NATURE'S NUMBERS
Please watch
"All
watched over by machines of loving grace:The use and abuse of
vegetational concepts"
Mathematics is perhaps not the first subject that springs to mind when thinking
of nature,but many modern researchers have endeavoured to explain nature
in mathematical terms, and some inroads have been made. There are certain
areas of mathematics that impact upon various aspects of nature.
In particular,the following areas are worthy of note:

Fractals  A term contracted from the words
'Fraction' and 'Dimensional'. Fractals explain why some systems in nature
are 'self similar',such as ferns.

The Butterfly Effect & Strange Attractors
 This type of mathematics gives us clues about the flow of water
and the nature of the weather system.

Affine Transformations  The processes
of reflection,rotation and scaling are seen to be operating
in nature to generate biological forms.

Geometry & Symmetry  Basic
geometrical figures can be found in nature,such as
the hexagonal arrangement of a snow flake.

Spirals  Mathematical spirals are seen to
occur naturally in snail shells, nautilus, galaxies and in DNA.

Fibonacci Numbers & The Golden Ratio
 Number series can indicate the pattern of growth
in rabbit populations and terms may be found in the number of petals
of a flower.

Population Dynamics 
Certain areas of maths such
as differential calculus
can be shown to govern how populations grow and
collapse.

Animal Skin markings  The mathematics
of 'Reaction Diffusion' can explain the patterns on animal skins.

Fractals
 In our world we are used to measuring
things in the three dimensions of length, width and depth. Some of
the things we find in nature do not sucumb easily
to such measurements and we are forced to explain their shape using
fractional dimensions.
Mountain
landscapes are a good example of a fractal, as, if
we use a one metre rule to measure around
a mountain surface we would miss
out all the gaps less than a metre in lengh. If we then
used a shorter rule to measure the surface we
would find that it would fit in all the gaps and
we would end up measuring a longer distance.
Fractals are also representative of
those types of structure
which exhibit "self
similarity". In the case of a mountain face,it is made
up of smaller and smaller rocks which are vaguely similar. 

A better example of such self similarity in nature is a
fern, which can readily be
seen to be made up of what appear to be copies
of itself at smaller and smaller scales.
Cauliflower heads and
Broccoli also exhibit patterns where copies
of a main pattern appear at smaller and smaller scales. Benoit
Mandelbrot discovered fractals whilst working for IBM and now
has a set of numbers named after him. 
The Butterly
Effect  The
butterfly
effect is the propensity of any dynamic system to be sensitive
to initial conditions. This type of mathematics pertains to such
things as water flow and weather systems, where any slight change in
one circumstance can radically alter the outcome. It may also be that our
current interaction with the Earth's environment is having a more and more
pronounced effect upon the weather
system and habitats, as we find that more and more creatures struggle
to find a niche in which they can continue to survive. 

The symbolic representation of this effect is engendered by the
Lorenz
Attractor, a diagram which is the result of modelling weather equations.
Such equations exist for many natural processes, including fluid flow. The
diagrams which represent the equations are called "strange
attractors" after their propensity to try and keep the variables in the
equations within certain extremes. In the case of the Lorenz attractor,the
red lines indicate normal weather patterns,and the other areas may be the
points that we don't normally see such as snow in the desert. Such attractors
are behind modern computing methods used to give us weather predictions and
can also be used to predict catastrophes such as volcano eruptions and
earthquakes. 
Affine
Transformations  These are the processes
of rotation, reflection and scaling. Many plant forms utilise these processes
to generate their structure. In the case of Broccoli and Cauliflower heads,
it can readily be seen that there is a type of pattern, which also shows
some spiralling in the case of Broccoli. 

It is perhaps not so obvious what is happening in a Cauliflower head,
but perhaps the process of taking a smaller copy and rotating it
is very evident in the case of a fern, where each branch appears to
be a smaller version of the main plant and so on, at smaller scales. 

The branching processes of trees which do not appear to the
naked eye as self similar can also be modelled with branching computer
programs called Lsystems which attempt to
understand the structure of trees in terms of
branching rules. 

Symmetry
 It is obvious to us as human beings when
we look in the mirror that to a large degree our own bodies
have an axis of symmetry and that the left and right hand side of us
are, to within a degree of error, mirror images of each other. Elsewhere
in nature we can see other such symmetries. The most obvious is
the sixfold symmetry of a snowflake crystal. The same symmetry can be found
in the combs of the social insects such as
bees. 
Many fruiting bodies such as tomatoes
or apples have rotational symmetries or carry segments such as those of an
orange which break the circle up into fractional parts. The propensity of
symmetries
and geometries to show up in natural objects can sometimes be traced
to the molecules that make up the object, or possibly because the arrangement
of matter
takes up the
least space or minimises the use of energy,or
maximises the
use of space whilst creating structural strength. 

The arrangement of atoms in a molecule is not visible to us under ordinary
every day circumstamces, but often the properties of any compound can
be due to the geometrical arrangment of the atoms in the substance.
Water
molecules are composed of Hydrogen and Oxygen atoms in the molecule
H_{2}O which has configurations of the atoms that may give rise to
the hexagonal structure of the snowflake. 
It is worth noting that some molecules come in left and righthanded versions
which are called Chiral molecules and before the human race new
of the consequences of this simple reflection, it was assumed that
the 

variation in the molecules symmetry made no difference to the
chemical properties of the actual molecule. In fact, we now know that it
very much does affect the properties,and in at least one case, lack
of knowledge of Chiral molecules led to a medical travesty,that of
Thalidomide. Chiral molecules are also at large,where there is a variation
in taste,such as lemon and orange fruit. 
Spirals
 The most wellknown, if little actually witnessed
spiral in nature is our own DNA inside our cells, this of course is known
as the "doublehelix" and a helix is slightly different than a spiral, but
as with the above geometry it is interesting that there are subtle mathematical
relationships between the atomic arrangment of molecules and the symmetries
of the forms that those molecules are involved in creating. 


Another, more readily observed spiral in nature, is the arrangement of the
outer skin of a Pineapple. On this fruiting body, we can see two interlacing
spirals wrapped around the surface of the fruit. We can see a similar
arrangement in the seeds of the seedhead of the sunflower,and we also find
that such spirals are sometimes a consequence of
Fibonacci
number sequences and an attempt by the flower or fruit to maximise the
number of seeds into the space that it has for them. 
More common spirals in nature appear in snail shells and a particularly
beautiful example is the shell of
the Nautilus,
whose flotation chambers are arranged in a mathematical sequence of
increasing size as the animal grows within it's shell. We might see from
this that the shell shape is a consequence of the process of increasing size
of the contained animal and that the shape is not 'designed'. 

Spiral galaxy  a consequence of the physical
laws of gravitation. 

Further out into the cosmos we still find spirals, and so we can see
that mathematics is at large in our universe, from the very smallest
scales of the atom right up to the largest scales that the universe
can provide. 
Fibonacci
Numbers
 Fibonacci
numbers are those that create a sequence, by adding up successive
numbers starting with 1 which is added to itself to get 2, then 2 is added
to 1 to get 3 and then 3 is added to 2 to get 5 and so on. This sequence
 which continues 8,13,21... very often is behind the arrangement or number
of petals in a flower head or the number of seeds or spirals in a fruiting
body. 


Because it is a relatively simple sequence where the next term is generated
from the previous one, it is something plants can do easily, and also easy
for computers to copy. Such sequences can also be behind the spiralling
arrangements often seen in biological forms. If we lay the terms out on
a twodimensional
diagram,we can generate spirals. 
Above: A crosssection through a bamboo with a hexagonal
outer edge,and right, the seed arrangement of pine cones.
Below: The seed head of a sunflower. 

Pine cones are another good example of where we see spirals and
Fibonacci
numbers,as with the Pineapple, the pattern is obvious to
the eye, but without a mathematical understanding we take these aspects
of nature for granted or attribute them to a 'designer'. 

The terms of the Fibonacci sequence are also present in the
reproduction rates of rabbits, which may account for the surprising increase
in the populations of them, so much so, that they are often used as a symbol
of fertility as when Easter cards carry images of
them.
Population growth and dynamics is another
area of nature that mathematics has rather a lot of impact upon. 
Population
Dynamics  This area of mathematics is sometimes
controversial because it starts to make statements about policies such as
culling and vaccination,which people often create emotional arguments about
without understanding the facts, or possibly go looking for causes which
don't actually exist.
Currently,for instance, the human race is witnessing a
collapse in the
population
of bees and is also concerned
about overfishing. There
have also been arguments about seal culling, whaling and also the hunting
of foxes. In the public domain, very few people are aware of the work that
has been done analysing how populations of creatures grow and develop and
sometimes collapse. In particular, the work of
Robert
May on Chaos
Theory as it applies to populations has shed some sometimes
counterintuitive light on population dynamics.
There are some computer models which capture some of the essences of either
populations of creatures or cellular systems.
John Conway's LIFE
program attempts to mimic the way a cell culture might grow or display
characteristics or even die off altogether. Most interestingly  the program
only has 3 rules and yet exhibits complex behaviour  including reproduction
and evolution and mutation.
Cells in his program can proliferate or die depending in what environment
they start off and it maybe we have lessons to learn about cause
and effect in the environment  if we see that the dinosaur population crashed
 we go looking for a meteor crash site or a virus or an environmental change.
Mathematical models tell us that this need not be the case,and also instruct
us that we might think we can keep fishing and fishing with given
"quotas"
and feel safe in doing so  but it may not in fact be the case.
Those of the country set seem to think they understand how to control
'pests' such as foxes or
maybe even
squirrels, but the populations
are subject to factors such as availability of food and the propensity
of pedators to kill them  all of this must be taken into account before
a crass 'cull' is taken up,for we may actually be shooting ourselves
in the foot when we discover a knockon effect,or indeed
Butterfly Effect because of unseen conequences
of our actions.
Simple
predatorprey
modelling is done using
differential
calculus and can easily be watched on a home computer,it becomes
apparent when doing so that simple meddling with a population with a
cull is illadvised and it behoves us as humans to understand the complexity
of population dynamics before trying to exhibit control over it. 
The following is an extract from
"A Mathematician
Reads the Newspaper" by John Allen Paulos 
Rodent Population Patterns Difficult to Fathom
Ecology, Chaos, and the News
News
stories on ecological issues always seem to be on the verge of making what
philosophers used to call a category.mistake, describing a situation as if
it belonged to one logical type or category when in fact it belongs to another.
In the case of ecology, there is a tendency to look primarily for culpability
and conflicts of human will rather than at the dynamics of a natural process.
(I'm reminded of the comedian who announced that an Iranian terrorist group
had claimed responsibility for the 1993 floods in the Midwest.) Often, however,
there are no discernible agents, no Whos to write about. Furthermore, the
process isn't instantaneous or precisely localized, so the journalistic When
and Where seem inappropriate as well. Moreover, the What, Why, and How are
generally complex and impersonal, often beyond the complexity horizons of
most readers and sometimes even of the ecologists themselves.
An intriguingly simple ecological example studied by the biologist
Robert May
and the physicist
Mitchell
Feigenbaum illustrates. Imagine that the population of a certain animal
species is given by the socalled logistic formula X' = RX(1X), where X
is the population one year, X' the population the next year, and R a parameter
that varies between 0 and 4. For simplicity, we take X and X' to be numbers
between 0 and 1, the true population being, say, 1,000,000 times these values.
If we assume the population X is now .1 (that is, 100,000) and R = 1.5, we
can calculate the population next year, X', by plugging .1 into the formula,
obtaining 1.5(.1)(.9), or .135. To calculate the population for two years
from now, we plug .135 into the formula and obtain 1.5(.135)(.865), or
approximately
.1752.
If you have a calculator handy, you might want to verity (or you might not)
that the population for three years from now will be .21676. I'll spare you
any more arithmetic, but the same procedure may be used to find the population
for any future year. You would find that the population grows rapidly and
then stabilizes at .3333. Moreover, you would find that whatever its original
value, big or small, the population would still stabilize at .3333. This
population is termed the steady state population for this value of R (see
diagram). If we perform similar calculations for a smaller R, say, R = 1,
we find that whatever its initial value, the population "stabilizes" at 0;
it becomes extinct. When R is bigger2.5, to be specificwe find in carrying
out these procedures that whatever its initial value, the population now
stabilizes at approximately .6, the steady state for this R.
So what? Well, let's increase R a little more, this time to 3.2. As before,
the original population does not matter, but now when we substitute iteratively
into the formula X' = 3.2X(1X), we find that the population of the species
doesn't stabilize at one value, but eventually settles into a repetitive
alternation between two values: approximately .5 and .8; that is, one year
the population is .5 and the next it's .8. Let's raise our parameter R to
3.5 and see what happens. The initial population is once again immaterial,
but this time in the long run we have the population alternating regularly
among four different values, approximately .38, .83, .50, and .88, in successive
years.
If we again increase R slightly, the population settles down to a regular
alternation among eight different values. Smaller and smaller increases in
R lead to more doublings of the number of values.
Then suddenly at approximately R = 3.57, the number of values grows to infinity
and the population of the species varies randomly from year to year. This
is an odd sort of randomness, however, since it results from iterating the
formula 3.57X(1X), and the sequence of populations is thus quite determined
by the initial population.) Even stranger than this chaotic variation in
the species' population is the fact that a slight increase in R again results
in a regular alternation of the species' population from year to year, and
a further increase again leads to chaotic variation. These orderly alternations
followed by random chaos followed again by windows of regularity are critically
dependent on the parameter R, which seems to be a gross measure of the volatility
of the
model.
A closeup view of the chaotic region reveals an unexpected intricacy, and,
as in the case of the economy discussed earlier, there is a lesson to this
intricacy. If a system as trivial as this single nonlinear equation can
demonstrate this sort of chaotic unpredictability, perhaps we shouldn't be
quite as assertive and dogmatic about the predicted effects of various ecologic
policies on the gigantic nonlinear system that is Planet Earth. The policies'
consequences, one would think are considerably more opaque than are those
of the value of R in this model.
Let me reiterate that there are no agents involved in the story; no greedy
corporate despoilers or overzealous environmentalists. Any news story that
purported to explain the variation in population would either have to attribute
it falsely to someone or something comprehensible to most readers or else
find some other way to present the account. As in reporting on the economy
and other complex systems, there is a tendency to settle upon some stylized,
conventional set of stories that focus on turf and clout rather than on the
intricacies of the impersonal system in question.

Animal
Skin markings  When we look at
a Leopard's spots or the markings of a Dalmation,or maybe even the wing's
of a Butterfly,we maybe apt to think that these patterns are intrinsically
random or perhaps entirely a property created genetically via
evolution
perhaps to enable a species to survive by creating a warning sign or by
echoing
some aspect of the environment for camouflage purposes. 


What may not be apparent is that work done by Alan Turing suggested
ways in which maths might explain such markings, but latterly we have begun
to understand the
patterning
on animals in terms of
Reaction Diffusion
equations. These can produce stable states which may offer a glimpse as to
how chemical systems in a biological system work to create a pattern. 
Such systems are equally capable of producing rings such as those
seen on certain species of butterfly, so when we look at the nature around
us, it might be worthwhile considering the phrase "I never did maths at school,
I didn't see what I would use it for"!! 

How did the leopard get its spots? 
LifeSonia Van Gilder Cooke, reporter
Rudyard Kipling was halfright, say William Allen and colleagues of
the University of Bristol, UK.
The team googled pictures of big cats and matched their coats to
computergenerated patterns. The resulting classification was used
to test the relationship between a cat's patterning, where it lives, and
how it hunts. The researchers found that nocturnal treedwellers have evolved
complex coats to match a dappled, dark forest, while cats stalking the open
plains tended to be more plain.
One exception is the cheetah, whose patterned coat contrasts with
its monochromatic hunting grounds. Allen said this might be explained by
the cheetah's hunting strategy, which relies not on camouflage, but on speed.
Low genetic diversity might also be a cause.
"It may be that there isn't variety in the population to become anything
other than spotted," says Allen. For other cats, however, camouflage evolves
in response to environment over short periods of time, suggesting a leopard
can actually change its spots.
Allen drew inspiration for the title of his study from Rudyard Kipling's
Just So Stories. "Rudyard Kipling was wrong by suggesting how leopards got
their spots as the fingerprints of a man," Allen told the Telegraph. "But
he was right about the reason because they provide the perfect camouflage
in a forest habitat with dappled light."
Journal reference: Proceedings of the Royal Society B, DOI:
10.1098/rspb.2010.1734

Natural food fractals  Cauliflower and Broccoli 


Images from Oaken Clough Garden 

The segments of this plant reduce by the same amount in length
in each segment.


A fern  an example of natural fractal.


A fern unfolds from a spiral.






Example of Pentagonal symmetry, natural spiralling and affine transformation.
Click on a plant below to see it's symmetry.

