Patterns in Nature - the Scientia Review

Patterns in Nature
Paul Harrington and Juliana Kew
Contents
Ratios in the Human Body………………………………..3
The Vitruvian Man………………………………………4
The Fibonacci Sequence………………………………….5
Fibonacci Sequence in Bees……………………………….6
Fibonacci Sequence in Rabbits……………………………8
Fibonacci Sequence in Flowers……………………………9
Spirals in Nature………………………………………..10
The Fibonacci Spiral……………………………………11
Fractals…………………………………………………12
Fractals in Bacteria Colonies……………………………..13
Fractals in Vegetables……………………………………14
Fractal Electric Discharge………………………………..15
Simple Natural Fractals…………………………………16
Tessellations……………………………………………17
Tessellations in Honeycomb……………………………..18
Tessellations in Rock……………………………………19
Glossary………………………………………………..20
Image Credits…………………………………………...21
About the Authors……………………………………...22
Ratios in the Human Body
Cartoons
Small children have
disproportionately
large heads, which is
part of what distinguishes them as
cute. A baby’s head
grows as its body
does, but not as
quickly or as much.
Eventually the body
catches up to the
scale of the head.
Some ratios appear
often in the study of human anatomy. Human
beings are highly variable,
and so the ratios that describe the average or ideal
individual may not hold
true for everyone. However, some are often accurate. Cartoon style toys
and drawings take
advantage of disproportionality to
be more appealing.
height : arm span :: 1 : 1
height : femur : : 4 : 1
height : head circumference : : 3 : 1
height : head height : : 8 : 1
foot length : palm width : : 4 : 1
face length : ear length : : 3 : 1
The Vitruvian Man
The ratios of the human body are often observed in art, where regular proportions seem
more natural or beautiful to the viewer. The
most famous study of human proportion is Leonardo DaVinci’s Vitruvian Man. The name
comes from the ancient Roman architect Vitruvius, whose writings on human proportion DaVinci referenced. The drawing shows that the man
standing with feet together and arms wide can be
inscribed in a square, which makes sense if his
height equals his arm span. The second image
of the man, spread-eagled this time, is inscribed
in a circle.
The Fibonacci Sequence
The Fibonacci sequence was discovered by Leonardo of Pisa, who was called Fibonacci (a contraction of filius Bonacci, Latin for “son of Bonacci”).
He introduced his work to western mathematics in
the book Liber Abaci, even though the pattern had
been described by earlier Indian mathematicians.
The pattern begins with two numbers, zero and one.
Other than the first two, any number in the sequence is the sum of the two numbers before it.
Some simple calculations yield the following series:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. This method of
generating the sequence is known as a recursively defined function. The two numbers are defined (F[0]
= 0 and F[1] = 1) and the rest are defined by the
previous numbers (F[n] = F[n-1] + F[n-2]).
Fibonacci Sequence in Bees
Bee birthing follows two simple rules. A male
bee hatches from an unfertilized egg, and a female
bee hatches from a fertilized egg. A single female
can lay an unfertilized egg, but a male and female
can lay a fertilized egg.
While following the bees related to a single
male, we find the Fibonacci sequence. We have to assume that all the parents are not related. First, we
have the single male bee (1). Then, we have its female parent (1). The female has two parents: a male
and a female (2). The male grandparent has a one
parent, and the female has two: one male, two females (3). The females are the offspring of two bees
each, and the male is the offspring of one: two
males, three females (5). Placing the numbers next
to one another, we have 1, 1, 2, 3, and 5, which are
the first five elements of the Fibonacci sequence.
Fibonacci Sequence in Bees
(cont.)
Following the ancestry of a female, a similar
pattern occurs. We start with the single female (1).
It has two parents, a male and a female (2). There
are three grandparents: one male, two females (3).
There are five great grandparents: two males, three
females (5). Placing the numbers next to one another, we have 1, 2, 3, and 5, the second Fibonacci
number to the fifth.
Fibonacci Sequence in Rabbits
Rabbit mating pairs follow a simple pattern.
For the first cycle that they exist, they bear no children. For each cycle after that, they have one pair.
Assuming that rabbits never die, and that the population can grow without bound, the Fibonacci sequence is found in the number of pairs of rabbits.
At first, there is one new pair of rabbits
(1). In the next cycle, they mature (1). Now the mature rabbits have one pair of offspring (2). After another cycle, the mature pair has another set of kids,
and each of the rabbits from the immature pair become adults (3). Now there are two mature pairs
and one immature pair. In the next round, there will
be three mature pairs and two immature pairs (5).
Placing these numbers next to one another, we have
1, 1, 2, 3, and 5, the first five numbers in the Fibonacci sequence.
Fibonacci Sequence in Flowers
The Fibonacci series can be found in flowers. A
good place to look is the number of petals. Unfortunately, the smallest numbers of the sequence
(mainly zero, one, two, and three) are not easily
found. However, the larger numbers, such as five and
up, are fairly commonplace. The columbine flower
has five petals. Bloodroot has eight petals. Black
eyed susan has thirteen petals. Various types of daisies have 13, 21, 34, 55, or even 89 petals. Of
course, saying that a flower has a number of petals
is measure of the mode, or the most common occurrence for that flower. It is possible for a flower to
over develop, or more commonly, under develop. For
example, black eyed susans can have anywhere between eleven and fifteen petals.
Spirals in Nature
Spirals can be found in many
natural formations. Many different
shells have the shape of an Archimedean spiral. Certain galaxies, such as
our own Milky Way, also have the
same form. Deoxyribonucleic acid,
also known as DNA, forms a helical
structure, which is a three dimensional spiral, inside the nucleus of cells.
Helixes can also be seen in goat
horns or climber plant tendrils. Spiral growth refers to an organism’s
growing into a spiral shape. It uses
the maximum amount of area within
a certain radius of the center of the
growth. For plants, that maximizes
the surface area that is exposed to
sunlight.
The Fibonacci Spiral
A Fibonacci spiral is a shape built from the relationships between Fibonacci numbers. Imagine a
square with unit-long sides. To the right place another unit square. Now beneath the two put a third
square whose side length is the sum of the lengths
of the previous two square side lengths. It will have
sides of length 2. Next go to the left and repeat the
process, for a square with side lengths of 3. Then
move above for a 5x5 square. The pattern can continue indefinitely, right, down, left, up, always adding the side lengths of the previous two squares. To
draw the spiral, inscribe a quarter circle in each
square.
Fractals
A fractal is a pattern within a pattern. At the
largest level, it looks like a single shape or a group
of shapes. But when the fractal is expanded, it can
be seen that it is made up of more of the same
shape. And each of those shapes is made of even
smaller ones. This pattern continues forever, so it is
impossible to draw a fractal completely correctly.
However, computers can draw fractals with extreme
accuracy down to a scale smaller than the human eye
can see.
Fractals in Bacteria Colonies
Usually when colonies of bacteria expand,
they form compact shapes, as seen to the left
above. However, when the growing colony is
stressed, when it does not have a soft base or
enough nutrients, it grows into fractal branches.
This allows it to expand quickly without extra
effort. All the branches give it more surface area further away from the center. At the tips of
the branches the bacteria have access to more
nutrients, so the colony can grow larger and
stay healthy.
Fractals in Vegetables
Evolution
Plants tend to follow
the Fibonacci Sequence because it has
useful geometric properties. Spirals are a
good way to pack
many units closely together, and the sequence lends itself
easily to spirals.
Plants that are arranged according to
the Fibonacci Sequence therefore have
an advantage over
those that are not and
over generations are
more successful. The
more successful plants
are the ones that can
survive better and so
become more common.
Romanesco broccoli is an extreme example
of a natural fractal, with a basic shape reminiscent of the Fibonacci Sequence. Its underlying
structure is a simple cone, but overlaid on that
is a spiral of new cones defined by the ratios of
Fibonacci numbers. Each of those cones is itself a spiral of smaller cones, and the pattern
continues down to a minute scale. This selfsimilarity is what defines the whole vegetable as
a natural fractal.
Fractal Electric Discharge
Electric discharge often forms
fractals. When the electricity enters a
medium, it is said to take the path of
least resistance. However, it doesn’t
take just one path. Instead, it travels
along multiple pathways. If its medium is uniform throughout, the electricity takes multiple paths in the
shape of a fork. Fractals form because
the discharge, after forking, reacts to
the medium in the same way, but now
with a lesser charge. So, it forks again,
but each leaf of the fork is smaller
than the original fork. As this continues, the fractal becomes more and
more detailed.
Simple Natural Fractals
Landscapes, when viewed from a large distance,
can be seen as warped fractals. Mountains, for example, are warped examples of the Sierpinski gasket. Each vertex in the fractal has been slightly
moved from its true position, but the method of
generating the fractal remained the same. River erosion patterns can form fractals. As the water runs
downhill, it splits into several smaller flows. Those
flows also split, and so on. Each step of the pattern
is a smaller version of the step before it, which
makes it a fractal. The image below is a landscape
computer-generated using fractals.
Tessellations
Development
Living organisms are
vastly complicated assemblies, with special
structures called tissues to perform different tasks. For instance, bone tissue is
specialized to provide
support and cannot
contract like muscle
tissue can. But both
muscles and bones,
along with all the other structures in an organism, came originally from the same cells.
They diverged and became specialized because during their development different
chemicals were secreted in the fluid around
them. In response, the
cells marshaled themselves into the forms
seen in the mature organism.
A tessellation is a pattern that completely fills a
surface with no gaps or overlaps. Bricks are a simple
type of man-made tessellation, but these patterns
can also be found in nature. Turtle shells are tessellations; they have hexagons or other similar shapes
instead of rectangles, but the tiles cover the whole
surface of the shell. The individual tiles are actually
outgrowths of the ribs and vertebrae of the turtle,
which is why the reptiles cannot leave their shells. It
also means that the tiles must butt firmly against
each other to lend the shell strength. Without such
tessellation, the shell would collapse into the component parts and the turtle would be left defenseless.
Tessellations in Honeycomb
Honeycomb is a natural tessellation and a very
efficient arrangement for storage. Bees build honeycomb with wax, which they produce at a rate of
about 1 pound for every 8.4 pounds of honey consumed. With so little wax, it is important that the
cells of the honeycomb use as little of it as possible
to cover the area of the hive. This is where the hexagons of the comb are useful; hexagons cover the hive
with less material than quadrilaterals or triangles.
Those three shapes (hexagons, quadrilaterals, and
triangles) are the only ones that can completely cover a plane by themselves with no overlap. And hexagons contain the most area with the least perimeter,
so they are what bees use to store their honey and
larvae.
Tessellations in Rock
Large areas of rocks can sometimes form tessellations. When lava flows out of a volcano, it covers the ground with a continuous sheet of molten
rock. The rock starts to stick to anything underneath it, so it is anchored to the ground as well as to
itself. However, it then begins to cool. Like most
materials, rock contracts when it cools. But in this
case, it cannot just shrink because it is stuck to the
ground. Instead, it cracks into hexagonal segments,
like the ones seen in the Giant’s Causeway in Northern Ireland. Other kinds of rock can crack into tessellations, too, even if they have not come from lava.
An example is the Tessellated Pavement in Tanzania.
Glossary
Ancestry
Colony
Contract
Cycle
Discharge
Disproportionately
Diverge
Femur
Inscribe
Medium
Petal
Proportion
Quadrilateral
Ratio
Recursive
Secrete
Sierpinski Gasket
a series of parents and older generations
visible cluster of bacteria , theoretically
descended from a single cell.
decrease in size
any complete event that repeats
to rid of an electric charge
in a ratio different from normal
develop in a different direction
bone of the thigh
draw a figure within another so that their
boundaries touch but do not intersect
the substance on which something occurs
one of the colored segments that
surrounds the head of a flower
The comparative measurements or size
of different parts of a whole
4-sided shape
proportion or fraction
a description of a cycle that depends on
previous cycles
produce and discharge
fractal formed by dividing a triangle into
4 equal segments
Image Credits
Cover: http://www.funnyphotos.net.au/fractal/
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http://www.miqel.com/images_1/fractal_math_patterns/natural-patterns/vonkarman_clouds_1.jpg
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Ratios in the Human Body: http://img107.imageshack.us/img107/8147/girl2prefectmw3.jpg
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Fibonacci Sequence in Rabbits: http://www.landlearn.net.au/newsletter/2008term3/images/rabbit-family-tree.png
Fibonacci Sequence in Flowers: http://flowerinfo.org/wp-content/gallery/columbine-flowers/columbine-flower-3.jpg
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About the Authors: http://www.couchcampus.com/television/tv-talk/top-twenty-tv-series-on-hulu/
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About the Authors
Paul Harrington is a junior at the Massachusetts Academy of Math and Science.
He lives in Peabody, MA. Paul can often
be found playing Super Smash Brothers
Brawl as his favorite character, Metaknight.
Also, Paul is a fan of Magic: The Gathering, which he plays almost as often as
SSBB.
Julie Kew is a high school student at the
Mass Academy of Math and Science.
She enjoys languages, mythology, and
connecting everything possible to astronautics.