f-1

ETA-LORE
by Douglas R. Hofstadter
Introduct1on~
-------- .. ·---
I
I
The sequences with which this paper deals are infinite sequences
composed of a ftnite numher of dtst;nct integersJ they hav~ the
property of being quast~pericdic sequences, hy which I mean that any
1 ~ n i t e 11 c h 'J n k '' wh i c h c c u r s s om e wh e r e i n a p a r t i c u 1 a r s e q u en c e wi l I
actually occur infinitely often in that sequence. Probably the most
important c6nsequence of th~s fs that the sequence can be thought of
as a sequence 0~ "chunks", as well as a sequence of integers; now
11 each distinct "chunk" has a name, then one can specify the
entfre sequence uniquely, simply by ~tating the names of the chunks
which compose it, in the order in which they occur in the sequence.
~~ integ~rs are chosen to be the "nam~s" of chunks, then the "chunk~
descriptiol"l '1 of the original sequence ;s itself a new sequence of
integers, and it f~ called the "derivative" 0~ the or~g~nal sequence
(nothing to do with calculus). The sequences that are most
interesting are those whose de~ivatfves also have der1vatives, which
a1so have der~vatives, whfch also~ ••••••
Eta~sequences constitute a
special case of this kind of "fnfinitely•dffferentiable" sequence'
in fact, the derivative of an eta~sequence is another eta-sequence,
The featyre w~ich character~zes eta•sequel"lces is that they only
contain occurrences of two distinct · integers -- in fact, consecutive
integers. In a ce~tain sense, there is a one•to·o~e correspondence
between the set of Jll e~a-sequences and the set of all real l'll,.lmbers -but at this point, 4nstead of continuing with general results about
eta•sequences, I will present eta•sequc"ces in the order Jn which I
developed an acquaintance with them, and the general results wfl 1 fit
smoothly 1nto that context, C1 was not the tfrst person to discover
eta•sequen~es: t6 the best of my knowledge, A. Markov and G, Christoffel were the first to investigate them, towards the end of the last
century, But since then, apparel"ltly no new work has been done on
eta~sequences -·at least I don't know of any published work on
eta~sequences af~er those articles.)
o
An
eta~sequence
crops up,
----------------··-------
I first came across an eta~sequence as I was workil"l9 on a problem
having to do with squares and triangular numbers. CA triangular
number is a sum of consecutlve tnteger~, beginning with 1 -- such as
1 + 2 + 3 + 4 + 5 = 1S; a square can be similarly described ~sa
sum of consecut1ve odd integers, beginl"ling with 1 •• for 1nstance
1 + 3 + 5 + 7 = 16.)
In th~ course of this problem, I asked myself
how many tr1angular numbe'rs there are, on the average, between
success~ve squares.
Below, I show the result of a simple emP~rlcal
~nvestigation, with small numbers:
G)
Ll
1
I
31
I
l
I
16
9
6
I
10 1S l
3o
49
64
'
136
l
't
I
25
l
21 I
I
I
I
I
I
I
28
45
ss
I
I
At
I
100
66 781
I
q
1. • •
I. I
I•••
I
•• •
I
•
•
:
'
•
.
•
f
'
'
.
•
•
I'
2 • • • • • 1• • • • • 2 • • • • • • 1• • • • • 1• • • • • • • 2 • • • • • • l • • • • • • • 2 • • • • • • 1• • • • • • • •
see, there seem always tc be either 1 or 2 t~fa~gular
successive squares. ( ~ here a triangular number and
a s~~are coincide Ce.g. 36), I have treated the tri~ngular ~umber
as tf· it were great~r than the squar!, Had I treated 1t as 1~ ~t were
smaller th~n the square, a similar sequence of l's and 2 1 s would have
resulted~~ more on that later,)
Below, I exhibit ma~y more terms
of the seq~ence show~ above:
As
numbers
yo~
ca~
betwee~
21211?.1212112t2112121211212112121t21212112121t212l2112121121212112 •• ~
Certainly this sequence gives a strong visual impression of being
.. periodic; in fact, one might naively guess that it is actually
a periodic sequence, As tt turn$ out, such is not the case~ A
natural thing to do i~ looking at this sequence is to break ft into
" c h u n k s '1 - ~ ~ r ob a b 1y " 2 t " and 11 2 1 1 " a r e t h a s i mp 1 e s t c h oi e e ,
B c 1ow ,
the same sequence is given, broken Into thase "chunks":
~uasi
21 211-21 21 211 21 211 21 21 211 21 211 2t 211 21 21 211 21 211 •••
After a ~~i1e of star i ng at this segmented sequence, it would occur
to most PeOPle that the 211 1 9 always occur singly, whereas the 21 1 s
sometimes occur si ~ gly, sometimes in pairs. So why not write down
the number of 21's between successive 211 1 s? This is done below.
(Note that thls oPeration, although based on "chunks", is dlfferent
from takfng the derivative. However, the two operations are actually
very closely r~l~ted; their re1at~6n w111 be covered soon.)
21 211 21 21 211 21 211 21 21 211 21 211 21 211 21 21 211 21 211 •••
*I**
*I** I* I* I** I* I ,
(1)
2
1
2
1
1
2
1
••••
We will ignore the parenthesized "1" at the front of the lower
sequence, because it occurs before the ~1rst "211", What do we have?
It is a sequence which reads "2121121 •• ~". It 6ccur1 t~ us that thls
new sequence may actually he Just the old sequencel Of course such
a hyPothesis needs to be checked further Cit ehecks),~.a~d then
~roved, i~ posslble~
H~w t6 ~r6v~ ft fs not · obvious.
We p6stp~n~
the proof to its correct chronological place t~ my personal develop~
ment of eta-sequences, and instead proceed to a second place where
an eta•seq~ence cropped up.
Another
I
'
eta~sequenee
crors up
------------~----~----~----·My c~r~osity was p~qued by this (empir~cal) dfsc6very, s~ I
tried to 1rwel"t simi tar problems. One of these was the following:
how ma~y powers of 2 are there between successive rowers of 3? We
can construe~ the ffrst few te~ms of the sequence as before:
3
I
1
1
2
I
8 I
14
'
.
243
I
)
128
2187.'.~
729
I
I
I
I 32
I
16
.,I
~
81
27
I
9
I 256
512 11024
I
20'~8, -• • ~.
I
•
•
.
.
'
~
I
I
2 • • • • • • • • 2 • • • - • • • 1 • • • • ~ • • • 2 • • • • • • • • 1 • • • • • • ·• • • 2 • • • • •
;
•
4 • • • •
•
l
;
2• • •~ • • • • •
Agai,.,, only l's and 2's apPear; it is natural to try the same tric!(
lo~k~ng at ch~nks, only here the ch~nks arc not "21" and "211", but
11 21"
and "221", Below, we g1ve the sequence, in "pre•chunked" form:
o~
221 21 221 21 221 21 21 221 21 221 21 21 221 21 221 21 21 221 21 221
Now we c6uld co~nt the 21's (which occur singly and doubly) betwe~n
221 1 s --but this time let us form the true derivative, by dssigning
integers as names ~f the two distinct chunks, We can g1ve to ehunk 21
the name "1", and to chunk 221 the name "2", The chunk-sequence (the
derivative) can be simply re~d off of the above line, and we get:
2
2
1
2
1
2
1
2
1
1
2
1
2
1
1
2
1
2
It's another ~uasi•periodic sequence, composed of Just two fntegersl
(I pvt an exclamation mark, beca~se I think it's surPrising, However,
it is prob~bly not as much of a surprise to you as ~t was to me,
because a few ltnes back you were told to expect lt~) Now thls
sequence ~s not the same as any of the previous ones we have seen,
even though it looks very similar to them. But it shares with the
other sequences the fact that it breaks up into natural chunks (21
anrl 211) so we c~n form its derlvattve (which wfll be the seco~d derivative o~ the original sequence)~ And as was saf.d earlier, thls
process e~n continue indefinitely.
4t is easy
to
derive a form~la for the kth term of the powersFtrst observe that the numb~r
Cand 1ncludfng) a number N ts -
of•2•between~powers·o~-3 sequence~
of powers of 2 up
to
1 t
[log Nl
2
Where
II
tx]
II
StandS for
11
the greateSt fnteQer leSS than
Or
equal
tO
.
)(
"
.
l
r
For insta~c~, up to q there are 4 powers of 2 (1,2 1 4 1 8) 1 and _
log Cbase 2) of q is a shade over 3, so that our exp~ession gives
the right answer. Between the kth power of 3 and the k+lst power
o t 3, the r·e to,., e, t 1'1 e number o i p 6 we r s of 2 i s
c1c g n
1 +
ktl
2
=
t(k+'l)
A general
n -
log 3J 2
k
1 -
n og c3 'n
2
tk
log 31
2
formul~ for eta-sequences
·----·--~-----------------------·-~
This ex~ression _provides us wfth a model that we can easily
generalize, as follows:
eta (alpha)
=
((k+U
alpha] ... tk alpha]
k
Here, we can take alpha to be any real number. For each value
get a characteristic sequence, eta(alpha),
(In fact,
the name which C h rf~toffel gave to etaCalpha) was "characteristic
sequence of alpha".) It is straightforward to show that always,
eta(alpha) contains the two integers between wh;ch alpha lies, and
only those two 1nt~gers. Cif alpha is itself an integer, et~Calpha)
is total Jy trivial, cons;sting merely of an endless sequence of
alpha's.) So suppose alpha is pi~ the~ etaCalPhal must be composed
exclusivelY o~ 3's a~d 4 1 s. In fact, eta(p~) runs like this:
ot alpha, we
33333343333334333333U333333 A 3333334333333433333343333334333333433~.~
The sequence ttself is comp~sed of 3's and 4 1 s: the "natural" chunks
wh i c M t h e eye b rea k s f t up i .n t o a r e " 3 3 3 3 3 3 4 11 and 11 3 3 3 3 3 3 3 4 " ( t he
first occurrence of the latter chunk is a bit further out than
what Is shown above.} To f~l"'m a derivative 6f etaCpi), we must gtve
1nte9er~names to these two chunks: one possfbflfty would be
l!Qld
Style''
333333~ --> 6
33333334 ·-> 7,
or
where the name tells the number of 3 1 s in the chunk,
else we
eould s1mply tell the total length of the chunk, like this:
"New
Style"
3333334 -~> 7
33333334 ••> 8,
Ooth styles are quite natural~ If we take the former cho1ce, our
derivative sequence will count 3's bet~een successive 4's. This was
the original notion o~ derivative, and fr6m tt s~rang the terms
"coun" -- here 3 -- and "sep" (~or "separ~t6r") -- whfch is 4 here.
In general, the coun will be tne closest integer to alpha, wh11e
the sep is tne second-closest integer
alpha. The eta-sequence of
aJpha will ~lways have seps o~currtng singly, and cou~s varidhly.
To make the derivative, you count couns . between seps. Notice thdt
; f y o U. l i t e r a 1 l y mP. a n '' b e t we en s e p s " , yo 1.1 h a v P. t o d i s r ega r d t he v e r y
first group of couns, sine~ they precede the first sep. This was the
definition of derivative tor a long time-- I will call it the
"old•stYle" de~tvative.
to
The new•style d~rivative is almost the same-- it's just that
(1) every term is one bigger than fn the old-style derivative, and
(2) there is a~ ~xtra term in the new~style derivative, corresponding
the first grouP of couns and sep.
to
Now syppose we take the old-style derivative of et~(pi). I will
leave out the actual s~que~ce, because you have seen enough of them
t~ get the picture; it fs c~m~osed 6t 6's and 7 1 s and . has t~at tYPica1
appearanc~ of quasi•reriodicitY which is so characteristic of eta~
sequences. It is itself ar1 eta-se~ue,ce •• but , to what value of alpha
does tt belong? C1~arly th~s is a key question~
Tho
Fundam~ntal
Theorem of Eta•5equences
--------------------~-·-~--------·-----TMe answer ts given by this simple, central result:
Fundamental Theorem of Et.a•Sequences~ eta'Calpha) = eta(alpha'),
where by eta'(alpha) is meant the old-style der1vative of eta(alpha),
and by alpha' is meant the quantity
·
s ,. alpha
~-------alpha ""' e,
where "s" stands f6r the sep ~f alpha, and "c"
*or
the coun o~ alPha.
In the case of pi, this tells us that eta'(pi) is the et~·sequenee
be1onging to C4•pi)/Cpi•3), which comes out
about 6.0625. Our
knowledge of eta-sequences so far tells us that we should expect the
eta-sequence of 6,0625 to consist m6stly 6f 6ts, with sparsely-spaced
7 1 s, which ls Just what e~a!(ptl looks like~ What ab6ut ~ pr~oi !~r
the above result? The Proof 1s given below~ Once tt is Proven for
val~~s of 31rha between o and 1/2, it follows quickly f~~ all
to
va1ues
of
alpha ·.
Whe~ 0 < alpha < 112, counCalpha) 1s zero, and sep(alpha) fs one,
so eta(alPha) consists of a row of zeros and ones. By referring to
the figure below, you can visualize where seps ("1") occur, and where
c ou r. s ( '1 0 " ) o c c u r • T h c r e a 1 a x i s i s p 1 ot t e d h o r ; z o n t a 1 1 y , · a n d ~ h r e e
integers CN~t,~, ~ +l) are shown. Also, multiples of alpha a~e
indtcated by the letter "a". Cinctdentally, this fjgure shows whY
I somet1rnes Cilll eta-sequences "sidewalk-sequences". If YOU take
steps of length alp ha d~~n a sidewalk whose cracks are 1 unit apart,
the numbe~ of c~acks you cross on the kth step is the ktn member of
eta(~lpha),
providP.d
the zeroth step begins exactly on a crack.)
.
.
•
•
•
I
I
r
I
•
·
·
.
I
!
,
1
•••• ,a •• la .... a ••• a ••• a •• la ••• a ••• a ••• a •• la •• ,a ••••••
I
I
I
N+l
Ste~P~ng
irom one "a" ~6 the next, you can cross either one
tnteger, or none. When you cross one, a "1" appears ln the eta•
sequence; whet" you cross none, a "0" appears. Sooner or 1 ater,
each inte9er •• s~y N •• gets straddled by two successive mY1tip1~s
o~ alpha.
~~en that haPPens, a "1" ~ppears in the eta-sequence7
in fact, 1t must be exactlY the ~th "1". Our goal is to count
the number of zeros until tne next "1". ·
Suppose that the multiples of 81Pha which straddle N are
p alpha and Cp+l)aJpha; and th~t the multiples of alrha which
straddle N+t are q alpha and (q+1)a1Pha. Then
= eta
ata Calpha)
p
s1mflarly,
q
1
bet.ween the pth and the qth are z~~os~
make? Exactly q~p·l~ And this wi1 1 be,
of the derivative of eta(alpha). Now
in terms of~; there are exactly p
which means
a!"d all terms of eta(alpna)
So how many zeros does that
by definition, the Nth term
we can spec1fy both p and q
mYltiples of alpha up to N,
p
=
(alpha)
q
=
=
CN/alphel;
[(N+l)/alphal.
Putting our rfeccs of knowledge together, we k~ow that the Nth term
of eta'Calpha) is eqiJal to q .. p•l; and this fs
CC~~ + l) b e t a l •
~here beta
term
of
= 1/alpha
eta(beta)~
• 1.
CN b e t a l ,
What we have is the ex~~ess1on for the Nth
moreover,
beta
1 - al~ha
- ............
..
0
alpha
alpha'
whfc:h proves our - theorem for 0 < alpha < 112.
Suppose alpha lies between 0 and 112, and
Then one c~n easily show that et~Calpha) and
complementary to each other, in the sense that
one, "l'' occurs In tne other, and vice versa.
their der~vatlves are the same sequence~ That
et.a'Cgamma)
Aut eta'<alpha)
is known,
alpha+ gamma= 1.
et<l(gamma) a,.e
where ~o" occurs in
Con,sequently
i s'
= eta'(alpha)
from above: eta' C-'lPha)
= eta(t/alpha ..
1).
A!"'d
1 .. alpha
1/alpha • 1
= •••••••••
alpha
gamma
=
~-----~-t • gamma
=
gamma'
So
we havt shewn that eta 1 (garnma) = eta(gamrna'), tor any gamma between
112 and 1. The onlY remaining values for which the theorem needs
be prov~n are those of the form "J + alpha, where 0 < alpha < 1;! . It
is tr;v;al to show that
eta( N + alpha) = N + etaCalPha) 1 and from
this 1t follows thet eta 1 (N +alPha) = eta 1 (alpha) ·. It 1s Just
~lgebra te show that
(N + alpha)' =alpha', and this completes the
proof.
to
I now mentiol"' two other results whose Proofs are extremely simple:
Theorem. As ~ approaches ~nfinity, the average
terms of eta(alpha) approaches alpha,
of
the first N
Theorem. If alpha = p/q (a rational number in lowest terms)
then etaCalpha) is a per;odfc sequence, with period q. If
alph~ is ;rratfona1, then eta(alpha) is not Periodic.
Triangles•between~squares
seen in a new light.
Let us now go back to the triangles-between•squ~r~s example.
As I pointed out 1 the operation we performed on the sequence was
not exactlY taking the derivative. The chunks we perceived were
21 and 211; to form the derivative, we should replace 21 by some
integer, a~d 211 by a different integer, Let 1 s do it fn the "old
style":
21 .... > 1
211
2
·->
The sequence and its derivative are
show~
below.
..
21 211 21 21 211 21 211 21 21 211 21 211 21 211 21 21 211 21 211 '
1
211
21
211
21
21
2 l 1
21
2 •••
---- -------
---- ------- -----
The underlining highlights th~ fact that the derivdtfve is al!o
compose<l
21l's and 21's.
It is, however, not identical to any
o~ the seque~ces we have exhibited so far,
But we can take another
derivative, thus gettir.g the second derivative of the original
sequence. It is
cd
2
1
2
1
1
1
2
••••
Now this sequence apPears famt!~ar •• it l6oks like the original
sequence1 It is 1r"'deed the origin<'l sequence. So, for the second
t~me, we have f6und the original sequence coded in Itself.
The
f~rst time, we got it by counting 21's between 211's; the sec6nd
t~me, We got tt by taking the second derivative~
It turns out
that the two Processes are really one and the same process, Presented
in two superficially different ways.
Consider the followfng ide~:
when we took the second der~vat~ve, we broke the ~irst derivative
int6 chunks; now each single term in the first derivative rePresent~d
a ch~nk in the tcP•level sequence, so that to each chunk ln the
ftrst derivatlve, there is a "chunk of chunks" tn the toP•level
sequence, Here it is visually:
21
211-21·21
2--1--1
2
211-21
2---1
1
211-21-21
2---1--1
211-21
2---1
t
2
211-21
2---1
1
211•21•21
2---1--1
211 ... 21
2-- ... 1
2
. ...
~. !.
1 '· • '·
Each number in the bcttom sequence reflects the occurrence in the
t6p sequence of ~ ~second·o~der chunk" (chunk of chunks) ·- e~ther
2112121, or 21121. Now when we counted 21 1 s between 21t's, we were
i n e f f e c t ~ s s i g n i n g '' 2 '' as a name to t he suP e r c h u,., k 2 11 2 1 2 1, and
"1" to the superchunk 21121~
And that is exactly what taking the
second derivative does, too.
the two Processes come down to the
same thing. This points out the important fact that what you get
for your iirst derfvative depends on what you choose ~or "chunks",
in the top•level sequence , If we'd chosen 2112121 and 21121, then
the first derivative would have given vs ~sck the original sequence,
but With 211 and 21 as chunks, YOU have to Walt Untfl the second
der~vat~ve to return the top~level sequence.
So
In eta~9equences, the usual choice f~r a chunk J~st c~ntains a
s~ngle se~, preceded or followed by some couns,
Such chunks are
called qf~rst 6rder chunks". The natural way t6 make "second~6rde~
chunks'' is to make sup~rcl'lunks in the top""1eve1 sequence which reflect
the 1irst""order chunks i~ the ffrst derivative.
Third~order and
htgher•order chunks are deftned analogously. There are always Just
two Nt~~order chunks,
matter what N ls. The higher the ~rder
o~ a chunk, the lo~ger it is bound to be.
As N goes to tnfintty,
the length of both Nth~order chunks goes to infinity. A 17•th ~rder
chunk ts a giant segment of the eta-sequence of alpha, and the
arithmetic ave~aae of its terms Provides a good approxtm~tton t~
alph~.
We will go into this tM m6re detat1 shortly~
no
r
fta•sequences in full generality
-~-------·-~--------~----------Gofng back to the sidewalk•tmage, recall that I pointed out
that the sidew31k-sequonce gives etd(al~ha) •• prov~ded y~u start
on a crack. What harnens if y6u don't start 6n o crack? Supp~se
tM~t the st~rting•point of your ~eroth step is displaced by
a distance delta from a crack~ Then you get someth~ng very much
11ke an eta•sequ~nce. In fact, I call it eta(alpha;delta), What we
have Prevlously called etaCalpha) is the sam~ as eta(alphaJO). The
form~la for the kth term of eta(a1pha;de1ta) 1s easy to derive, and is:
[(k +
1)
delta] •
alpha t
tk alpha+ delta]
P~om now ~~,
by "eta~sequence", I mean somethtng 6f the above f~rm~
Now suppose delta equals minus alPha~ Then we have
eta
(alpha;•alph~)
k
=
((k + l)alpha • alpha] •
=
[I<
tk alpha
~
alpha]
alpha] "" t(k • 1) alpha]
= eta
(alpha70)
k•l
As a matter of fact, this is intuitively obvious: shi1t;ng every steP
t6 the le~t by alpha 6nlY p6stpones arriving ~t a given spot by
exactly one step. In general, shifting every steP to the left by
m times alpha h~s the effect of rostpon1ng the moment of arr;val ~t
a given spot by m steps:
=
eta (alpha;•m alpha)
I<
eta
k,..m
Calpha;O)
to
tMe right or left by one s~uare
know the difference. Th1s ts say~ng that you can
add or ,ubtract any integer to delta and the eta-sequence won't
change, In other w~rds, only the tracti~nal part of delta •• which
ls denoted as "{delta}" --matters. In symbols,
You can move the whole sidewalk
~nd nobody will
eta Ca1Pha;de1ta)
k
= eta
Calpha1(de1ta})
k
There's a generaltzat~on ~f the Fundament~l Theorem of
Eta-Sequences, which holds far ~11 eta-sequences. I state it
w~thout proof, since the proof follows the lines
the earli~r
theorem.
of
r
General~zed
dS
Fundamental The~rem of Eta•Sequences.
eta' (alPha; de1ta) = eta (alpha'; delta'), where alpha' is
before, and delta' can be defined the followfng way:
delta'
I
=
f({alpha},{delta}),
f f {a I.P h a} < 1/2
I
<-alpha}, {•del ta}),
\
f(
I
•y/x
i t 1/2 < {alpha}
where
Hxr y)
=
for x+ Y < 1
1
\
Cl·y)/x
for x+Y > 1.
So that you are not deprived of the experience of seeing an
eta•sequence wh~se delt~ is n6n-zer6, I now exhi~lt eta (sqrt 21 1/2):
21211212121121211212121121211212112121211212112121211212112121211.~,
Do
you recognize t.hi3 sequence? It ;s our ~ld friend, Triangles•
When I discovered this, twas really amazed.
How is it possible to prove this? Actually, tt is qu~te easy~
First, let us derive a formula for Triangles•between•squares,
The nth triangular numher is equal to n(n+l)/2; 1t we invert
this fi.Jnction, we w111 get a function that tells us how many
triangular numbers tMere are up
a given size. In other wo~ds,
between•square~!
to
ts<1rt(2N • 1/4) •1/2)
is the number of t~iangu1ar numbers less than or equal to N. Therefore
we should evaiuate this qu~ntity using the square of k+l as N, and then
the sqyare of k, and then take the difterenca:
2
tsqrt(2(k:+l)
'2
• 1/'0 •1/21 -
CsqrtC2k
- 1/4) - 1/2l
•
I
This expression gives the kth term of trian91es-between-squ~res. It
ean be stmP1f~1ed, wfth the helP ~f the fol1owing identltY (whose
not-wtoo•tricky proof is omitted, sfnee it is not central
eta .. th~~rY) .l
to
cs q r t c2 k
2
- 111n -
112 J
--
Ck sqrt 2 • 1/2l
With it, we get a revised exPression for the ktn term of the
triangles~be~ween•squares
sequence:
t(k+l) sqrt 2
= eta
I
~
1/21 ""
tk sqrt 2 "" l/2l
(sqrt 2;
~1/2)
(sqrt 2;
1/2)
k
I
= eta
k
which is the desired amazing res~lt ~~perhaps less amazing for fts
scrutabilitY. With 1t, we can at last prove the observation that
when you c~unt 21's between 211 1 s in Tr1angles•between•squ~res, you
get the same sequence b~ck. We now know that counting 21 1 s between
211 1 s is the same as taking the second derivativeJ and so the
question amo~nts to whet~er or not eta(sqrt 2; 1/2) equals its own
second derivative. A few man~Pulattons show that alPha' and al~ha 1 t
are both equal to ~qrt 2, and that delta' equals (3 ... sqrt 2)/2, and
de1ta 1 ' equals 1/2; and that is that.,
Now to round out our digcussion on triangular numbers between
we can take a look at what happens when we handle coincidences
of triangles and squares (such as 36) in the other way than before,
This means counting the triangular number as if tt were less than
the square of the same magnitude:
squar~s,
1
~
9
I
1I
I
I
3
6
16
I
110
I
•
•
I
21
151
128
I
I
~
•
o
t
I
45 I
361
1
f
•
I
78 I
I
55 166
I
•
I
•
•
100 '
I
I
I
8l
64
I
I
•
49
36
I
25
I
1
i
1
~
•
•
'
f·
•
•
..
f
'
c1 ) • • • 1 • • • • 1 • • • • • • 2 • • • • • 1 • • • • • • 2 • • • • • • 1 • • • • • • 1 • • • • • • 2 • • • • • • t - • • • • • • • •
We h~ve a new quast•pertodic sequence o* 1 1 s and 2's wh~ch di~fers #r6m
tho nriginal 6ne only tn a ¥ew scatte~ed places~
(Actually, saying
"a few" fs a dfst6rtion; there are in fact infinitely many squares
which coi~cide with triangular numbers, but such coincidences are
~uite sparsely spaced-- or, to Justify my earlier terminology, they
are "few and ~ar between"l) If you count the parenthestzed "1" 1 this
seq u en c e - beg i n s wi t h a t r1 o o _f 1 ' s , · l t • s d ; sp 1aye d m r e t u 11y b~e 16 w, ·
together with iis first and second derivatives, using 12 and 112 as
o
"cht.~nks",
. . '
(1)112121121211212112121211212112121211212112121121212112121121212112.~
2
1 2
1
1 2
1
1 2
1
1 1 2
2
1 2
1
1 1 2
2
1 2
1
1 2
1
1 1 2
2
1 2
1
1 1 2~.~
2
Well, not surprts1ng1y by now, ~t equals ~ts second dertvatlve (but ts
nevertheless dtiferent from the other versi6n o~ tr~angles~between~
squares).
A~ \
Finite segments
of
v
eta•seque~ces
-----------------~--------------
r
I
Supp~se y6u had a wtnd~w slx units wtde, through whfch you could
look at some eta-sequence. What I mean by thls is that you c6uld see
exactly six consecutive terms of that eta·s~quence. How m~ny
different scenes could you be entertained by? Naturally, it ought to
depend on which eta-sequence you hav~ got, so let us try it with
eta (sqrt 2; 0). Here are the poss1b1e views through a 6•unit windowl
112121
121121
121211
121212
211212
212112
212121
Seven, they are seve~~ Wh~t m~y surprise you fs that thfs result h~1ds
whatever the eta-sequence-- there are always seven different v~ews
througn a o•unit window. And there fs nothing sPecial about 61 ;1 you
have a ~indow of width n, there are always n+t distinct views to be
savored. Actuijlly, the cla~m is not qu~te true; ~t requires alphd to
be trrat,onal.
Thus the theorem can be stated formally this way;
Given an eta~sequence belonging to an irrational alpha, there a~e
exactlY nt1 distinct segme~ts of length n. This may s~em 11ke a
remarkably simpie answ~r to a c~mplex combinatoric problem; but
ulthough it ca~ be looked on combinator;ca11y, there is an easy route
to the answer which avoids any combin~tortal analysis~ The pr6o1 is
as toll ow$,
Each of the dist1nct segments of lengt~ n can be produced by
using an aPPropriate val~e of delta, and generating the first n terms
of eta(alpha; delta). Since there are only a fin~te number of
distinct segme~ts, but an uncountable number of delta's between 0
and 1, many delta's yield the same segment, A good 9uess is that
for each distinct segment there is a little interval fnsfde C0 1 1l
where all the delta's produce that segment. This idea is pictured
below, usfnq segments of length 2 and alpha = sqrt 2,
lall delta's!
lhere yield I
I
" 11"
I
al 1 del ta 1 s
here yield
0. 1 71. ·••
0
II
all delta's
here yfeld
12 11
"21"
o.sa1, ·••
1
This ~s exactly the way things work~ The internal demarcatton~ltne~
are determined by the first n multiples of alpha Cas it turns out),
To see tn1s, imagine that the first n multiples of alpha, and zero,
have been marked on a transparent Plastic sheet which we can slide .
above the "sidewalk" defined bY the integers, Set the ~lastic sheet
on the sidewalk
that their zeros coincide, whfch sets delta to zero,
so
~nd g~ves
Until o~e
a segment of length n~ Now slide the sheet to the right,
of the m~r~s on the sheet crosses an intener~l Inc Cc~ack
in the sidewalk), none of the terms in the segment wtll change.
When a mark does eventuallY cross a line, the segment will change~
Remember the kth t~rm of the segment is given by the number of lines .
crossed betwee~ the k•lst and kth marks. Therefore, when the kth mark
crosses a line, the kth term ~n the segment increases by 1, and the
k+lst term decreas~s by 1. CExceptlons: when the zero-ma~k crosses
~line, only the first term of the segment change9, decreasing by 17
and when the rightmost mark crosses a line, only the last (i,e, the nth)
term of tho segment changes, ~ncreasing by 1,) As the sheet c~nt~nues
sliding to the righ~, 6ne by one, the marks w~ll cross tn~eger~lt~es.
No two will do so simultaneously, beca~se alpha is irr3tiona1, Now
eventual Jy, the zero 6n the plastic sheet wi11 reach the ~nteger 1 ~n
the sidewalk. Once that has happened, the whole thing starts over
agafn. But in the meantime, each mark on the sheet will have crossed
exactly or.e integer, (Tn1s must be so, beca~se the sheet has moved
on~ integer unit to the right~)
Since there are n+l marks on the sheet,
there have been n+l distinct segments generated~ That's the pr~of,
and ft corroborates the Piet~re we had of n+l little intervals in CO,ll~
Extraction
---------We are about to wind up the "f~rst phase" ~four dtscussl6n of
eta-sequences: this phase has consisted largely of explorations of
the horizontal aspect of eta•sequenc~s. "Hor~zontal" properttes are
those Which involve a single eta•sequence, and which make little Or no
explicit reference to its derivatives. They are horilontal because,
obviously, a single eta•sequanee is thought 6f as extending out to
tnfin~ty horizontally.
"Vert~cal'' properties are com~ng up soon,
Now I do not mean to imply that ~here is a clean separation between
horizontal and vertical properties; in f~ct they are very tangled up
together and Pr~bably it is a silly distinction-- but the distinction
perhaps can aid one's intuition, as one grows used to it, Whe~ we
eome to vertical properties, I am sure that Y~Y will get a clearer
idea of this distfn.ction~
But now I would like t6 give an example par excellence o~ hori~
zonta1 Properties, a ~r6perty which I call "e~tractlon"~ The idea ~s
this, To beg~n with, write down etaC~lpha,o), Now choose some
arbitrary term ~n it, called the "starting point"~ Begtnning at the
starting po1nt, try to match etaCalpha;O) term by term. Every time
you f~nd a match, circle that term, So6n you wfll come t~ a ~erm
which diffe~s from eta(alpha:O)~ When this happens, Just skip over
~t with~ut circling 1t, and look for the earliest match to the term
of etaCal~ha,ol y6u are seeking~ Continue this Process indetlnttely~
In t~e end ycu nave circled a great number of t~rms _ after the startfng
point, and 1elt s6me uncircled, We are interested tn the unclrcled
terms, whtch are now "extracted" ~rom eta(alpha:O), The ~irst
interesting fact is that the ~xtracted sequence is itse1f an etaw~at 1 s more, It ts the subseque~ce of _ eta(alph~~O)
which beg1ns two terms earlie~ than the starting-point& To decre~se
confusion, I now show an example, where instead of cfrcl fng I underline
the terms which match eta(alpha;O). In this e~ample, alph~ equals
Jog (base 2) of 3.
sequ~ncc;,but
r
have chosen this ~2" as the starting point,
I
I
I
I
..
- - -- - -- . . -- - --- . - - - - -- -- - -
2 1 2 1 2 2 l 2 1 2 2 1 2 l 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2
~
"
~
. . -.
1 2 1 2 1 2 2 1 2 1 2 2 l 2 1 2 2 1 2 l 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
~
~
~
~
The ~nderltned sequence matches the full eta~se~uence, term by
Now what ls the extracted sequence? It is:
2 1 2 1 2 2 1 2 1 2 1
term~
•• • • •
And you will fin~ that thls matches with the sequence which begins
two Places earlier than the starting-point, Carrying it further is
tedious, and does nothing but confirm our observation. ~hy does
this extraction-property hold? At this point, I must adm~t that I
don't know, It is a curioys property which needs further investi•
gation. Fo~ example, who knows what happens if, instead of using
etaCalPha;O) as the sequence to be matched, you use eta(alpha;delta)?
"A1tsum"
-------~
As a tran~iticn Into vertical properties, I cite one last problem
which gives the "appearance of being a "horizontal" pr6blem, but whose
~nswer turns out to be very fnt~~ate1Y related to vertical properties~
This fs the question 6f "altsum" •• an abbreviation f~r "alternating
sum", The de~initton ts:
a1tsum
(k)
= eta
' eta
1
+ eta
2
~
3
•••• + C•l)
k+l
eta
k
Cin the above, afp~a and delta are assumed kn~w~ and fixed,) N~w 1i
it makes sense to guess that the terms of such
~1Pha is ~rrat1onal,
a sum tend to cancel each other cut, more or less, over a long span~
ri
Therefore, the expected approximate behavior of alts~m is that it
will hover near zero, stray1ng away occas1cnal1y, !omewhat like the
''random walk" of a drunk away from n;s lamppost. 13ut one exPects
the fluctuations to be small, in the following sense:
altst,Jm
11m
(k)
--·~-----·
k
= o.
This follows from a general theorem which one can assert about
the eta~seouence of any irrdtional nu~ber:
Theor~m:
The ar}thmetlc averaqe 6f terms ~n etaSJlp~a' delta)
whose subscripts form an arithmetic progression ;g alpha.
The proof of the t~eorem depends on a famous property of the
multiples of any irrational -·namely, that thP.y are uniformly
d'strfbuted~ modulo 1.
This means that out of the first N
multiples of alpha, the proportion whose fractional parts 11e
in any interval inside ro,ll is ~symptotfeally equal to the length
of that interval. Now from the uniform d;stribution of the numbers
(n alpha}, an immediate corollary fs the uniform distribution 0~
the numbers {n alpha+ delta}, whatever delta is. We can now apply
this fact to Prove the theorem, First we make the remark that the~e
is a dividing•ltne tnside the interval [O,ll such that the positi6n
of <n alp~a} w1th resp€ct to that line determines whether the nth
term
t~e eta•squence is a coun or a sep,
(This just says for
segments of length 1 what was said a couple of pages earl;er, for
segment9 of arbitrary length~) It fs illustrated below~
ot
I
I
I
0 •
"c:cun country"
''septal i a"
. . . ' , . . . ' . ' . . ... '
Suppose the arithmetic progress~on of subscripts is qk+p, with k
varying. Then what matters 1n the above diagram is where the
number {(~k+p)alPha +delta) ~a11s, Numbers of this form aror.
however, also of the form {k beta + gamma) where beta = ~ alpha,
and gamma= p alPha+ delta, Now beta is irrational whenever alpha
~s~ which allows us to say that the multiples of beta, shifted by
gamma, are uniformly distribU.ted in tO,tl. Therefore, the proportion
o~ sU.ch numbers which land tn "septalia" is asymptotically equal to
the length of septalia. But the Proportion of nU.mbers of the form
{k alpha+ delta) f~ septalia fs also the length of sep~~lfa -- whloh
~eans ~hat the proportion of seps in the subsequence defined by the
a~ithmetic progression qk+p is ~aymptotlca11y the same as fn the
etawsequence ~ts~lf, COf course the propo~ti~n 6f couns ~s the
same t6o.) Consequently, the average of the subse~uence must be
equal to the average of the se~uence itself.
FinallY we ca~ prove that altsum(k) becomes neglfgible in
comparison to k~ . To get altsumCk), you add up k/2 terms of eta whose
subs c r i r; t s are o <.J d, t hen y e u ·sub t r ~ c t f rom t hat t he sum of k /2 terms
of eta whose subscripts are even. From the just-proven theorem, ·
both the subsequence whose subscripts arP. odd, and the subse~uence
whose suhscrfpts are even, h~ve average alpha~ There~ore both
SYm3 will be of magnitude tk/2) alpha, With correction terms whtch
necessari1Y become sma1l compared to k, ask approaches infinity~
When the even sum is subtracted from the odd, all that ls left ts a
quantitY wh1ch is small compared to~
so the lfmitfng value of
a1tsumCk)/k is Z9ro, as we s~t nut to prove,
••
how doe~ dltsum act, in more det~il? When does it have
fluctuations? Below are the first 100 terms of the altsum
be1onging to eta Csqrt 2; O), so that you can see tor yourself.
la~gP.
Bu~
•2 .. t -2
1 ... 1
0
0 •l
0 •2 -1 -3 .. 2
•2 •3 .. 1 -2
1
. :, •3
0
w2 -1 '""3 -2 ... 3 -t ·2
... 4 ·2 •3 ·1 .. 2 -1 -3 -2 -4 ""3 •4
0 ·2 -1 •3
0
""1
1
•2 .. 1 ... 3 .. 2
"'I~
0
1 -1
0
0 -1
0 .. 2 ... l •2
0
0 -1
·2 .. 3 ·1 ""2
1 ·1
0 .. 2
0 •1
1
1 ... t
0
.,1 ·3 •2 "'3 -1 ... 2
0
""1
.. s ·4 ·5
·3 .. q ·2 .. 3 •1 .. 2 -1 ... 3 •2 "'14 ""3
Aside f~om the ~nlt~al term oi lr all the terms are n~n-postt~ve.
And s~ccess1ve minima ar~ reached at the 2nd, 4th, 16th, 28th, and
~8th terms,
Wh~t are these numbers, and how do they conttn~e?
As a matter of filet, they c~ntinue as follows:
~~,
2,
.
~
l6, 2R, 98, 168, 576 1 984, 3362 1 5740 1 • • • •
You will notic~ that the differenc~s between elements occur twice:
2
12
12
70
70
408
408
2378
~
.
r
2378 • • • •
Thts ~s another cur~o~s effect, whose explanati6n will ~e slightlY
postPoned, untfl we have bl,dl; , UP a repertoire of "vertical" concepts,
in terms ~f which an expl~nation becomes very natural.
Vertlcal c~ncepts are those related to the resylts oi re~e~ted
differentiation of an eta ... sequence, The first vertical eoncepts we
describe are the "Vertical Coun and Sep~sequences" CVC and VS
sequences)., The nth term of VC is ~he coun of .t he nth derivative
of etaCalPha). The andlogous definition goes for vs. Tl'\e sq~are
root of two Provides us with an easy (but needless to say, atyp~cal)
e•amp1e:
n=O
n=l
n=2
r
I
vc
vs
1
2
1
1
1
n=3
D
2
2
2
etc.
Tne~e ls no reason that we should expect s~ch simPle beh~vior in the
VC ~nd VS sequence~ b~l6ng1ng
randomly chosen values o~ alpha,
In fact, the VC and VS of Pi exuQe an utterly d~fferent aroma:
to
n=O
n=2
VC
vs
3
6
4
7
n=3
n=4
n=5
~and
so
abovel
on" •• It you can find any rhyme or reason to the sequences
On th~ other h~nd, e has very regul~r VC and VS sequences:
vc
vs
3
3
1
4
1
6
1
8
2
2
2
3
2
5
2
7
VS·s~qyence.
What w6u1d you say ts its pattern? Ii Y~u
are slightly nafv~ fn number tMeory, you might guess, optlmistieally,
that the Patte~n of the VS•sequence is: Prfmes alternating with 2 1 s~
However, that would be too SPectacular, Natu~e nev~r hands YOU the
primes on a Platter, The actual Pattern is more humble, but the very
faet of a Pattern being there at al1 is remarkable, when you think of
p~~
It consists of the successlve odd numbers Cbut w~th "2" replaclng
"1"), alt~rnattng wtth 2 1 s, And the VC•sequence, a~ter a shaky ~tart,
consists of an alternation between the even numbers and t's~ -
Look at the
If YOU Mave ever ,een the simple continued fraction for e, yoY
may have noticed a similarity~ Here it fs:
e
=2
+ 1
1 t 1
...
r
2 + 1
1
t
..1
1 + 1
~
+ 1
"'1
+ 1
..
1
+
1
-+
6
•
•
•
The denominators go: 2, 1, 4, 1, 1, 6, 1, 1, a, 1, 1, 10, , ••••
The similaritY is so st~fking th~t one wonders if these aren't two
representat1o~s of one th{ng,
Eta-sequences and Continued Fractions
There
--------·-----·-----·----·------·----
is
a defin;te relation between eta•sequences and continued
ShOW it, let us go back to the definition
t~e
Recall that I spoke of two "styles" of assigning names
to chunks, the old style and the new. In the old StYle, a chunk like
" 2 11 " wo u l d get n am <e d '' 2" be r. au s e o f t he t wo 1 1 s , b 1.1 t f n t he new s t y 1 e
i t g e t s n am e d " 3 '' be c au s e t ha t i s 1 t s 1eng t h , l. e t u s c on s i de r wh a t
h a P P ~ n s 1 f ~ ~ t a k e de r i v <1 t i v e s i ,., t h e new s ~ y 1e
Eye r y , t e .r m f t h e
derivattve fs increased by 1' hence the derivat1ve is no longer the
eta•9equ~nce belonging to a1pha 1 , but
l+alpha', We could make
matters con~using, by calling 1+~1pha 1 the "new-style" alpha' •• ~but
instead of that we wtll write "DCalphal" for ~t~
1~~ctions.
derivativ~,
To
of
t
o
to
s • alpha
DCalpha)
= •-·---···
alPha • e
e and s being the coun and sep
Nat~rally,
X is a function
Then we ean write
of
of
alpha, as before,
~et US PUt
alpha, aMd e~uals plus or minus one~
X
D(a1Pha) =
+ 1
-~·-·····
alpha ,. e
Inverting this
equatio~,
alpha
r
=
we get
x(~lpha)
c(alpha)
t
----~---
DC alpha)
Here, alpha is expressed as an ~nt~ger Plus Cor mtnus) a
nymerator 1~ The trick is to express DCalpha) likewise:
D(alpha)
=
fr~cti~n
wlth
x(D(alpha))
c(D(alpha))
t
-----------
DCDCalpha))
This trick can be repeated for DCDCalph~)), then DCDCOC~lpha))), etc ••
Each t ·ime t~e trick ts done, ~t corresp~nds t~ 6ne more level o~
d~ffere"ttation of the eta"sequence~
So we are looking at the vert1ca1
strYctyre of an eta~sequence this way, Now the whole thing can be
summed up i, o,e grand continued fraction; but before we write that
down let us make some changes in convention. I Just lntr~~uced the
VC and VS sequences and s~, presumably, you are not s6 used to them
that YOU Wf1 1 vigorously protest if I change my def~n~t~on of them,~,
All . I pr6~ose ts that everYthfng sh6uld be as before, except that ~tl
dertvatlves should be taken acc~rding to the new style, That has
onlY 6ne ef~~ct: it raises ~ach term 6t VC and VS Cexcept ~~e z~roth)
by one, The~efore, our revised VC and VS for the ~quare root of two
go:
vs
vx
1
2
tl
2
2
2
etc.'
3
3
3
tl
vc
n=O
n=l
n=2
n=3
tt
+1
I have put }n an extra col~mn for the X-sequence. The nth element of
VC Cor VS) fs now equal t~ the c6un (6r sep) ~f DCDC,~,O(al~ha)~,.)),
where the number of D's is n,
N6w we can wrtte down the continued fraeti~n for alPha, ust"g
the redefined vc~sequence, and the vx~se~uenee:
alpha :
VCCO) + VX(O)
----~
VCCl) + VX(l)
VCC2) +
•
t
•
•
Here, ''~C(n)~ stands ~~r the nth element of the VC belonging t6 alpha,
6f
course~
For instance,
r
sqrt 2 =
\\
~ e..
+ 1
-2 +
))
.1
2 + 1
"'
2 +
•
t
•
Now I must hasten to mention that this k1nd of continued fraction
ts slightly ditf,rent from the most commonly exhibited kind, the
difference being that the ~sual ones only have +1 1 s in their numerators,
never •lls (which wf11 happen to our fractions whenever VXCn) = .. 1)~
The. usual type of continued . fraction is . called "s~mple"1 our tractions
also have a name: "nearest•integer continued fractions" (this is the
best trans1~tton I can giye for the German phrase "Kettenbrue~he nach ~
naechsten Ganzen", which is what Oskar Perron calls them ~n hfs class1c
work on continued fractions, "Die Lehre von den Kettenbruechen", whlch
is, unfortunately, out of Print).
Many of the theorems which hold for simple c~ntfnued fractlons
carrY over to nearest~integer continued fractions, f~r instance, a
famous theorem
simple contfnued fractions asserts that the $equence
of denominators in the contlnued fraction for a real number will be
pertodic if and only if that number fs a quadratic trrationaltty.
(Here, the ~ord "Periodic" means t hat after a while, t~e sequence
repeats over and ove~ agaf~; but the block which is rep~ated need not
st~rt with the v~ry first term~) A slight modificat~on of this
statement holds tor VC and VX sequences, namoly~
on
If the VC and VX s~quences belonging to alpha are both periodic
(in the above se~se), then alpha is a q~adratie irratfona1ity7
and conversely, tt alPha fs a quadratic ~rra~ionallty, then its
VC and VX seque~ces are necessarilY periodic.
This the6rem can be equtva1~nt1y restated, us~ng "VS" fn place ~f "VX".
A co~ollary is this th~orem:
Al~ha ls a quadrattc 1rrat~onalfty if and only f~ there are
unequal fntegars m and ~ sueh that the mth and nth derivatives
of eta CalPha' o) are i-dentical.
~.-~
The funct1on INT
--------------- ..
l
There fs no criterion by which one can d1stinguish a VC·s~quence
from a VS~3equence. For i~stance, the vc~sequence of p~ could very
easily be the _ vs~sequence of some other number (in fact ft ls the
VS•sequence of many other numb!rsll~ Becau5e of the indistfngutsn~
ab;lity of vc and vs-sequences, one is ndtura11y led to ask, what ff
t interchange the VC and vs~sequ~nces of alph~ -- what number has for
its VC•sequence the YS•sequence of alpha, and for its VS~sequence the
VCwsequence o~ alPha? Wh~t number beta has the following VC ~nd VS?
vs
vc
n=O
n=l
n=2
n=3
n:=4
•
2
3
3
3
•
•
•
1
2
2
2
;3
2
.•
•
•
•
•
i s , wi 1 1 b e c a 1 1e d " I NT ( 91 r t 2 ) '' , b e c a u s e
its VC and VS are those of sqrt 2, tnterchanged~ Generally, ~or
beta to be equal to INTCalpha) means that for all non-negative n,
Th 1s n l.J mb e r , wh a t e v 9 r I t
VS (beta) : VC (alpha),
n
n
VC (beta) : VS C~lpha)~
n
n
Dbvto~.Jsly,
It so
,,
happ~~s
beta= INTCalpha), then
that
I~TCsqrt 2)
whe r e '' ph i 11 s t an rj s f o "" t he ''
go l den
Ph1 : (1 +
alPha= INTCbeta), as well,
= P~~,
rat f
s~rt
o" ,
S)/2,
CThe proo~ ~f this comes from the fact th~t DCphil ; 1 + ~hf.l
.
One ca~ ea$i Jy see that INT of any q~adratfe must be another quadratic,
s~mp1y because the new VC jnd VS sequences are ~eri6d~c~
You may hav!
noticed that INTCalpha) is not yet well-defined for rational values of
alpha~ This ~s because the VS~sequence for any ratfonal n~.Jmber . hits
~ snag after a finfte number of steps ~~ in fact, on the mth stePr
where m fs defined by:
·
DCDC.~.D(al~ha)~.~))
\
I
\
I
m D's
= an
1nte9er
·.
( 2 /7 :
~
N~ttce that s~ch an m is guaranteed t6 exfst,
f6r only trrati~n~ls have
infinitely ~ifterentiable e~a-~e~uences 4 On the mth step, the~, VC
is well-defined . (being the in~eger itsol f), hut vs fs not, ~f~ce the
second•closest i~teg~r to an integer is not well•deffned: there ~re
two vying . cand1dates. The most e~thettcally pleasing solution, perhaps,
is to define I NT at a rational value as being the arithmetic average
of two values, 6ne calcul~ted by taking the mth vs to be on~ less than
the mth vc, the other by taking the mth vs to be one gr~ater than the
mth VC, ~nder this defin~tfo~ 6f INT at ratt~nals, it ~s easy to see
that tNi of a rational is also a rational, So we may now say:
It alpha is algebraic of
degree
0 Can integer), then
I NTCalpha) is likewise algebraic
o~
degree
Q~
If alpha is algebratc o~ degree 1 Ca r~ttonal n~mber), then
alg~bra~c
INTCalPha) is likewise
oi degree
1~
If alPhe is algebraic of degree 2 (a quadratic), then
I NT(alPha) is likewise algebraic of degree 2_
The pattern is suggestive, ls it not? ~owever, I ~m not sure if the
extension of this pattern is valid
not, A m~st interesting question~
Incidentally, ;f ft were valid, then one would nave as a corollary
the following statement:
or
If
alp~a
1s transcendental ~not algebraic of any de9ree) 1
then INTCalpha) is likewise transcendentAl~
Certainly it is provocative t6 ask ~hat mathematical sfgniffcance
a constant such as INTCel or INTCpi) has. I have not been able t6
f~nd any for either.
Their values are, roughly:
INT(e)
= 2 ••• ~ ••• ~.~~
I NTCpi)
= 3,86~.~.~.~.: ••
Now before going any further fn the descripti6n of INT, it is vital t~
All the information ab~ut INT is c~ntained fn
a plot where alpha runs only ~rom · 0 to L, To get the value oi INT ~or
any other value of alpha, subtract the integer Part of alpMa, consult
the graph between zero and 6ne, and then add back the integer part:
exhibi~ a plot of it,
I NTCalpha)
=
INT
Calpha~N)
+ N
of
This means that the graph of INT consists of infinitely many copies
the c~nte~ts of a single "b~x", touching each 6the~ at t~eir c6rnersf
as snow~ on , the ~ext page. What rappe~s inside each - of the boxes is
then shown on the page after that,
. The most . striking fa<:t, at ffrst gla~c:e anyway, ts , how the . graph
each bcx ~·henceforth called a "box~graph" -~consists of ~cads
of little "subgraphs~~ All the subgrap~s seem to resemble each 6ther,
and Gre aligned more 6r less ~arallel to each 6ther, t~e onlY d~ffe~ence
betn~ that as they recede into the corners ~f the b6x, they 9et smaller,
and smaller, and smaller •• ~ But the next level of observation brfngs a
yet greater surprise: all the little subgraphs themselves seem als6
be composed of subgraphs cf their own, And, to the extent that the
~ns1de
to
t
t
t
I
beta
•
,•
............ 4 ..
~-
I
I 3
beta
= INTCalpha)
-----·----1
I
.•
•
2 ,.
I 2
----------1
I
.•
I 1
•
1--~------ 1
I
I 0
;
1
1
•
•
•
r
•
•
•••••••
•••••••••••••••••••••••••• -·
•5
•3
•4
w2
~1
0
..
I
I .
.
,
•
~
•
,
,_
alpha
l
t
•
1
2
3
4
5
1·1
.. ---------1-.1
I
I
'
~
1... 2
-----.. ---·1
• 2
·....
1·3
---------1
·'
.•
•
The graph
of
INT fs confined to thfs d;agonal ~hain of
of boxes, whose edges are defined bY tht fntegers on
the horizontal and vertical axes, The integer lf-, each
box
is
equal to both talphal and tiNTCalpl'la)l~ What
looks like fs shown on the
the g~aph inside any
next page.
box
,.
~
• l • • • • • ~.' • • • • • • • • • • • • • • • • • •
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.· ..
graph allows you tc peer into the level beyond that, the same thing
seems to be happening. It never comes to an end. It may seem mind"
boggling at , first, but after it has settled, it makes a little sense,
because it is reminiscent of the w~y that an eta~sequence yields
~nother eta-sequence, on being differentt~ted, which yields anoth~r
one, and so on. Of co~rse, infinite differentiability r~quires alphq
to be irrat1ona1 -- but that brings us back to INT, whicn seemed to
be more natural to de~ine ~t ~rrationals, anyway,
And then the idea of INT hits you: the 11ttle subgraphs are
6f the box•graph, That Poses a question, howev~r: ~Haw can
the subgraphs be copies of the box-graph when the box~graph is stralght,
and the subgraphs are all curved?n The answer ~s that they are copies
in an extended sense o1 the word •• they are not only of a different
size than the original, but they are also a little distorted, The
dtstortfo~ fs not chaotic or random, though, but ~uite ne~t and
systematic.
~coptesn
11 YOU look back at all the boxes touch;ng each other along~
dlagonal, you will see that the large"scale strycture of INTis
Just 1fke the structure in each box: many repeated parallel copies of
one item. The total graph contains identical copies, and fs therefore
0~ in~tntte extenstonJ a bo~-graph, on the other hand, Jnvolves the
s~ueezing o1 an tnftntte number ~f coPies into a +inite space, and
therefore causes shrinking to occur, near the corners~
You can
probably imagine a giant with infinite reach pfckfng up the whole
INT~graph, rotating it 90 degrees, and then compressing it so that ~t
will fit inside a txt box (and in the process, slightly distort~n9 the
pfeces composing it), Such a squeezing•process, if done right, would
transform the total graph of INT into one single box~graphl A proof
of this wculd establish~,, the earlier speculations about the nested
structure of each individual box-graph, We will prove t~at such a
g~ant exists, in the +ol low~ng sense: we will provid~ a monotone mapp~ng
which compresses boxes 2 through infinity down fnto the upper left half
of a box-graph, (The lower right can be taken care of by symmetrY,)
Proof of the Nestfng•Property of INT
·---------·--~---------------------Earlier, I showed a way that INTCalpha) can be calculated from
just one box-graph: shift alpha into the relevant region, use th~
b~x•graph, then un•sh~ft the result~
This has the general form~
INTCalpha)
:
g(INTCfCalpha))
where f is a function that shifts any alpha into the relevant region
of the x•axis, a~d g is a function that 1hifts valves of INT obtatned
lrom the bo~·graph back tnto the c6rrect part o~ the y•ax~s~ . When we
d~d this be~ore, fCX) was X•N and g(y) was y+N; they were inverse
functions. The resulting e~uat~on told us that one b~x~gra~h 1~6ks
exactly the same as any other box•graPht because the funettons f and g
are simple translations: they do not shrink or expand, they merely
shi~t.
To prove our nesttng~~roperty, We wtlt need an e~uat1on of ~he
above form, but where f is a "shr~nking~functton", one w~leh carr~es
values of alPha betw~en 2 and infinitY into values lYlng between
0 and 112~ and where g is an expandlng•function, which carrfes the
small interval [0, 1/2] baek onto the half•lfne from 2 to inftnity,
Fortunately, such an equatton ~s not hard to come by; tn fact
practically falls in our laps, one~ we know we are looki"g for itl
It all comes from looking
the VC and VS•sequences of alpha and
INTCalpha). Let us exhibit them. First, the table for ~lpha~
~t
b
at
vc .. seq
vs .. seq
-----A(O)
n=o
n=l
n=2
.•
•
AC1)
AC2)
I
•
----~-
alpha
DC alpha)
DCD(alpha))
.
.
•
.•
B(O)
BC1)
BC2)
•
•
•
•
•
"
{.')The reason I have included the numbers alpha, DCalpha),.~, in the
~dle is the following~ The VC and vs .. sequences which begin at any
given level and continue downwards are vc and vs~sequenees ~n thefr own
r~ght, wh~ch b~long to the n~mber written at the correspo"ding level
~n the middle.
For instance, the VC and VS•sequences belonging to ihe
17th derivative of alpha are, respectively, ACt7), AC18lt A(19) 1 ,~~ and
AC17), nC18), 8(19),.,.
.
Now
dnalogous
INTCal~ha) -~ let us
t~ble:
call It "beta"-~ has the follow~ng
vs .. seq
vc-s.,q
-----n=O
n=l
n=2
B(O)
B( 1)
'..
•
B(2)
~----~
DCD ,Cbeta))
,.
•
A (1)
AC2)
•"
,•
•
•
ACOl
beta
DCb~ta)
•
I
Now SYPPOSe we chop off the top level ~n t~e table for alpna '
The remaining saqu~nces 8re A~1), A(2),~ •• and 6(1), 6(2),,~~, whlch
are the VC and VS•sequences of DCa1~ha)~ If we now interchange, we .
have the VC and VS~se~uences of INTCD<alpha)) -~ by definltlon ~f INT~
~ut they are Just the same as the VC and VS~se~uences of D(beta), as .
you ean see by looking at the taPle for beta, Therefore, tn calculatin~
D(INTCalpha)), we can go either of two routesz first get INT(alpha),
then t~ke D oi ~t,
else take D ftrst, and then get INTo~ that~
Symbolically,
·
or
INT(D(alphall
= DCINT(a1phall 0 ~ ·
·rnis is close to, but not exactly, what we want~ ~et us write
gamma= D(alpha), Then what js alPha 1n terms of gamma? Prom the
dtscyssion on continued fr~ctions, we have
.
s ,. e
alpha
= c + •--~~
gamma
c__:_~·>'
Now, it gamma alone is gi~en, then ~1pha st11 1 has not been determined~
we have the freedom to choose the coun and sep of alpha as we like~
Let us nenote the value of this express1on, with c=a and s:b, by
•1
D
(gamma)
a,b
(Of course, a and b must differ by one.)
the earlier equation, we get
! NT(gamma)
:
DCI NTCD
•1
If we substitute this into
(g~mma))
a,b
Lo and behold, we have some candidates for the shrinkfn~ and expandlng
functtons& For notational simplicity, let us use "X" tor the argument
o~ the shrink~ng•functi~n D~tnverse, and "x" ~o~ ~ts value7 lfkew~se,
~y" for the argument of the expanding~tunetton D, and "Y" for its value :
Let us take a=o, and b=l in the shrink1n9•function:
"'1
)(
:=..
D
=
(X)
0,1
1/X
N6w if we let X vary ir~m 2 t~ ~n1fnity, the shr1nklng•iunct1onis
ran9e . wi11 be the , interval tO, 1121. Suppose we see ~hat haPPens when
X varies ~nside box number 2, The values of X are shifted lnto
the range tl/2, l/3J by the shrfnk1ng..,functfon' thel'l INT Provides y's
between 1/2 and 2131 finally, these y's ~~e transformed by the outer
D•functt~n fnto Y1 s between 2 and 3, ~ittin9i as e~Pected, ~nto the
original box. Th~s we see very directly how a particular b~x is repre~
sented by a particular ~ubgra~h~ A stmllar argument holds ~or any ·
other box to the right of the one Just considered~ The box betwe~n
N and N+1 is mapred onto a subgre~h located between 1/N and 1/Nt(~
As P r ~ m1,s e d , ~ h i s s ho ~ s how .t h e . s !.1 b g r a ph s ,f ~ o ~ 1 ( or a;, y t he r b o ¥)
are "copies" of the box•graphs from 2 out to infinity, Since each
box•graph is symmetric with respect
both
tt, dfag~~als, the Pr~of
~or the lower rfght half ts implicit ~n wh~t we have done~
o
to
o
of
An additional fact about the lfttle copfes, which ls furnished to
us by the sh1ftfng~equation, is how a box~graph must be compressed
ho~tzontallY and vert~c~lly in ~rder to be br~ught to ~ofnc~d~ with
a given suboraph. Consider once again the mapPfni betw~en box ? and
the subgraph between l/2 and 1/3. The shrfnkil'lg~ful'lctfo~ f(X) fs
X
=
a 1'1 d t he expand i n g• fun c t f on g Cy ) f ·s
1/X
r
Remember t~at 1 maps from the box to the subgraph 1 while 9 maps
from the subgraph back to the box. Consider two nearby points on the
X•axts, and what f does to them~ If they are X and X+dX Cw~th dX
infin;teslma1) then f c~rrfes them ~nt6 fCXl ~nd j(X+dXl respeetive1YJ
·and the latter, by Taylor's theorem, equals f(X) t dX f'CX), Tf'le
~eparatlon between the tmage~~olnts, therefore~ ts muttlplted by the
factor f'CX), This is called tbe "local e6mpressi6n '~ctor~ (and
nottee lt ts wrltten a~ a functton of the variable Bfg~X •• th~
variable belonging to the bo~·graph, not little~x, of the subgrap~),
'The local expansf~n•factor ~ue t~ g is 9 1 (y). •• but we~~~ tn~erested
fn compress~on 1 not exPansion, so we must take the reciprocal, ·
Sec9ndly, w~ wa~t to wr;te ft i~ terms
B'g Y, the bo~89raph
variable, not 1itt1e y, Thls gfves us:
"
•
'
'
•
I
of
fI
~X)
·;;
·- 2
·•1/X
'
l,.ocal vertical compression f'ctor ;;:
- ., 2
1/g'~y)
=
~1·y~
•
·2
1/Y
Notice that these ~omPre~sion f~ctors both .vary betw~e" 1/~ ~nd
1/9 when X and Y range ever box .27 bu ,t "(hen ~ . ~nd Y Y~ry over bo~
say, tnen Ca~ there 1s much _greater .compression, ·and (b) h ts m~.o~c~ .
c 1o s e r t o yn { to r m, s 1nc e b o t h 1ac t r s r' ma i n ·a l mos t con s t an t I : v a r Y~· "g
·botweon 1/400 and 1/~1.11~ Tn~o~s, we obtaln en ·answer
how tht .. . ·
c~.o~rvature . of the aubgraphs comes abou~~ an~ secondly, w~ learn why ·
the ~ubgraphs closest
the c~rners of ~ny box .are so much .less . .
: ~yrved tr~n tM• ones i.n the center, so m~eh .more · f~'thfu~ , ~a · "~oPl~s"
- o~ the box•grapl'le th~mselv~s, .
..
. ·· - ..
. .·
ao,
o
'to
to
/
---·t-;
.......
-: -J-
\
.· /
:-