a d i s c i p l i n e w i t h
such l a b i l e p r i n c i p l e s ) o t h e r r a t i o n a l e s which m i g h t seem
t o v a l i d a t e o u r a p o c a l y p t i c number.
I n thousands o f
f u r l o n g s an edge o f New J e r u s a l e m measures 12 x 22 = 26.4|
b e c a u s e a cube has 12 edges the t o t a l l e n g t h 1 o f the
edges o f the c i t y i s 1 * 12 x 264 *= 144 x 22 = 3168.
T h i s l e n g t h 1 can be t a k e n as b e i n g i n some sense a
measure o f s i z e . A n o t h e r approach i s t o c o n s i d e r the
g r o u n d p l a n o n l y - i t s p e r i m e t e r p. i n k i l o f e e t i s
p = 4x3x12x22 = 3168 ( I )
We s h o u l d say t h a t t h e s e l a s t
t h r e e arguments,which may savour l e s s o f i n g e n u i t y t h a n
o f d i s i n g e n u o u s n e s s , h a v e been devised*.by^ the: p r e s e n t
w r i t e r s and n o t , as f a r a s we are aware, by Mr. M i c h e l l 1
t h e y seem however n o t t o be e n t i r e l y a l i e n t o the s p i r i t
o f n u m e r o l o g i c a l cosmology, whose a t t i t u d e t o numbers
r e s e m b l e s t h a t t o words o f Humpty Dumpty i n A l i c e through
the L o o k i n g G l s s s — t h e r e i s no doubt a s t o who i s t o
be m a s t e r .
As r e g a r d s 1 as an i n d i r e c t measure o f
volume we may n o t e t h a t the a r e a o f any p l a n e p o l y g o n o f
p e r i m e t e r P ^ i s , because o f the i s o p e r i m e t r i c i n e q u a l i t y ,
l e s s than p / M M ) .
A p p l y i n g t h i s and the c o r r e s p o n d i n g
volume i n e q u a l i t y we see t h a t the volume o f the C i t y i s
l e s s than l3/6(Pi) .
These bounds, a l t h o u g h t h e b e s t
p o s s i b l e , p r o v i d e however o n l y v e r y p o o r e s t i m a t e s o f
size.
2
32
Even the ahove a n a l y s i s , t e d i o u s though i t may he,
n e g l e c t s one a d d i t i o n a l p o i n t o f d i f f i c u l t y .
The
measure o f the "Sity o f R e v e l a t i o n i s g i v e n i n the King
James A u t h o r i z e d V e r s i o n o f the B i b l e as 12,000 f u r l o n g s ,
a s i t i s i n the New E n g l i s h B i b l e .
However, the t r a n s l a t i o n s
a r e made-from the o r i g i n a l Greek, where the word used
i s , o f c o u r s e , s t a d i a , the p l u r a l o f s t a d i o n — the o r i g i n a l
o f stadium - the d i s t a n c e which gave i t s name to the
Stadium a t Olympia and to a l l stadiums ever s i n c e .
To
c l i n c h the m a t t e r we need n o t l o o k up the Greek New
Testament;
the Revised Standard V e r s i o n says "twelve
thousand s t a d i a " .
Oddly enough i t g i v e s a f o o t n o t e ,
"About f i f t e e n hundred m i l e s " which would be c o r r e c t
i f a s t a d i o n were equal to a f u r l o n g . However t h i s i s
n o t the c a s e .
The l e n g t h o f a s t a d i o n has been v a r i o u s l y
estimated;
v a l u e s g i v e n range from 19*+.17 to 200.25 y a r d s .
W
—
^
—
I n c i d e n t a l l y , on page 103 o f The View Over A t l a n t i s ,
i t i s c l a i m e d t h a t the s t a d i o n i s 0.1 o f a n a u t i c a l m i l e ,
i . e . about 202.9 y a r d s , which suggests i t has been s t r e t c h e d
a b i t i n the i n t e r e s t s o f n u m e r o l o g i c a l cosmography.
Be t h a t as i t may, i t i s h e l p f u l as i n d i c a t i n g t h a t John
M i c h e l l i s aware o f the d i f f e r e n c e between the f u r l o n g
and the s t a d i o n .
C l e a r l y t h e r e f o r e he has d e r i v e d 31680
from t h e , i n f a c t e r r o n e o u s , concept o f an edge o f the
C i t y as b e i n g 12,000 E n g l i s h f u r l o n g s .
Within the g e n e r a l
p h i l o s o p h y o f n u m e r o l o g i c a l cosmography t h i s d e r i v a t i o n
i s , because o f the a p p a r e n t l y g e n e r a l p r i n c i p l e o f
s u b s t i t u t i o n o f u n i t s , not n e c e s s a r i l y i n v a l i d .
Even i f i t i s o n l y a c o i n c i d e n c e , the f a c t noted by
M i c h e l l t h a t the t o t a l area o f f o u r o f the c i r c l e s i n the
B a r b u r y C a s t l e f o r m a t i o n approximates to 31680 square
f e e t i s a remarkable one, because he drew a t t e n t i o n to t h i s
number b e f o r e and not a f t e r the f o r m a t i o n was made.
Let
R/ and R^ be the r a d i i i n f e e t o f the c e n t r a l c i r c l e
( c e n t r a l o u t e r r i n g ) and o f the o u t l y i n g c i r c l e ( N ) .
Suppose t h a t R
and R z ^ i n r e s u l t o f e r r o r s o f f o r m a t i o n
and o f measurement, are d i s t r i b u t e d i n p r o b a b i l i t y n o r m a l l y
w i t h r e s p e c t i v e means
and m and e q u a l v a r i a n c e s ,
then the l e a s t squares (op maximum l i k e l i h o o d ) e s t i m a t e o f
m i s c'ft/(i 3+ T^J/h* •
observed v a l u e s quoted by John
M i c h e l l are K, = 71, R = 41, and there i s no r e a s o n n o t
to a c c e p t them.
I n p a s s i n g we may note t h a t 71/41 =
1.7317073, which i s a good a p p r o x i m a t i o n to
» / T = 1.7320508.
A l s o , i f we t r y to f i n d v a l u e s o f R and Rz, t h a t s a t i s f y
= 31680,
the j o i n t t h e s i s t h a t R, - RZV3 and 2 7TT R,
;
m
T
r
*"
h
e
a
t
1
i t t u r n s out t h a t they would he r e s p e c t i v e l y 71.0072 and
40.9960, w h i c h a r e i n d i s t i n g u i s h a b l e i n p r a c t i c e from
71 and 41.
A l i t t l e more a n a l y s i s i s however i n o r d e r .
As 71 and 41 a r e g i v e n t o the n e a r e s t f o o t o n l y , each
f i g u r e c o n c e a l s an unknown r o u n d i n g - o f f e r r o r , i . e . R/
l i e s between 70.5 and ft.5 and R^ between 40,5 and 41.5.
We r e p e a t the e s t i m a t i o n o f m, d o i n g i t f o r the two
extreme c a s e s (a) R = 70.5. R
= 40.5. and (b) R/
= 71.5.
R = 41.5.
We o b t a i n (a) m = 40.652, Area = 31,151.10
square f e e t . ( b ) m = 41.336, Area = 32,20686 square f e e t .
Each a r e a d i f f e r s from 31680 by 1.67% o n l y . I t would
seem t h a t the agreement w i t h the a p o c a l y p t i c number i s good
enough t o a l l o w t h a t the a s s u m p t i o n t h a t i t was b e i n g
aimed a t by an i n t e l l i g e n c e , human o r o t h e r w i s e , i s an
a d m i s s a b l e one.
Of course, the h y p o t h e s i s i s n o t
proven thereby.
2
The V e s i c a P i s c e s .
I n The C e r e a l o g i s t (No 3. S p r i n g 1991). John M i c h e l l
commented b r i e f l y on the s o - c a l l e d 0Z0 f o r m a t i o n d i s c o v e r e d
a t Fordham P l a c e , n e a r C o l c h e s t e r , E s s e x , i n 1990.
As w i l l be seen i n F i g u r e £T
. a diagram o f the formation
can be completed t o g i v e a f i g u r e w h i c h i s a good
a p p r o x i m a t i o n t o t h a t known t r a d i t i o n a l l y as the v e s i c a
pisees.
T h i s p h r a s e sounds r a t h e r g r a n d , b u t j u s t
means t h e " f i s h b l a d d e r " , i n p u r e r e f e r e n c e t o i t s shape.
However i t has been o f •• s y m b o l i c i m p o r t a n c e t o C h r i s t i a n s
from t h e b e g i n n i n g o f the C h r i s t i a n e r a .
The o*s i n the
0Z0 f o r m a t i o n a r e n o t t r a d i t i o n a l .
The e s s e n t i a l p a r t o f
the symbol i s t h e p e r i m e t e r only? w h i c h c o n s i s t s o f two
c i r c u l a r a r c s o f equal r a d i u s c u t t i n g a t an a n g l e o f
120°.
( c o r r e s p o n d i n g l y the c e n t r e o f each a r c l i e s on
the p t h e r a r c ) .
Another c h a r a c t e r i s t i c f e a t u r e i s t h a t the
l e n g t h i s e x a c t l y e q u a l t o i t s w i d t h m u l t i p l i e d by
\TS
( i . e . by 1.7320508).
&s shown i n F i g u r e ^
which i s
t a k e n f r o m The View Over A t l a n t i s t h e v e s i c a c o n t a i n s
two e q u i l a t e r a l t r i a n g l e s ; a l s o a o n e - t m r 3 s i z e r e p l i c a
o f i t s e l f can e a s i l y be c o n s t r u c t e d w i t h i n i t by d r a w i n g
two c i r c l e s each w i t h r a d i u s o n e - t h i r d o f the bounding a r e a s .
C u r r e n t l y M i c h e l l says the v e s i c a o r "The V e s s e l " has been
u s e d as a symbol f o r the V i r g i n Mary.
He a l s o e q u a t e s
i t w i t h t h e o v a l nimbus around some e a r l y r e p r e s e n t a t i o n s
o f the V i r g i n .
As many r e a d e r s w i l l r e c o g n i z e , the v e s i c a
i s i d e n t i c a l w i t h t h e o u t l i n e o f the symbol o f the U n i t e d
Church o f Canada.
Here the d e r i v a t i o n i s p r o b a b l y from
p a r t o f the F i s h Symbol o f the e a r l y C h r i s t i a n s b e i n g
3<f
Vesica
m
W1WCATE EXISTS ^ ^ u ,
FORMER EflST WILL
t h e body o f the F i s h sans t a i l .
I n The View Over A t l a n t i s
M i c h e l l used t h e v e s i c a a s the s t a r t i n g p o i n t o f a new
t h e o r y o f the G r e a t P y r a m i d .
He s u r m i s e s t h a t the
v e r t i c a l m i d s e c t i o n o f the P y r a m i d has the shape d e f i n e d
by t h e h e a v i l y p r i n t e d t r i a n g l e ( F i g u r e £> ).
One o f
the p r e s e n t a u t h o r s has examined t h i s h y p o t h e s i s i n an
e a r l i e r p a p e r (New H o r i z o n s . 1, No. 2, Summer 1973. 102-108).
The r a t i o o f t h e bs^e t o the h e i g h t o f t h e t r i a n g l e was
f o u n d t o be ( . / 3 T - — 3 ) / S3,which
i s numerologically
amusing, as i t c o n t a i n s o n l y the d i g i t 3;
the f o u r 3's
t o t a l 12, w h i c h by c a b a l i s t i c r e d u c t i o n i s 3 once more.
The v e s i c a seems to have a c h i e v e d some prominence i n
c o s m o g r a p h i c a l a r c h a e o l o g i c a l and a r c h i t e c t u r a l theory,
a s a r e s u l t o f B l i g h Bond's e x c a v a t i o n s and n u m e r o l o g i c a l
musings a t G l a s t o n b u r y i n the e a r l y y e a r s o f t h i s c e n t u r y .
Q u o t i n g W i l l i a m K e n a w e l l (The Quest at G l a s t o n b u r y ) " B o r i d
t h o u g h t he had f o u n d two r i v a l systems f o r d e t e r m i n i n g the
ground p l a n s o f m e d i e v a l b u i l d i n g s .... They were (1) A
system o f commensurate s q u a r e s , (2) A system o f e q u i l a t e r a l
t r i a n g l e s , w h i c h , where combined i n p a r a l l e l o g r a m s gave a
rectangular f i e l d or setting"
To q u o t e Bond h i m s e l f
( a r t i c l e e n t i t l e d "The G e o m e t r i c C u b i t . . ) the p r i n c i p l e
i n v o l v e d was "... one o f g e o m e t r i c a l p e r f e c t i o n , the
o b j e c t b e i n g the r e p r o d u c t i o n o f t h e f o r m o f the Rhombus
o f two e q u i l a t e r a l t r i a n g l e s i n t h e g r e a t e s t degree o f
a c c u r a c y consonant w i t h p r a c t i c a l methods o f b u i l d i n g ,
and h a v i n g harmonious s c a l e s o f measurement .... from
very e a r l y times a p e c u l i a r respect
even a s a n c t i t y
— a t t a c h e s t o t h o s e p r o p o r t i o n s w h i c h most c l e a r l y
a c c o r d e d w i t h the m a t h e m a t i c a l p r i n c i p l e s known t o
M a s t e r Masons".
The Rhombus.is, o f c o u r s e , t h a t formed
by t h e two e q u i l a t e r a l t r i a n g l e s i n s c r i b e d to t h e v e s i c a
pisces.
Bond saw i n e f f e c t t h a t a r e c t a n g l e c i r c u m s c r i b e d
t o a v e s i c a w o u l d be q u i t e w e l l p r o p o r t i o n e d .
This i s
shown i n F i g u r e 7
, w h i c h i n c i d e n t a l l y shows a n o t h e r
i n t e r e s t i n g f e a t u r e o f the v e s i c a — t h a t the c i r c u m s c r i b i n g
r e c t a n g l e can be c o n s t i t u t e d f r o m d i a g o n a l s o f a r e g u l a r
hexagon.
(En p a s s a n t , i t may be n o t e d t h a t i n the
r e c t a n g l e the r a t i o o f l e n g t h t o b r e a d t h ,
/T
as i n the
v e s i c a , d i f f e r s from t h a t i n a r e c t a n g l e based on the
" d i v i n e p r o p o r t i o n " o r " g o l d e n segment" — namely
P h i = 1.618 ... a s w e l l a s from the " s a c r e d cuJrt" o f
Tons B r u n e s . However the eye may n o t f i n d i t easy t o
d i s t i n g u i s h between these shape_s. (The p r e s e n t w r i t e r s
knew an a r c h i t e c t who used
-Jz - 1.4142 as an a p p r o x i m a t i o n
Phi.)
1
36
In B o n d ' s work t h e r e i s much numerology i n c l u d i n g use
o f 666 = 18 x 37 = 9 x 74, so t h a t l e n g t h s such as
74 f e e t a r e encountered;
In addition gematria establishes
c r o s s c o n n e c t i o n s with the G r e a t Pyramid and Stonehenge.
Thus the Greek word f o r a house o r a temple ( i . e . the
d o m i c i l e o f a d e i t y ) i s o i k o s = 370 = the circumference o f
Stonehenge i n the u n i t MY, i . e . m e g a l i t h i c y a r d s , where
1MY = 2.72 f e e t , b e i n g a u n i t d i s c o v e r e d by P r o f e s s o r
Thorn to have been a p p l i e d to the l a y o u t o f many p r e h i s t o r i c stone
circles in Britain.
(See A . Thorn, M e g a l i t h i c S i t e s i n
Britain).
A l s o , as shown i n F i g u r e 8
. a vesica pisces
can be f i t t e d on to the ground p l a n o f Stonehenge.
7
y
m
I f we speak o f "The Temple" which i s o O J K P S i n Greek,
gematria y i e l d s 440 = the s i d e o f the Pyramid i n c u b i t s ,
p r o v i d e d the c o r r e c t choice i s made from the v a r i e t y o f
c u b i t s i n use i n a n t i q u i t y .
The name c u b i t f o r the u n i t
i n q u e s t i o n i s d e r i v e d from L a t i n , c u b i t u s — the lower arm.
The l e n g t h i s n o t o n l y d i f f i c u l t to s t a n d a r d i z e but even
to d e f i n e , a c c o r d i n g as to whether the f i n g e r s , and, as
w e l l , the back o f the hand, are i n c l u d e d .
Thus, while
the Hebrew c u b i t i s o f t e n quoted as about 19.05 inches =
1 f o o t 7.05 i n c h e s , a t l e a s t two o t h e r c u b i t s were i n use
i n B i b l i c a l times.
The l a t e s t to be adopted was c a l l e d
the "new c u b i t " and was about 2 0 . 6 i n c h e s , b e i n g the same
a s the E g y p t i a n c u b i t , i n which the N i l o m e t e r , a p o l e
i n d i c a t i n g the h e i g h t o f the N i l e was marked.
However
i t seems t h a t a sacred o r Royal c u b i t o f 20.88 i n c h e s was
a l s o used i n E g y p t i a n h o l y c o n t e x t s .
T a k i n g the s i d e
( i . e . edge o f base) as 755 f e e t , M i c h e l l c o n v e r t s i t to
440 E g y p t i a n c u b i t s .
That i s he t a k e s the c u b i t as
1.71591 f e e t = 20.5909 i n c h e s , which goes w e l l w i t h the
f i g u r e quoted above f o r the "new" o r o r d i n a r y E g y p t i a n
cubit.
(See a l s o S i r C h a r l e s Warren).
The Mandelbrot Set a t
Ickleton.
The most s u r p r i s i n g f o r m a t i o n i n 1991* and i n f a c t
d u r i n g the whole h i s t o r y o f crop c i r c l e s , o c c u r r e d a t
I c k l e t o n , a v i l l a g e i n the s o u t h e a s t e r n c o r n e r o f
Cambridgeshire.
At f i r s t s i g h t i t s ..outline ( F i g u r e
)
resembles t h a t o f the cosmopolitan b e e t l e A d a l i a b i p u n c t a t a .
T h i s i n s e c t , which i s judged h a r m l e s s , and indeed
b e n e f i c i a l to man because i t e a t s a p h i d s , i s g e n e r a l l y
p o p u l a r , b e i n g neat, c l e a n , and s h i n y i n appearance.
Known as a ladybug o r l a d y b e e t l e i n N o r t h America, i n
England i t i s the "ladybird"}
i t i s the s u b j e c t o f a
n u r s e r y rhyme and i t s name has been g i v e n to a s e r i e s o f
e d u c a t i o n a l books f o r c h i l d r e n .
I t c o u l d be presumed
t h e r e f o r e t h a t t h i s f o r m a t i o n should be c l a s s i f i e d as an
' insectogram*.
However, u n l e s s the m o t i v a t i o n f o r the
I c k l e t o n f o r m a t i o n i s overdetermined and i n v o l v e s a k i n d
o f v i s u a l pun on the resemblance to the l a d y b i r d , the
f i g u r e i s n o t p r i m a r i l y i n the form o f A d a l i a .
Instead
i t r e p r e s e n t s a good f i r s t a p p r o x i m a t i o n to a famous
mathematical o b j e c t
the s o - c a l l e d "Mandelbrot Set".
The Mandelbrot Set i s a r e g i o n o f the p l a n e whose boundary
i s a curve o f a s p e c i a l k i n d known as a " f r a c t a l " curve
or set.
I t may be noted t h a t the word set o r curve may
o f t e n ( w i t h r e g a r d to context) be used i n t e r c h a n g e a b l y .
Thus a c i r c l e i s a set o f p o i n t s e q u i d i s t a n t from a g i v e n
p o i n t (the c e n t r e o f the c i r c l e ) .
(While a curve can
l e g i t i m a t e l y be c a l l e d a s e t , the term "set" i s more
comprehensive than "curve".
Thus the p o i n t s i n t e r i o r
to a c i r c l e c o n s t i t u t e a s e t , as do the p o i n t s w i t h i n
o r on the c i r c l e , i . e . the i n t e r i o r p o i n t s p l u s the
boundary).
F r a c t a l s e t s a r e d e f i n e d a b s t r a c t l y by mathematical
r u l e s , and, o f c o u r s e , cannot be drawn e x a c t l y ;
they
can o n l y be approximated t o .
But i n t h a t r e s p e c t they
a r e no d i f f e r e n t from b e t t e r known curves such as c i r c l e s ,
which, as p o i n t e d out by P l a t o more than two m i l l e n i a ago,
can o n l y be drawn approximately and n e v e r e x a c t l y .
Fractal
s e t s ( o r curves) a r i s e i n many d i f f e r e n t ways.
The
e s s e n t i a l f e a t u r e i s t h a t , regarded as a curve, a f r a c t a l
s e t d i f f e r s from c l a s s i c a l curves such as c i r c l e s ,
e l l i p s e s , e t c . by t o t a l l y l a c k i n g the p r o p e r t y o f
smoothness.
5 c i r c l e i s not o n l y continuous i t i s a l s o
smooth.
A t every p o i n t we can d e f i n e the slope o f the
c u r v e , i . e . we can draw a tangent.
A tangent i s a s t r a i g h t
l i n e which meets the curve o n l y i n the p o i n t o f c o n t a c t .
I t meets the curve i n no second p o i n t .
(There are
e x c e p t i o n s to t h i s l a s t remark, b u t they are o f ng>
importance i n the p r e s e n t c o n t e x t ) .
A curve i s s a i d to be
"smooth" a t any p o i n t where a tangent can be drawn.
At such a p o i n t the f u n c t i o n ( i . e . the mathematical
formula f ( x ) d e f i n i n g the c u r v e d i s s a i d to be
d i f f e r e n t i a b l e ; the slope o f the tangent there i s c a l l e d
the d e r i v a t i v e
o r d i f f e r e n t i a l c o e f f i c i e n t , and i s
written f'(x).
In most o r d i n a r y cases the d e r i v a t i v e
can be c a l c u l a t e d from formulae d i s c o v e r e d independently
by S i r I s a a c Newton and G o t t f r i e d W i l l i a m L e i b n i z i n
the 1 7 t h c e n t u r y .
Thus i f x and y are the C a r t e s i a n
c o o r d i n a t e s o f a g e n e r a l p o i n t on a c i r c l e o f r a d i u s 1
u n i t w h o s e _ _ £ e n t r e i s a t the p o i n t with c o o r d i n a t e s ( 0 , 0 ) ,
then y = v/?7~-X*) . a n d the slope o f the tangent,
i.e.
the d e r i v a x i v e o f the f u n c t i o n
/ T ^ X * J iS. ( - x V ^ - x ^
T h i s i s i l l u s t r a t e d i n F i g u r e lO
. Here
s
An i m p o r t a n t f e a t u r e o f the t h e o r y o f f u n c t i o n s and the curves
they d e f i n e i s t h a t smoothness ( i . e . d i f f e r e n t i a b i l i t y )
i s l o g i c a l l y and f a c t u a l l y independent o f c o n t i n u i t y .
Wherever the curve has a sharpv bend i n i t i s a p o i n t
o f c o n t i n u i t y , b u t the curve i s n o t smooth t h e r e .
& s i m p l e example i s c o n s t i t u t e d by f ( x ) = x i f x i s
p o s i t i v e , f(x) = 0 i f x i s zero, f(x) = -x i f x i s
negative.
The graph i s continuous everywhere but not
smooth a t x = 0, (see F i g u r e It
)•
(From a mathematical
p o i n t o f view a s t r a i g h t l i n e i s a l s o a "curve).
It is
a l s o easy to c o n s t r u c t f u n c t i o n s where curves are "unsmooth"
a t an i n f i n i t e number o f p o i n t s , such as those whose
c o o r d i n a t e s are i n t e g e r s , i . e . x = 1 . 2 , 3
, or
11,1,
" r a t i o n a l numbers" e . g . f r a c t i o n s '(\J/%, fr,fo>*-fa'/$7)Y^/s,
T^'S*.»
I t i s f a i r l y easy to go f u r t h e r s t i l l and c o n s t r u c t
fInjections which are not d i f f e r e n t i a b l e a t an i n f i n i t e
s e t o f p o i n t s , the set b e i n g "everywhere dense",
i.e.
t h e r e i s no i n t e r v a l o f x v a l u e s , however s m a l l , which
i s f r e e o f such p o i n t s .
(See T i t c h m a r s h , The Theory o f
F u n c t i o n s , f o r the method known as "the condensation o f
s i n g u l a r i t i e s " , invented by George C a n t o r . )
x
I t was i n the second h a l f o f the n i n e t e e n t h c e n t u r y t h a t
pure mathematicians formed the a m b i t i o n o f d e f i n i n g
c u r v e s t h a t were continuous b u t n o t smooth a t any o f
their points.
T h i s i s the same as d e f i n i n g f u n c t i o n s
t h a t are nowhere d i f f e r e n t i a b l e .
The problem was
s o l v e d "by the g r e a t mathematician K a r l W e i e r s t r a s s e
(1815-1897) who announced h i s d i s c o v e r y i n a paper read
to the B e r l i n Academy i n J u l y 1872. ( I t was g i v e n
g r e a t e r fame however by P a u l du Bois-Reymond i n a paper
i n 1875).
W e i e r s t r a s s gave a formula f o r a f u n c t i o n
( d e f i n e d as a s p e c i a l type o f F o u r i e r s e r i e s ) which was
c o n t i n u o u s but n o n - d i f f e r e n t i a b l e f o r a l l v a l u e s o f x.
W e i e r s t r a s s l e f t the theory somewhat i n c o m p l e t e .
Other
eminent mathematicians such as Bromwich and D i n i c o n t r i b u t e d
improvements, b u t a major advance i n u n d e r s t a n d i n g the
W e i e r s t r a s s f u n c t i o n had to wait u n t i l P r o f e s s o r G e o f f r e y
Hardy o f T r i n i t y C o l l e g e , Cambridge, a p p l i e d new and more
p o w e r f u l mathematical methods i n a paper p u b l i s h e d i n
1916.
I n 1918 Knopp gave a g e n e r a l method o f c o n s t r u c t i n g
n o n - d i f f e r e n t i a b l e functions;
a f u r t h e r method was g i v e n
by B . L . van d e r Waerden i n 1930. (A f u n c t i o n somewhat
s i m i l a r to t h a t o f W e i e r s t r a s s was d e s c r i b e d by Darboux
i n 1875; both had been a n t i c i p a t e d by the mathematicians
Bolzano and C e l l e r i e s i n d e f i n i n g nowhere d i f f e r e n t i a b l e
f u n c t i o n s , b u t t h e i r r e s u l t s were not p u b l i s h e d , p r o b a b l y
because they d i d not understand them.)(See Hawkins).
To r e t u r n to the W e i e r s t r a s s curve, i t needs to be s a i d
t h a t i t i s extremely d i f f i c u l t to v i s u a l i s e i t .
I t has
been d e s c r i b e d as c o n s i s t i n g o f an i n f i n i t e number o f
infinitesimal crinkles.
What we can say i s t h a t between
any two p o i n t s on i t there are an i n f i n i t e number o f
"wiggles", thus there cannot be any unique l i n e which i s a
tangent to the c u r v e .
(As w e . s h a l l see, t h i s i n f i n i t e
c r i n k l i n g o r c r e n u l a t i o n i s i n t i m a t e l y r e l a t e d to the
c o n n o t a t i o n s o f the word " f r a c t u a l ) . F i g u r e 12. shows
an a p p r o x i m a t i o n to a s e c t i o n o f a W e i e r s t r a s s c u r v e .
A l t h o u g h some p a r t s appear to be smooth t h i s i s an i l l u s i o n .
I f the curve c o u l d be p l o t t e d e x a c t l y (which i s , o f
c o u r s e , i m p o s s i b l e ) , and then m a g n i f i e d , the p a r t s t h a t
now appear r e l a t i v e l y s t r a i g h t would be r e v e a l e d as j u s t
as c r i n k l e d as the "jagged" p a r t s do on the p r e s e n t s c a l e .
I t i s c h a r a c t e r i s t i c o f f r a c t a l c u r v e s , i . e . o f nond i f f e r e n t i a b l e f u n c t i o n s , t h a t each p a r t on a s m a l l e r s c a l e
r e p e a t s the same k i n d o f c r i n k l i n g as i s v i s i b l e i n the
l a r g e r scale presentations.
Such curves are s a i d to
be a p p r o x i m a t e l y s e l f - s i m i l a r .
(Some, indeed, are
e x a c t l y s e l f - s i m i l a r , e . g . von K o c h ' s "snowflakes",
b u t these c o n s t i t u t e a m i n o r i t y . )
R e v e r t i n g to the b a s i c f e a t u r e o f a W e i e r s t r a s s c u r v e ,
the f u n c t i o n may be d e s c r i b e d as one unbounded v a r i a t i o n
(or fluctuation or o s c i l l a t i o n ) .
The " v a r i a t i o n " o f
a f u n c t i o n o v e r an i n t e r v a l o f x, i . e . f o r x r a n g i n g between
the extremes x = x
and x = x ^
may be v i s u a l i s e d
i n the f o l l o w i n g way.
L e t A and B be the p o i n t s on the
curve y = f ( x ) c o r r e s p o n d i n g r e s p e c t i v e l y to x = x and
x =
.
LetX/.Xj
X « „ , be any p o i n t s on the
curve between A and B .
L e t L be the l e n g t h o f the
polygonal arc
AX,X
. . . X ^ . B.
Then, i f L i s l e s s
than a constant independent o f the mode o f d i v i s i o n ,
( i . e . o f the p o s i t i o n s o f X, , X ,
. . . X & and the choice o f
N) the f u n c t i o n f ( x ) i s s a i d to be o f bounded v a r i a t i o n
o v e r the i n t e r v a l (AT©,***-) .
The f a c t s o f i n t e r e s t are
t h a t a f u n c t i o n o f bounded v a r i a t i o n s i s smooth a t a l l
b u t an i n f i n i t e s i m a l f r a c t i o n o f p o i n t s , i , e .
it
p o s s e s s e s a de f i n ed *^lope o r d e r i v a t i v e " f \ x ) , f o r
"almost a l l " v a l u e s o f x.
I n a d d i t i o n the curve y = f ( x )
is "rectifiable"j
i . e . any s e c t i o n o f i t has a d e f i n e d
and f i n i t e l e n g t h .
However no l e n g t h can be d e f i n e d
f o r the a r c o f a f u n c t i o n o f unbounded v a r i a t i o n ;
i n o t h e r words the l e n g t h i s i n f i n i t e .
By making the
d i s s e c t i o n AX/ X . . . . X^!E>
' f i n e r and f i n e r the
l e n g t h L can be made to exceed any p r e s c r i b e d v a l u e .
0
0
Z
z
u
z
The curve i s n o t r e c t i f i a b l e , and any p i e c e o f i t has to
be regarded as o f i n f i n i t e l e n g t h .
( " R e c t i f i a b l e " means
t h a t a s e c t i o n o f s t r a i g h t l i n e can be found with the
same l e n g t h as any s e c t i o n o f the c u r v e ) .
An elementary
way o f d e s c r i b i n g a continuous* curve which i s not o f
bounded v a r i a t i o n i t to say t h a t any p a r t o f i t , however
s h o r t , has an i n f i n i t e number o f maxima and minima.
(zv^^
TJiis c h a r a c t e r i z a t i o n . h o w e v e r , i s inadequate because
c o n t i n u o u s curves can be d e f i n e d t h a t o s c i l l a t e i n f i n i t e l y
o f t e n i n any i n t e r v a l and y e t p o s s e s s a d e r i v a t i v e ,
(See Kopcke, Pereno, B r o d e n , i n Hawkins;
Broden showed
t h a t such f u n c t i o n s can be d e f i n e d v i a C a n t o r ' s method
of condensation o f s i n g u l a r i t i e s ) .
Mathematical t r u t h
i s e l u s i v e and belongs to a f i e l d i n which i n t u i t i o n —
o r p r e j u d i c e — i s a bad guide and which over the c e n t u r i e s
has been o f t e n d i s c r e d i t e d .
R e t u r n i n g however to n o n - r e c t i f i a b l e curves o f i n f i n i t e
l e n g t h — why a r e they c a l l e d " f r a c t a l ? "
This
name r e s u l t s from an attempt to c h a r a c t e r i z e them
m a t h e m a t i c a l l y i n terms o f a q u a n t i t y c a l l e d "HausdorffB e s i c o v i t c h dimension".
The two mathematicians
commemorated i n t h i s phrase are p e l i x H a u s d o r f f , who
a t the beginning o f t h i s c e n t u r y made g r e a t c o n t r i b u t i o n s
t o t h e thjory o f s e t s , and P r o f e s s o r A.S. B e s i c o v i t c h
o f T r i n i t y C o l l e g e , Cambridge, who i n t h e p e r i o d 19301950 made p r o f o u n d c o n t r i b u t i o n s t o t h e t h e o r y o f
f r a c t a l s . The d e f i n i t i o n o f H a u s d o r f f - B e s i c o v i t u h
d i m e n s i o n , (v/hich f o r convenience we s h a l l j u s t c a l l
H a u s d o r f f o r H-dimension) i s r e l a t i v e l y s o p h i s t i c a t e d .
B u t , o f c o u r s e , i t h a s some r e l a t i o n t o o u r o r d i n a r y
o r i n t u i t i v e idea o f dimension.
I n ordinary parlance
we would a l l a g r e e t h a t a p i e c e o f s t r a i g h t l i n e i s
o n e - d i m e n s i o n a l and would be p r e p a r e d t o say t h e same o f
an " o r d i n a r y " c u r v e such a s t h e a r c o f a c i r c l e .
S i m i l a r l y we would r e g a r d t h e i n t e r i o r o f a square o r
o f a c i r c l e as two-dimensional.
L i k e w i s e we would
g r a n t t h a t any p a r t o f t h e s u r f a c e o f a sphere i s o f
d i m e n s i o n 2.
T h i s i n t u i t i v e approach t o
d i m e n s i o n a l i t y o f spaces o r o f s e t s i s i n f a c t p e r f e c t l y
sound.
I t h a s been v i n d i c a t e d b y a v a r i e t y o f d i f f e r e n t
approaches by mathematicians.
However i t needs t o be
s t r e s s e d t h a t t h e dimension a s c r i b e d by the i n t u i t i v e
o r n a i v e approach r e p r e s e n t s o n l y one k i n d o f d i m e n s i o n a l i t y .
The " i n t u i t i v e " d i m e n s i o n i s c a l l e d t h e " t o p o l o g i c a l
dimension" o f the set.
I t i s i n f a c t t h e case t h a t b o t h
o r d i n a r y c u r v e s l i k e c i r c l e s and t h e m a j o r i t y o f f r a c t a l
c u r v e s such a s " s n o w f l a k e s " and W e i e r s t r a s s e c u r v e s
have a l l t h e same t o p o l o g i c a l d i m e n s i o n , namely d i m e n s i o n 1.
However t h i s seems t o l e a v e t h e s i t u a t i o n i n c o m p l e t e l y
described. N o n - r e c t i f i a b l e curves, being o f i n f i n i t e
l e n g t h , are^we f e e l , " t h i c k e r " t h a n s i m p l e a r c s , and,
i n a sense, occupy more space than t h e l a t t e r , even, though
t h e y have t h e same t o p o l o g i c a l d i m e n s i o n and c o n t a i n t h e
same number o f p o i n t s ( i . e . i n t h e sense o f t h e C a n t o r i a n
infinite).
H a u s d o r f f ' s work l e d t o a way o f c h a r a c t e r i z i n g
t h e " t h i c k n e s s * o f t h e s e c u r v e s , a s w e l l as t h e " t h i n n e s s "
o f o t h e r anomalous s e t s such a s C a n t o r ' s " t e r n a r y s e t " ,
w h i c h i s o f t o p o l o g i c a l d i m e n s i o n 1, and c o n t a i n s t h e
same number o f p o i n t s a s any s i m p l e l i n e o r a r c , b u t i s
" t h i n n e r " t h a n such c u r v e s .
A p r e l i m i n a r y a t t e m p t had been
made i n 1914 b y C a r a t h e o d o r y , b u t t h e f i r s t e f f e c t i v e
approach was by H a u s d o r f f i n 1919.
H a u s d o r f f drew
on t h e t h e o r y o f "measure" d e v e l o p e d i n the e a r l y y e a r s o f t h i s
c e n t u r y by t h e F r e n c h m a t h e m a t i c i a n H e n r i Lebesgue^
who p u b l i s h e d i m p o r t a n t p a p e r s i n 1901 and 1902. -or
F o r Lebesgue t h e i n t r o d u c t i o n o f "measure" a s a more
s o p h i s t i c a t e d form o f " l e n g t h " o r " a r e a " was e s s e n t i a l l y
a s t e p p i n g stone t o an improved d e f i n i t i o n o f t h e
m a t h e m a t i c a l p r o c e s s known as i n t e g r a t i o n .
"Lebesgue
ifl^§»gration" p r o v e d t o be a p r i n c i p l e which i n c r e a s e d
the u n i t y and power o f modern mathematics to an
a s t o n i s h i n g degree (see Hawkins o r T i t c h m a r s h ) ,
I n t e g r a t i o n need not concern u s » "but Lebesgue's
d e f i n i t i o n o f measure i s h i g h l y r e l e v a n t .
Consider f i r s t
a set S of-points located e n t i r e l y i n 8 section o f a
straight line.
Bach p o i n t can be l a b e l l e d by a c o o r d i n a t e
x, i t s d i s t a n c e from a f i x e d o r i g i n 0 .
Form a " c o v e r i n g
set" J o f n o n - o v e r l a p p i n g i n t e r v a l s .
(An " i n t e r v a l "
i s any p i e c e o f the l i n e such as the p o i n t s with x between
say, 0.316 and 2 . 8 2 7 ) .
A set o f J such i n t e r v a l s
"covers" the s e t S which we wish to "measure" i f ,
and o n l y i f , e v e r y p o i n t o f S/ i s i n s i d e one o f the
intervals of J .
We now imagine J to "shrink" as much
as p o s s i b l e , s u b j e c t o n l y to the c o n d i t i o n t h a t Jalways
covers S / .
The minimum v a l u e M o f the t o t a l l e n g t h L ( J )
o f J ( i . e . the sum o f the l e n g t h s o f the separate
i n t e r v a l s o f J) i s then the "Measure" o f the set S.
A c t u a l l y t h i s l a s t statement r e q u i r e s to be q u a l i f i e d i n
two r e s p e c t s .
F i r s t l y * L ( J ) may n o t have a minimum
v a l u e i n which case the "infimum" o f L ( J ) i s taken to
be the measure o f Sf,
An infimum i s however f o r
p r a c t i c a l purposes h a r d l y any d i f f e r e n t from a minimum.
The infimum i s , as i t were, a " f l o o r " beneath which L ( J )
cannot s i n k .
(A minimum i s an infimum which happens
to be a t t a i n e d ) .
The second q u a l i f i c a t i o n i s t h a t S
be 'measurable".
F o r t u n a t e l y most
sets are!
Indeed
i t i s n o t known f o r c e r t a i n whether any non-measurable s e t s
exist;
the q u e s t i o n c o n s t i t u t e s a r a t h e r arcane
b r a n c h o f s e t t h e o r y i n v o l v i n g the s o - c a l l e d A x i o j » o f
C h o i c e , i n t r o d u c e d i n t o mathematics by Zermelo.
#
;
(
I n the f o r e g o i n g M i s the Lebesgue measure o f the set S/
which i s o f t o p o l o g i c a l dimension 1 .
I f however the S/
were spread o v e r p a r t o f a plane and t h e r e f o r e o f
t o p o l o g i c a l dimension 2, the c a l c u l a t i o n o f i t s Lesbesgue
measure would have to be m o d i f i e d .
We imagine such a
set — c a l l i t S
covered by a s e t J c o n s i s t i n g o f
squares i n s t e a d o f i n t e r v a l s .
The Lebesgue ( p l a n a r )
measure o f S
i s then the infimum o f the t o t a l area o f
the squares which J comprises, as we imagine J s h r i n k i n g
w h i l e s t i l l c o v e r i n g 3 2. •
I f by t h i s means we
c a l c u l a t e d the Lebesgue p l a n a r measure not o f S j , b u t o f S /
the r e s u l t would be z e r o .
T h i s simple f a c t g i v e s
the c l u e to a method o f d e f i n i n g dimension v i a
c a l c u l a t i o n o f measure.
F o l l o w i n g H a u s d o r f f , f o r any
Set S we can d e f i n e ( a d m i t t e d l y i n a s l i g h l y c o m p l i c a t e d
2
a
t'c
•»»
way) a H a u s d o r f f d - d i m e n s i o n a l measure which we denote
by H ( d , 3 ) .
I t can be shown t h a t K ( d , S ) i s ^ c a p a b l e o f
o n l y t h r e e p o s s i b l e v a l u e s , namely, 0, 1 o r ' i n f i n i t y
F o r a segment S o f a s t r a i g h t l i n e o r o f any " o r d i n a r y "
smooth curve H ( 0 , S ) i s i n f i n i t e , H ( l , £ ) equals one,
and H ( 2 S ) = 0 .
Also H ( d , S , ) . , i s i n f i n i t e ^ f o r d
l e s s t h a n 1 b u t zero f o r D g r e a t e r than 1, "fhus d = 1,
i s c o r r e c t l y the H a u s d o r f f dimension o f any smooth c u r v e ,
a g r e e i n g with i t s t o p o l o g i c a l d i m e n s i o n .
S i m i l a r l y f o r «£.
S z , such as the i n t e r i o r o f a c i r c l e H(2> S2)
1 with
H ( d . S j ) = 0 f o r d g r e a t e r than 2 and i n f i n i t y f o r d
l e s s than 2.
H a u s d o r f f a p p l i e d t h i s method to C a n t o r ' s
t e r n a r y set which, although i t has as many p o i n t s as a
line-segment^has Lebesgue measure zero because i t i s
"too t h i n " .
However the H a u s d o r f f dimension i s d = ( l o g 2)/{
( l o g 3) = O.6309, so t h a t i t has some t h i c k n e s s ;
indeed
i t i s t h i c k e r than the "everywhere dense" set o f " r a t i o n a l
p o i n t s " — '( /fij%
whose H a u s d o r f f
dimension i s z e r o .
( F o r tne t h e o r y o f H ( d , S ) see e i t h e r
Falconer or Barnsley, or H a r r i s o n ) .
;
=
1
t
Von K o c h ' s "snowflake", a l t h o u g h o f t o p o l o g i c a l dimension 1,
i s , l i k e a l l non r e c t i f i a b l e c u r v e s , " t o o t h i c k " .
Its
H a u s d o r f f - B e s i c o v i t c h dimension i s twice t h a t o f C a n t o r ' s
set and i s ( l o g -+}^og 3)= 1.2619.
When speaking above o f
the W e i e r s t r a s s curve we were b e i n g s l i g h t l y i m p r e c i s e .
There i s i n f a c t a whole f a m i l y o f them.
Any i n d i v i d u a l
one i s c h a r a c t e r i z e d by the v a l u e s o f two c o n s t a n t s
and g.
The f i r s t constant X
i s g r e a t e r than 1 w h i l e €
l i e s between 0 and 1.
The W e i e r s t r a s s f u n c t i o n i n
F i g u r e IZ has
A * 2 and S = 0 . 5 .
On the b a s i s o f a
formula g i v e n by Hardy i n h i s 1916 p a p e r the H a u s d o r f f B e s i c o v i t c h dimension o f a W e i e r s t r a s s curve i s j u s t
(2 - £ ) which, p r o p e r l y enough, i s g r e a t e r than 1 ( i t s
t o p o l o g i c a l dimension) but l e s s than 2
— the
t o p o l o g i c a l dimension o f any p l a n a r r e g i o n .
The name " f r a c t a l " f o r curves and s e t s whose dimensions
a r e n o n - i n t e g r a l and t h e r e f o r e " f r a c t i o n s " was proposed
by P r o f e s s o r B e n o i t B . M a n d e l b r o t .
B o m i n France and
educated i n P a r i s he i s a s c i e n t i s t w i t h a v e r y b r o a d
and d i v e r s e knowledge o f many t o p i c s o f c u r r e n t i n t e r e s t
and has h e l d many r e s e a r c h and t e a c h i n g p o s i t i o n s i n
F r a n c e , S w i t z e r l a n d , and the U . S . A .
I n the l a t e 1960's
he became i n t e r e s t e d i n the shapes o f v a r i o u s n a t u r a l
objects.
F o r example, i f one t e a r s a sheet o f p a p e r
the b r o k e n edges w i l l be h i g h l y i r r e g u l a r .
Furthermore,
o p t i c a l m a g n i f i c a t i o n i n c r e a s e s the i r r e g u l a r i t y .
C o n t i n u i n g to i n c r e a s e the m a g n i f i c a t i o n , one w i l l get
the impression^ t h a t the p r o f i l e i s a p p r o x i m a t e l y s e l f similar.
Indeed i t i s v e r y l i k e t h a t o f a n o n - r e c t i f i a b l e
curve.
S i m i l a r r e s u l t s a r e got by b r e a k i n g a r o c k o r
l o o k i n g a t the edge o f an a c t u a l r e a l snowflake.
On
a c c o u n t o f these resemblances Mandelbrot proposed t h a t
such curves and p r o f i l e s be c a l l e d " f r a c t a l " , both
those m a t h e m a t i c a l l y d e f i n e d and the approximations found
in nature.
The term would seem to be a p p r o p r i a t e i n r e l a t i o n
to the jagged o r broken c h a r a c t e r o f these c u r v e s .
I n F r e n c h f r a c t i o n means breakage o r f r a c t u r e , as w e l l
as a r i t h m e t i c a l f r a c t i o n .
I n E n g l i s h there i s rthsoalmost
A r c h a i c word "anfractuous" = " t o r t u o u s , having many
w i n d i n g passages or grooves, sinuous", l a s t used by T . S .
E l i o t i n h i s poem The Waste Land —
" P i c t u r e an
a n f r a c t u o u s waste shore . . . "
However t h i s i s not
Mandelbrot's intention;
he wishes " f r a c t a l " to r e f e r
to the H a u s d o r f f - B e s i c o v i t c h dimension, p a r t i c u a r l y to the
f a c t t h a t t h i s dimension i s not a whole number, b e i n g an
"improper f r a c t i o n " , ( i . e . g r e a t e r than 1) f o r the curves
o f i n f i n i t e l e n g t h and a p r o p e r f r a c t i o n ( i . e . l e s s than 1)
f o r s e t s o f zero Lebesgue measure such as C a n t o r ' s .
M a n d e l b r o t ' s own p r e f e r e n c e went f u r t h e r . He wished
to r e s t r i c t " f r a c t a l " to the former c l a s s ( g r e a t e r than 1)
approximated by rocky c o a s t l i n e s , the edges o f snowflakes,
and t o r n m a t e r i a l s .
However t h i s r e s t r i c t i o n was
i m p o s s i b l e to m a i n t a i n , and a l l the curves and s e t s we
have mentioned are lumped t o g e t h e r as ?!fractals".
B e s i d e s M a n d e l b r o t ' s own books, o f which the most r e c e n t
one i s The F r a c t a l Geometry o f N a t u r e , numerous o t h e r s
have made f r a c t a l s p o p u l a r w i t h a p a r t o f the l i t e r a t e
p u b l i c o u t s i d e the ranks o f pure mathematicians, and
a l s o w i t h photographers, a r t i s t s , and computer b u f f s ,
because o f the beauty and i n t e r e s t , b o t h o f the n a t u r a l
o b j e c t s , and o f computer s i m u l a t i o n s o f f r a c t a l s ,
(see
a l s o Schroeder, Peitgen, Barnsley, H a r r i s o n , G l e i k ) .
The e x i s t e n c e o f what came to be known as the Mandelbrot
Set was f i r s t announced i n 1979 by Brooks and M a t e l s k i
who p u b l i s h e d a computer drawing o f i t .
Independently
and a t about the same time Mandelbrot a l s o d i s c o v e r e d the
Set.
I t was however Douady and Hubbard'who i n 1982
named i t i n h i s honour.
The Mandelbrot Set emerged from
a l i n e o f r e s e a r c h q u i t e unconnected w i t h n a t u r a l o b j e c t s
and h a v i n g no r e l a t i o n to the W e i e r s t r a s s , Koch, Cantor
t r a d i t i o n s of n o n - r e c t i f i a b l e curves, or lacunary sets.
The f i e l d jof i n v e s t i g a t i o n can be c a l l e d the t h e o r y o f
functional iteration.
We can i l l u s t r a t e t h i s i n i t s
s i m p l e s t form by an a p p l i c a t i o n o f "New*ton's r u l e " f o r
e x t r a c t i o n o f a square r o o t .
Suppose t h a t we r e q u i r e
the square r o o t
y/HT o f a number K .
Then Newton's
r u l e i s summarized by the e q u a t i o n
A"/*>iJ ,
L e t XQ be any r e a s o n a b l y good a p p r o x i m a t i o n to
then we can use the above formula to c a l c u l a t e
\TK7
x
l - "ifc^ + ^ A * )
which, i n f a c t , i s a b e t t e r
a p p r o x i m a t i o n to /K".
If
k
i s n o t good enough >
we can proceed i n the same way and o b t a i n the s u c c e s s i v e
f u r t h e r approximations <*z X>, Xq.;
. . . ad l i b i t u m .
A n i c e example i s p r o v i d e d by K = 10.
Take Xo = 3»*
we o b t a i n
t
;
C o r r e c t to f i v e decimal p l a c e s we i n f e r \flO = 3.16228.
Newton's r u l e f o r square r o o t s i s i n f a c t a s p e c i a l case
o f what i s c a l l e d the Newton-Raphson p r o c e s s , Raphson
b e i n g a younger contemporary o f S i r I s a a c Newton.
Applied
to f i n d i n g a v a l u e o f x s a t i s f y i n g an e q u a t i o n F ( x ) = 0
the p r o c e s s c o n s i s t s i n d o i n g the f u n c t i o n a l i t e r a t i o n s
d e f i n e d "by the e q u a t i o n
,
1 *,
where F ' ( X ) i s the d e r i v a t i v e o f the f u n c t i o n F ( x ) ,
which c l e a r l y has to be smooth f o r the p r o c e s s to be
applicable.
A p p l y i n g t h i s to the square r o o t o f K i s to
s o l v e the e q u a t i o n X — K = 0, which i s F(x) = 0, with
F(x) =
X* - K .
U s i n g the well-known r e s u l t t h a t
F ' (x) = 2xjthe i t e r a t i v e e q u a t i o n i s
1
which i s Newton's r u l e f o r
4 K.
us
m
^
m
U n f o r t u n a t e l y a l l problems are n o t so simple as
b u t r e q u i r e v e r y complicated i t e r a t i v e p r o c e d u r e s , not
o n l y to g e t an answer, but o f t e n m e r e l y to f i n d out
whether an answer e x i s t s , o r what k i n d o f answer e x i s t s .
The f i e l d o f a p p l i c a t i o n comprises a l m o s t every area o f
human i n t e l l e c t u a l endeavour whether t h e o r e t i c a l o r
p r a c t i c a l — mathematics, p h y s i c s , c h e m i s t r y , b i o l o g y ,
meteorology, the environment, s t a t i s t i c a l i n f e r e n c e ,
economics, as w e l l as p r e d i c t i o n and f o r e c a s t i n g o f a l l
kinds.
Of course the contemporary dominance o f n u m e r i c a l
methods r e s u l t s from the advent o f h i g h speed computers,
b u t the e x p l o s i o n o f s c i e n c e i n the l a s t decades o f the n i n e t e e n t h
c e n t u r y engendered a t t h a t time an i n t e r e s t i n the
"convergence" problems t h a t antedated the modern computer
era.
Thus, i t had been r e a l i z e d t h a t the Newton-Raphson
p r o c e d u r e had' i t s q u i r k s .
I f our c h o i c e o f
X
i s "bad"
i . e . i f we do n o t s t a r t the i t e r a t i o n " s u f f i c i e n t l y close"
to the s o l u t i o n x o f the e q u a t i o n F ( x ) = 0, then the
sequence X ><, ^ X3. ,
may n o t converge to x but
to a n o t h e r s o l u t i o n ( i r r e l e v a n t to our p r o b l e m ) , o r i t
might n o t converge a t a l l , o r i t might converge to a number
which i s not a s o l u t i o n a t a l l l .
(See Brftnner f o r an
example).
I t was i n I897 t h a t A r t h u r Cfiyley, P r o f e s s o r o f
Mathematics i n the U n i v e r s i t y o f Cambridge, and o f T r i n i t y
C o l l e g e , some t h r e e c e n t u r i e s a f t e r I s a a c Newton, s t u d i e d
the Newton-Raphson i t e r a t i v e , f o r m u l a i n the form
0
mt
0 s
^
•
Here
Hi
—
Z ^
}
}
=r
X n ~ir Ifa
.
O
I n the e i g h t e e n t h century mathematicians had g e n e r a l i z e d the
concept o f a number. The m o t i v a t i o n f o r t h i s was the d i s c o v e r y
t h a t e v e r y a l g e b r a i c e q u a t i o n can be s o l v e d i f we a r e
p r e p a r e d to a l l o w s o l u t i o n s i n terms o f s o - c a l l e d complex
numbers as v a l i d .
The eq,action
j <
o'x- + 2 S s o
i s e q u i v a l e n t to
(yc— 3 ) - =r — >
which as we know from our s c h o o l days "has no r o o t s " i . e .
there i s no o r d i n a r y number x which i s a s o l u t i o n because
(-16) has no square r o o t — the square o f any o r d i n a r y
number i s p o s i t i v e .
I t was then d i s c o v e r e d however, t h a t
i f one chose to work with a new v a r i a b l e , namely, « ? »
where c o n v e n t i o n a l l y i i s t r e a t e d a s an o r d i n a r y number, 0
with the one i d i o s y n c r a s y t h a t i t s square i
i s equal to
( - 1 ) , then any a l g e b r a i c e q u a t i o n i n V has s o l u t i o n s
i n j u s t the p r o f u s i o n t h a t decency would r e q u i r e .
,
z
w
2
—
mt
Thus i n the f i e l d o f complex numbers the equation
Z ^ - 6Z + 25 = 0 has the two s o l u t i o n s (approximate
to an e q u a t i o n o f "degree" two), namely Z = 3 +i and
Z=3-4i.
P r i o r to C a y l e y ' s time, mathematicians had developed
an e x t e n s i v e t h e o r y o f f u n c t i o n s o f the complex v a r i a b l e .
This theory i s very elegant;
i t i s a l s o very"powerful"
i n the sense t h a t i t embodies many a p p l i c a b l e theorems.
Of course every r e s u l t o b t a i n e d f o r i t e r a t i o n s o f Z
can be d i s p l a y e d as r e l a t i n g to the simultaneous i t e r a t i o n
o f s e p a r a t e o r d i n a r y v a r i a b l e s x and y .
Thus i f
+i
the c o r r e s p o n d i n g i t e r a t i v e e q u a t i o n s f o r
z
F o r example, Newton's r u l e f o r z r f - ~
2
w i t h f -Xv?
u
+ ^ because
i
''jz*
-
and V are
is
£ X * - £-J* )/f*li .
I n working w i t h complex mumbers i t i s convenient to use
a g r a p h i c a l r e p r e s e n t a t i o n as terms o f " C a r t e s i a n c o o r d i n a t e s "
x and y .
The complex number z = x + i y i s r e p r e s e n t e d
by a p o i n t P i n a plane (see F i g u r e fO ) ;
the d i s t a n c e
o f P "east" o f a g i v e n o r i g i n 0 i s x;
the d i s t a n c e "north"
of 0 i s y.
'We w r i t e r f o r /^c-H-y* , the d i s t a n c e OP.
Supposing an i t e r a t i o n to s t a r t al; the p o i n t Po r e p r e s e n t e d
by Zo, the r e s u l t s o f the i t e r a t i o n can be v i s u a l i z e d as the
sequence o f p o i n t s Ti^T-x^Ty, . » ,
where
corresponds to
Zn i n the i t e r a t i o n Zn+1 = F ( Z n ) .
The systematic
study o f such complex v a r i a b l e i t e r a t i o n s was i n i t i a t e d
by G a s t o n J u l i a i n 1919I t was found t h a t even the
s i m p l e s t p o s s i b l e f u n c t i o n s F ( Z ) were c h a r a c t e r i z e d by
e x t r e m e l y c o m p l i c a t e d b e h a v i o u r s o f the p o i n t
Often i t t u r n e d out t h a t the p l a n e i s d i v i d e d i n t o two
d i s t i n c t regions.
I f Po i s i n one o f the two r e g i o n s
then the sequence T o / P - f \ .
e x h i b i t s "chaotic"
behaviour;
the p o i n x Pn n e i t h e r converges towards a
l i m i t i n g p o s i t i o n , n o r moves on a p e r i o d i c o r almost
periodic orbit;
i n s t e a d i t wanders i n no d i s c e r n a b l e
p a t t e r n — w i t h o u t rhyme o r reason
like a particle
i n a t u r b u l e n t f l u i d , e . g . i n a saucepan o f b o i l i n g water.
The r e g i o n o f c h a o t i c b e h a v i o u r i s c a l l e d the " J u l i a set",
i n honour o f Gaston J u l i a .
The shapes o f t e n a d v e n t i t i o u s l y
have a chance resemblance to n a t u r a l o b j e c t s .
Thus F i g u r e / 3
shows ( i n b l a c k ) the J u l i a s e t o f the i t e r a t i o n
which, i n the t r a d e , i s nick-named "Douady's R a b b i t '
i n honour o f A . Douady (See K l e e n ) . (Yes, V i r g i n i a ,
mathematicians do have a sense o f h u m o u r l ) .
While there are numerous and d i f f e r e n t J u l i a S e t s there
i s o n l y one Mandelbrot S e t .
T h i s s e t i s r e l a t e d to the
J u l i a s e t s , although i t i s n o t , s t r i c t l y speaking, a
J u l i a set.
The Mandelbrot Set i s d e f i n e d i n terms o f
the f o l l o w i n g i t e r a t i o n :
and
^
-
<?v» +
^»
where C i s a d i s p o s a b l e (complex) c o n s t a n t .
The
Mandelbrot Set i s unique except f o r s c a l e and o r i e n t a t i o n .
Figure
1*+
shows three Mandelbrot Sets d i f f e r i n g
o n l y i n these r e s p e c t s .
The convention i s to show the
Set i n b l a c k
c a l l e d the " f i l l e d - i n " Mandelbrot S e t .
I t i s o f dimension 2, i . e . both i t s t o p o l o g i c a l and
H a u s d o r f f - B e s i c o v i t c h dimension e q u a l 2.
The Set i s
c h a r a c t e r i z e d by the f a c t t h a t i f Zo(=C) i s o u t s i d e i t ,
then -^TVx , the d i s t a n c e
u l t i m a t e l y "tends to
i n f i n i t y " , i . e . to p u t i t i n more p a r l i a m e n t a r y language
" i n c r e a s e s without l i m i t " .
C o r r e s p o n d i n g l y f o r any
Po n o t i n the Mandelbrot S e t , the i t e r a t e o f //D?h
approaches zero as n i n c r e a s e s .
However, i n t e r e s t
a t t a c h e s to the boundary o f the Set, as shown i n F i g u r e
The boundary i s a p p r o x i m a t e l y s e l f - s i m i l a r .
Every
m a g n i f i c a t i o n d i s c l o s e s d e t a i l s w i t h a g e n e r a l resemblance
to the main p a r t o f the c u r v e .
The curve i s i n f i n i t e l y
c r e n u l a t e d and n o n - r e c t i f i a b l e .
I t s f r a c t a l dimension
i s n o t known to us a t time o f w r i t i n g but i t i s between
1 and 2.
Figure
/5(a)
shows the whole boundary a t
low m a g n i f i c a t i o n ;
figure
/ S ( b ) shows a d e t a i l i n a
higher magnification.
The importance o f the C i t y o f Cambridge i n the g e n e r a l
scheme o f t h i n g s may be judged from the f a c t t h a t , u n l i k e
Oxford, i t i s served by two r a i l w a y s from London.
One o f
them, on the west, approaches through the towns o f B6yfetph
and H i t c h i n , and c o u l d be regarded as the more p l e b i a n o f
the two.
The journey i s u s u a l l y r a t h e r d u l l u n l e s s one
d e p a r t s from London f a i r l y e a r l y on a day when r a c e s are o
57
Filled Julia set of. 2w*/ .74486177/("Douady's Rabbit',)
2
v i -0.12256117+
13.
Blow up of the marked area
quasi-self similarity.
mi
Ik
a t Newmarket.
A t these times announcers over the
p u b l i c a d d r e s s system a t K i n g ' s C r o s s r a i l w a y s t a t i o n
warn passengers to beware o f c o n f i d e n c e - t r i c k s t e r s
and c a r d - s h a r p e r s .
Indeed i n p r e v i o u s decades one
c o u l d g e t , f o r q u i t e a modest investment, an e x c e l l e n t
d e m o n s t r a t i o n o f the t h r e e card t r i c k l
Ickleton is,
however, f a r from this r a i l w a y l i n e .
I f from a l o c a t i o n
on the western r a i l w a y l i n e , e . g . S h e p r e t h , o r Foxton,
one chooses i n summer to d r i v e eastward to I c k l e t o n ,
'
one w i l l approach the v i l l a g e over r o l l i n g downs which
a r e among the most b e a u t i f u l a g r a r i a n landscapes i n
Britain.
P l a n t e d with g r a i n s , p r i n c i p a l l y b a r l e y (whose
l o c a l v a r i e t y i s the h i g h e s t r a t e d by B r i t i s h brewers)*
on a sunny day the view cannot be b e t t e r e d .
On coming
to I c k l e t o n one f i n d s t h a t one has a r r i v e d almost a t
the o t h e r r a i l w a y l i n e , which has an important stop a t
Audley E n d .
Of fame i n the p a s t ( e . g . the M a s t e r s h i p
o f Magdalene C o l l e g e i n the U n i v e r s i t y o f Cambridge was,
and perhaps s t i l l i s , i n the g i f t o f the L o r d o f the Manor
t h e r e ) A u d l e y End has f o r some decades been a p l a c e f o r
f i n a n c i a l persons i n the C i t y o f London ( i . e . E n g l a n d ' s
Wall S t r e e t o r Bay S t r e e t ) to use a s the centre o f a
d o r m i t o r y suburb.
The a r e a , which i n c l u d e s the h i s t o r i c
borough o f S a f f r o n Walden, has l o n g seen the c o n v e r s i o n
o f m e d i e v a l c o t t a g e s to b i j o u r e s i d e n c e s , as w e l l as the
r e h a b i l i t a t i o n o f Tudor-esque h a l f timbered houses.
Indeed i f one chose to go to London moderately e a r l y i n
the day, and r e t u r n l i k e w i s e , one c o u l d enjoy the
dominance o f the t r a i n by these f i n a n c i a l p e r s o n s ,
d r e s s e d comme i l f a u t i n bowler h a t s , b l a c k j a c k e t s ,
p i n s t r i p e d t r o u s e r s , and w i t h t i g h t l y r o l l e d u m b r e l l a s .
Perhaps nowadays t h e i r costumes a r e d i f f e r e n t — d e s i g n e r
jeans ( ? ) , b u t t h e i r a t t i t u d e s are as g e n i a l as e v e r .
a
These gentlemen seemed always e x t r o v e r t and happy to the
point of "jolliness".
We cannot judge whether f i n a n c i a l l y
they had much to be j o l l y about, b u t even i f t h i s was n o t
the c a s e , they have to be g i v e n c r e d i t f o r p u t t i n g a good
face on i t .
P o s s i b l y two decades ago these gentlemen
were n o t e s p e c i a l l y n u m e r i c a l l y l i t e r a t e
b u t times
have changed, and d o u b t l e s s a h i g h p r o p o r t i o n o f these
b a n k e r s , i n v e s t o r s , b r o k e r s , e t c . e t c . n o t o n l y employ
computer o p e r a t o r s , but are themselves f i n a n c i a l a n a l y s t s ,
and a l s o w e l l r e a d i n contemporary mathematics and
computer s c i e n c e .
Taking these c o n s i d e r a t i o n s i n t o
account we have to r e t a i n i n our mind t h a t some g e n i a l
IS
15 la)
TILE
M E C H A N I C S
Since the mid-1970s, Roger Penrose, one of the
world's top mathematicians, has been fascinated by
patterns of tiles. It is easy, for example, to cover a floor
with square tiles but impossible to cover a surface with
pentagons all of the same size. Gaps would occur
because the spaces between pentagons are not
pentagon-shaped.
Penrose found, however, that he could cover a floor
with pentagons and three extra shapes—a star, a
diamond, and a hat The pattern atfirstappears to be
regular. Butacloser look shows that the motif never
repeats itself, though it is always close to doing so.
The phenomenon is of interest to those studying
crystals, which grow—atom by atom—with the sort of
regularity found in tile patterns. If a person were tiling a
floor with pentagons, stars, diamonds, and hats, he
would have to step back and look at the entireflooras if
it were a jigsaw puzzle, before deciding where the next
oteca should oo.
—
*
_
"'
of A u d l e y End and i t s
a j o v i a l jest*
e n v i r o n s may have p e r p e t r a t e d
Other p o s s i b l e c u l p r i t s i n r e s p e c t of . I c k l e t o n a r e , of, c o u r s e ,
the s t u d e n t s and graduate students o f Cambridge U n i v e r s i t y .
B e s i d e s mathematicians and p h y s i c i s t s the U n i v e r s i t y
has many b r i l l i a n t engineers and computer s c i e n t i s t s , as
w e l l as b i o l o g i s t s / m e t e o r o l o g i s t s ,
g e o p h y s i c i s t s , and
economists, a l l o f whom are h i g h l y numerate and p l a y
on computer keyboards as e a s i l y as a Mpzart on a p i a n o .
Even i n e a r l i e r y e a r s the U n i v e r s i t y has been the s e t t i n g
f o r many b r i l l i a n t student f e a t s o f an amusing n a t u r e ,
b u t which a l s o demonstrated c o n s i d e r a b l e t e c h n i c a l a b i l i t y .
Twentyfive y e a r s ago an unknown group o f students
(presumed to be engineers) i n the course o f one S p r i n g
n i g h t s e t an automobile ( a d m i t t e d l y s m a l l , but beyond
most p e o p l e ' s a b i l i t i e s to h o i s t ) on top o f the r o o f
o f the Senate House.
A f i r m o f c o n t r a c t o r s took some
days to get i t down.
Apocalypse Soon?
T u r n i n g back to the g e n e r a l p i c t u r e , we f i r s t g i v e a t t e n t i o n
to the d i r e s t i n t e r p r e t a t i o n s o f the crop p a t t e r n s .
It
i s n o t h a r d to s t i t c h t o g e t h e r the images i n t o an a p p a r e n t l y
c o o r d i n a t e d p r e d i c t i o n , i f n o t o f doom, a t l e a s t o f stormy
waters ahead f o r humankind!
To paraphrase a well-known
h y m n
>
"On Avon's bank the p r o p h e t ' s c r y
Persuades us t h a t the End i s nigh"*
The argument goes as
First,
followsi-
simple geometry — a c l e a r c u t message —
"We a r e r a t i o n a l l i k e youI"
Then, v a r i a t i o n s to show t h a t the communicators have
a programme
a n n u l i , keys, s i d e e x c r e s c e n c e s .
Next,
h i s t o r i c a l r e f l e c t i o n s on the p r o g r e s s o f humankind —
"These were thy gods, 0 I s r a e l i " . — sun god, f e r t i l i t y
goddess"
CAlso perhaps the B r a i n = snake c d e i t y ; c . f .
Serpent Mounds i n Ohio o r a t P e t e r b o r o u g h , O n t a r i o ) ,
Perhaps a l s o the i m i t a t i o n o f the s p i r a l on thejplain
o f N a s c a , P e r u (a p a t t e r n i n l o o s e s t o n e s and v i s i b l e
as a whole o n l y from above) has a message, "Did
you b e l i e v e i n gods i n the sky?".
A l t e r n a t i v e l y , do they mean to say, "We can see y o u l "
Or i s t h i s symbol o v e r d e t e r m i n e d w i t h y e t o t h e r l a y e r s o f
implication?
S p i r a l s w i l l i n e v i t a b l y b r i n g to mind the
symbolism o f W i l l i a m B u t l e r Y e a t s ' poem o f 1921 (which
f o r many o f us o n l y became a l i v e about 1937 i n the H i t l e r
years,when i t seemed to exemplify the s u p e r i o r i t y o f the
p o e t ' s i n t u i t i o n over t h a t o f the p o l i t i c i a n s ^ .
Yeats'
poem The Second Coming commencesi
?
T u r n i n g and t u r n i n g i n the widening gyre
The f a l c o n cannot hear the f a l c o n e r . . . .
Things f a l l a p a r t ;
the c e n t r e cannot h o l d
Mere anarchy i s l o s e d upon the w o r l d ,
S u r e l y some r e v e l a t i o n i s a t hand
"
The poem goes on to say what many o f o u r contempories
would agree w i t h , i n view o f the v i o l e n c e o f so many
current fanatacisms.
"The b e s t l a c k a l l c o n v i c t i o n , while the
Worst burn with a p a s s i o n a t e
intesnity."
On t h i s l i n e o f thought i t might be p o s s i b l e to g a t h e r
i n even the I r m i n s u l as h a v i n g an admonitory purpose.
The descendants o f Hermann's people r e s i s t e d the Emperor
o f Rome and c e n t u r i e s l a t e r the Emperor o f the F r e n c h , b u t
then a i d e d the i m p e r i a l i s m s o f the K a i s e r W i l l i a m and
A d o l f H i t l e r , the r i p p l e s from whose f a i l u r e brought down
a l l colonial imperialisms.
We have now to contejta w i t h
a l e g a c y o f nascent n a t i o n a l i s m s .
I f we are l o o k i n g f o r rebukes to humankind among the
crop p a t t e r n s , then what o f the Whale?
I s i t an
i n j u n c t i o n to "save the whales" o r a m o n i t i o n to the
e f f e c t t h a t "whales are people too"?
Science f i c t i o n
fans o r t h e i r p a r e n t s may r e c a l l the charming movie S t a r
T r e k l V i n which e x t r a - t e r r e s t r i a l e n t i t i e s check on the
well-being o f E a r t h ' s cetecea.
Following this l i n e of
thought we must admit t h a t the "insectograms" are somewhat
ambivalent i n t h e i r i m p o r t .
Should we beware t h a t
i n s e c t s do n o t become'-the meek t h a t s h a l l i n h e r i t the
earth?''
Or i s the megsage a warning n o t to g i v e o u r s e l v e s
airs?
Or i s i t t h a t the communicators, whoever they
may be, wish to i n d i c a t e t h a t t h e i r view o f us i s l i k e t h a t
expressed to the D e i t y by Mephistopheles i n G o e t h e ' s F a u s t i "The l i t t l e god o f e a r t h ( i . e . humankind)
To me he seems to be,
w i t h deference to y o u r G r a c e ,
One o f those c r i c k e t s ,
jumping o v e r the p l a c e " .
...
L a t e r , M e p h i s t o p h e l e s s p e c u l a t e s as t o whether we a r e
" f i t f o r o v e r t h r o w " so t h a t our " t o y - w o r i d " can be
sterilized!
What o f the M a n d e l b r o t Set a t I c k l e t o n ?
I t i s not
a l t o g e t h e r e a s y t o read an a p o c a l y p t i c meaning i n t o i t ,
P o s s i b l y i t s a y s " I n s p i t e o f the i n t e l l e c t u a l s u c c e s s o f
y o u r s p e c i e s as e x e m p l i f i e d by y o u r mathematics, p h y s i c s ,
and t e c h n o l o g y , you need t o be humble!
In a l l f i e l d s
o f endeavour you have a l r e a d y e n c o u n t e r e d l i m i t s t o the
e f f i c a c y o f y o u r r e a s o n i n g and have t o r e s o r t to r e l i a n c e
on the computer"
Perhaps i t i s a l s o s a y i n g "Beware
t h a t s l a v e does n o t become y o u r m a s t e r , i n the way t h a t
s l a v e s can b e l "
Again, p o s s i b l y , I c k l e t o n says "We a r e
more l i k e computers than you a r e " ( T h i s i s a s c i e n c e f i c t i o n
theme though n o t one o f those most commonly e n c o u n t e r e d ) .
A l t h o u g h i t i s J o h n M i c h e l l ' s a n a l y s i s o f the B a r b u r y
C a s t l e f o r m a t i o n t h a t has i n s p i r e d the h y p o t h e s i s t h a t the
c r o p c i r c l e s , t a k e n i n sequence, c o n s t i t u t e g p r o g r e s s i v e l y
d e v e l o p i n g s e r i e s o f warnings, i t i s f a i r to eay t h a t the
r e a s o n i n g i s c i r c u i t o u s and the c o n c l u s i o n s v e r y n a r r o w l y
b a s e d , d e p e n d i n g i n f a c t on a s i n g l e n u m e r i c a l c o i n c i d e n c e .
I f the f o r m a t i o n i s c o r r e c t l y i n t e r p r e t e d as r e l a t e d t o
R e v e l a t i o n , t h i s o n l y i m p l i e s t h a t i t s a u t h o r s , whoever
t h e y may be, a r e making a m o n i t o r y use o f the p r o p h e c i e s .
A f t e r a l l we have t o r e c o l l e c t t h a t R e v e l a t i o n was b a s e d
o n l y on a v i s i o n ( a l b e i t o f a good and s a i n t l y p e r s o n ) .
He d o u b t l e s s t o o k i t l i t e r a l l y i n a l l r e s p e c t s , b u t i t
was c l e a r l y c o n d i t i o n e d by the e x p e c t a t i o n o f the imminent
end o f the w o r l d as i t was, and the Second Coming o f J e s u s
( c . f . the Gospels, 8
Mark, 16 Matthew, 9 L u k e ) .
As
t h i s d i d n o t happen (a d i f f i c u l t y a s awkward f o r the
C h r i s t i a n F a t h e r s p r i o r t o A u g u s t i n e , as the d e l a y i n t h e
coming o f *tyesaiah was f o r the Hebrew p r o p h e t s a f t e r J e r e m i a h )
we a r e a t l i b e r t y t o a c c e p t R e v e l a t i o n as p u r e l y
metaphorical.
As e v e r , we s h o u l d beware o f the f i r s t
o f the f o u r horsemen — —
"the man on the w h i t e h o r s e " ,
the c o n q u e r o r "who rode f o r t h t o conquer and he conquered",
(though i t seems d i f f i c u l t t o p i c t u r e P r e s i d e n t Bush,
the l a s t r e m a i n i n g c a n d i d a t e f o r t h i s r o l e i n w o r l d
h i s t o r y , a s r i d i n g through Washington o r Badhdad i n t h a t
p e r s o n a as r e p r e s e n t a t i v e o f the w o r l d o r d e r as a Pax
Americana).
We may a l s o be a d v i s e d o f the p o s s i b i l i t i e s
o f war, p e s t i l e n c e , and d e a t h w h i l e forms o f economic
i m p e r i a l i s m , r e g i o n a l c o n f l i c t s , and arms s a l e s p e r t u r b
the T h i r d W o r l d .
Before resigning our f l i r t a t i o n with the apocalyptic view o f
the crop c i r c l e s we should v i s i t once more with our good
S i r Isaac Newton o f T r i n i t y College, Cambridge.
Newton
was i n many respects a successor to the Puritans o f the
mid-^ seventeenth century, "those Cromwellians o f the
eastern counties o f England who constituted the backbone
o f the Afew Model Army. In r e l i g i o n they were Independents
and the forerunners o f modem denominations such as
the Congregationalistg and Unitarians. In the f i e l d they
defeated both Anglican and Presbyterian armiesi
they
gave to the world what some would regard as the moat
p r e v i o u s o f a l l ideas — that o f t o l e r a t i o n . Long a f t e r
the C i v i l War some scholars and divines i n the U n i v e r s i t y
o f Cambridge, even though ordained i n the Church o f
England, kept a l i v e the s p i r i t o f t o l e r a t i o n .
Ralph
Cudsworth and Henry More attempted to bring the ehurch
to "latitudinarianisra" i . e . a breadth o f t o l e r a t i o n i n
religious belief.
This a t t i t u d e l e f t i t s mark on
Cambridge which i n both "town and gown" i s remarkably
free o f the "odium theologicaura".
In r e l i g i o n Newton
was a "Subordinationist", a Unitarian, i n f a c t an Arian,
and therefore t e c h n i c a l l y a h e r e t i c .
Gn t h i s account
he refused ordination i n the Anglican Church and thereby
s a c r i f i c e d appointment to the academically p r e s t i g i o u s
post (then as now) o f the Mastership o f T r i n i t y College.
Inspired by Cudworth, More, and also Joseph Mede (see,
A.R.G. Owen, 1963), Newton put together a book which was
published only posthumously, Observations upon the Prophecies
of Daniel, and the Apocalypse o f St. John. For us the
most i n t e r e s t i n g o f S i r Isaac's findings may be the
p r e d i c t i o n that the Beast o f Revelation, which was equated
sometimes with Milton's "She o f the Seven H i l l s "
i.e.
the Church o f Rome
would come to an end i n 1867 A.D.
Newton based t h i s p r e d i c t i o n on the 1260 years i n
Revelation and the date 607 A.D. when, i n h i s opinion,
the worship o f A n t i c h r i s t reached i t s peak.
As t h i s
date has come and gone, we may perhaps breathe again
(or not, i n case o f e r r o r ) .
F i n a l Remarks.
We are i n a p o s i t i o n l i k e David Hume re the universe —
i s i t a machine, a work o f a r t , a theatre f o r moral
improvement?
I f we do not know what i t i s , we cannot
deduce i t s purposed
The reader may object that t h i s i s
a truism
i f we do n o t know i t s p u r p o s e then we do
n o t know i t s p u r p o s e ;
yesi
q u i t e so — t h a t i s t h e
p r o b l e m — w h i c h i s j u s t what Hume s a i d t
U n l e s s we
can d i v i n e the p u r p o s e , i f any, o f the c r o p c i r c l e s
we have t o a c c o u n t them "a r i d d l e , a m y s t e r y , an enigma".
I f we can c a t c h l a r g e numbers o f p r a n k s t e r s i n f l a g r a n t e
d e l i c t o , w e l l and good!
U n t i l then serious students
s h o u l d r e s e r v e judgement.
I n the h i s t o r y o f seifence
t o o many counter-examples a r e on r e c o r d o f c a s e s where
a genuine phenomenon has been r e j e c t e d a s f a l s e .
Thus
we s h o u l d keep i n mind t h a t the c r o p f o r m a t i o n s may be
an o c c u l t phenomenon. Here we use t h e word " o c c u l t "
i n i t s p r i m a r y and c l a s s i c a l s e n s e , w h i c h was"".also
t h a t o f Newton's c r i t i c s r e the " f o r c e " o f g r a v i t a t i o n .
" O c c u l t " meant w i t h o u t known cause.
I n t h i s sense
" o c c u l t " f o r c e s may, f o r a l l we know, t o the c o n t r a r y ,
be w i e l d e d by humans who w i s h t o be s e c r e t , whether o r
not they are the " i l l u m i n a t i " — R o s i c r u c i a n s , freemasonic,
e t c . o f t h e k i n d r e c e n t l y b u r l e s q u e d i n Umberto Eco's
amusing n o v e l F o u c a u l t ' s Pendulum. (Among those l i s t e d
a t v a r i o u s t i m e s as b e i n g concerned i n t h a t b e n i g n
though h y p o t h e t i c a l c o n s p i r a c y have been F r a n c i s Bacon,
and, more r a r e l y , Newton h i m s e l f ;
Bacon was, o f c o u r s e ,
an alumnus o f T r i n i t y C o l l e g e , b u t we do n o t w i s h t o
p u r s u e t h a t avenue o f e n q u i r y i ) .
r
A r e t h e communicators v i a t h e c r o p f o r m a t i o n s non-human. —
a l i e n , though n o t i p s o f a c t o o r n e c e s s a r i l y i n i m i c a l
to man?
P e r h a p s s c i e n c e - f i c t i o n b u f f s are. as w e l l is ant*,
q u a l i f i e d t o answer; a c e n t u r y of so o f s c i e n c e - f i c t i o n
w r i t i n g has p e r f o r m e d a s a "thought l a b o r a t o r y " f o r
" t h o u g h t e x p e r i m e n t s " c o n c e r n i n g t h e n a t u r e and p s y c h o l o g y
o f non-human l i f e .
T r i v i a l p u r s u i t a s t h i s may seem,
i t h a s t h e m e r i t o f e n c o u r a g i n g open-mindedness, w h i c h i s
p o s s i b l y humankind's c h i e f v i r t u e .
Science f i c t i o n
p a r a d e s many t y p e s o f a l i e n s b e f o r e us. Kumanoidg, v e r y
l i k e o u r s e l v e s i n temperament • — a l l t o o humanl
A n d r o i d s . v e r y m o r a l , untemperamental, —• i n c l i n e d t o
s u b o r d i n a t e e m o t i o n t o r e a s o n . Cyborgs, b e i n g s s h a r i n g
a group mind and t h e r e f o r e w i t h o u t i n d i v i d u a l judgement.
E n e r g e t i c lifeforms«with a mode o f e x i s t e n c e p r o f o u n d l y
d i f f e r r i n g f r o m b i o c h e m i c a l l y based l i f e and w i t h v a r y i n g
d e g r e e s o f empathy w i t h humanoids. T h e i r a t t i t u d e s v a r y
a c c o r d i n g t o t h e i r s p e c i e s . Some a r e t o t a l l y b e n i g n t o
humans and w i l l h e l p them i n a n emergency. O t h e r s a r e
b e n i g n l y i n q u i s i t i v e as t o t h e n a t u r e o f human l i v i n g ,
and may t r y on o c c a s i o n t o u n d e r s t a n d i t .
Yet o t h e r
l i f e f o r m s w i t h o u t b e i n g p e r se e v i l , a r e y e t j u v e n i l e
o r immature i n t h e i r o u t l o o k and may do us m i n o r
m i s c h i e f on a c c o u n t o f i n s e n s i t i v i t y o r i g n o r a n c e .
A f e w non-humans, o f c o u r s e , a r e r e p r e s e n t e d tfn S t a r Trek
and o t h e r s c i e n c e f i c t i o n a s t o t a l l y c a l l o u s and'
d e s t r u c t i v e t o o t h e r species;, h u t t h e s e are^_a m i n o r i t y
and u s u a l l y humourless.
The humour o f t h e a u t h o r s o f
t h e c r o p p a t t e r n s i s a r e a s s u r i n g f e a t u r e . On t h e
f i v e hundredth a n n i v e r s a r y o f the d i s c o v e r y o f America,
i t i s t o o l a t e f o r t h e p r e v i o u s i n h a b i t a n t o f t h e New
World t o p r a y t h a t t h e y w i l l n o t be d i s c o v e r e d . I f t h e
c r o p c i r c l e commtoiicators a r e from e l s e w h e r e i n o u r
space, o r i n some o t h e r d i m e n s i o n , t h e n i t i s t o o l a t e
f o r u s t o u t t e r t h e same p r a y e r ^ a n d we must hope f o r
the best{
S u p p o s i n g t h a t t h e crop c i r c l e communicators r e p r e s e n t
a h i g h e r c i v i l i z a t i o n than o u r own, and choose t o i n d i c a t e
t h e f a c t t o u s , how c o u l d t h e y do i t , w h i l e s t i l l
r e s t r i c t i n g t h e m s e l v e s t o such a c r u d e , i n d e e d clumsy,
mode o f communication a s t h e y a r e now u s i n g ?
I t would
be v e r y s i g n i f i c a n t i f t h e y would i n d i c a t e some
m a t h e m a t i c a l o r s c i e n t i f i c f a c t w h i c h we would r e c o g n i z e
a s s u c h , b u t would be o f a r e v o l u t i o n a r y o r s t a r t l i n g
nature.
I t would need t o be o f t h e s t a t u r e o f B e c q u e r e l * s
d i s c o v e r y o f r a d i o a c t i v i t y i n I896, o r o f Go'del's theorem
o f 1930 on f o m m a l l y u n d e c i d a b l e p r o p o s i t i o n s .
However
i t i s h a r d t o see how a b s t r a c t m a t t e r s c a n be a p p l i e d
by s i m p l e d i a g r a m s .
The o n l y k i n d o f t h i n g which comes
t o mind i s t h e r e c e n t d i s c o v e r y o f i r r e g u l a r t i l i n g
o f a n i n f i n i t e p l a n e by P r o f e s s o r Roger Penrose o f the
U n i v e r s i t y o f O x f o r d (See F i g u r e f& ).
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