THIS EBOOK WAS PREPARED AS A PART OF THE COMENIUS PROJECT WHY MATHS? by the students and the teachers from: BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM) EUREKA SECONDARY SCHOOL IN KELLS (IRELAND) LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY) GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND) ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL) IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)

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Maths in Nature and Biology
Maths in Nature
and Biology

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THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA
(ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ
(POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA
(PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.

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II.. SSYYMMMMEETTRRYY
IIII.. PPAATTTTEERRIINNGG
IIIIII.. FFIIBBOONNAACCCCII SSEEQQUUEENNCCEE IINN NNAATTUURREE
IIVV.. TTEESSSSEELLAATTIIOONNSS IINN NNAATTUURREE
VV.. TTHHEE GGAAMMEE OOFF LLIIFFEE
VVII.. CCHHAAOOSS TTHHEEOORRYY AANNDD MMAATTHHSS

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HHOOWW IISS NNAATTUURREE RREELLAATTEEDD TTOO MMAATTHHSS??
Mathematics might seem an ugly and irrelevant subject at school, but it's
ultimately the study of truth - and truth is beauty! You might be surprised to find that
maths is in everything in nature from rabbits to seashells. Mathematics is everywhere in
this universe, even though we may not notice it. In this chapter, we are going to explore
a few properties of mathematics that are depicted in nature, mainly in:
I. Symmetry:
• Bilateral;
• Radial.
II. Patterning:
• Bacterial population growth;
• Snowflakes;
• Diamonds.
III. Fibonacci sequence
IV. Tesselations

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II.. SSYYMMMMEETTRRYY
Symmetry is when a figure has
two sides that are mirror images of
one another. It would then be possible to
draw a line through a picture of the object
and along either side the image would look
exactly the same.
There are two kinds of symmetry:
One is bilateral symmetry, in which an
object has two sides that are mirror images of each
other. The human body would be an excellent
example of a living being that has bilateral
symmetry.
The other kind of symmetry is radial
symmetry. This is where there is a center
point and numerous lines of symmetry could
be drawn. The most obvious geometric
example would be a circle. (which can be
found, for example, on a spider web).

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IIII.. PPAATTTTEERRNNIINNGG
Patterns in nature are visible
regularities of form found in the
natural world. These patterns occur in
different contexts and can sometimes
be shaped mathematically. Patterns in
living things are explained by the
biological processes of natural and
sexual selection.
These are some good examples of
patterns found in nature:
Bacterial population growth
Under favorable conditions, a growing
bacterial population doubles at regular
intervals. Growth is by geometric progression:
1, 2, 4, 8, etc. [or 20, 21, 22, 23.........2n (where n
= the number of generations)]. This is called
exponential growth.
Bacteria are everywhere around us.
Given good growing conditions, a bacterium grows slightly in size or length.
A new cell wall grows through the center forming two daughter cells, each with the same
genetic material as the parent cell: if the environment is optimum, the two daughter
cells may divide into four in 20 minutes. So in a very short time we can have many
duplicates of the parent cell as their growth is like this:

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We can notice that the shape of the curve we obtain is the one of the exponential
function and the growth is very fast. Then why isn't the earth covered with bacteria?
The primary reason is that the conditions in which bacteria live are rarely optimum:
scientists who study bacteria try to create the optimum environment in the lab that is
culture medium with the necessary energy source, nutrients, pH, and temperature, in
which bacteria grow predictably.
Let's have a look at this short video that shows us an example of bacterial growth:
LINK
But in the real world the grow curve is the same only at the beginning of the observation,
then it looks like this:
Let's analyze what happens when bacteria are grown in a closed system (also called
batch culture) like a test tube, the population of cells almost exhibits these growth
dynamics:

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LAG PHASE: Growth is slow at first and the cells have to adapt to the new
environment and to acclimate to the food and nutrients in their
new habitat. In this phase cellular metabolism is accelerated, cells
are increasing in size, but the bacteria are not able to replicate and
therefore there is no increase in cell mass. The length of the lag
phase depends directly on the previous growth condition of the
organism: when the microorganism growing in a rich medium is
inoculated into nutritionally poor medium, the organism will take
more time to adapt with the new environment. Similarly when an
organism from a nutritionally poor medium is added to a
nutritionally rich medium, the organism can easily adapt to the
environment, it can start the cell division without any delay, and
therefore will have less lag phase it may be absent.
EXPONENTIAL
PHASE:
Once the metabolic machinery is running, they start multiplying
exponentially, doubling in number every few minutes until they
run out of space or nutrients. The growth medium is exploited at
the maximal rate, the culture reaches the maximum growth rate
and the number of bacteria increases exponentially and finally the
single cell divide into two, which replicate into four, eight, sixteen,
thirty two and so on (That is 20, 21, 22, 23.........2n, n is the
number of generations: this will result in a balanced growth. The
time taken by the bacteria to double in number during a specified
time period is known as the generation time, that tends to vary
with different organisms.
STATIONARY
PHASE:
As more and more bugs are competing for dwindling food and
nutrients, booming growth stops and the number of bacteria
stabilizes. As the bacterial population continues to grow, all the

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nutrients in the growth medium are used up by the microorganism
for their rapid multiplication: this result in the accumulation of
waste materials, toxic metabolites and inhibitory compounds such
as antibiotics in the medium. This shifts the conditions of the
medium such as pH and temperature, thereby creating an
unfavourable environment for the bacterial growth. The
reproduction rate will slow down, the cells undergoing division
tends to be equal to the number of cell death, and finally
bacterium stops its division completely. The cell number is not
increased and thus the growth rate is stabilised. If a cell taken
from the stationary phase is introduced into a fresh medium, the
cell can easily move on the exponential phase and is able to
perform its metabolic activities as usual.
DEATH PHASE: Toxic waste products build up, food is depleted and the bugs begin
to die, so the number of bacteria decreases quite quickly. The
depletion of nutrients and the subsequent accumulation of
metabolic waste products and other toxic materials in the media
will facilitates the bacterium to move on to this phase in which the
bacterium completely loses its ability to reproduce: individual
bacteria begin to die due to the unfavourable conditions and the
death is rapid and at uniform rate. The number of dead cells
exceeds the number of live cells.
LONG TERM
STATIONARY
PHASE:
A small number of bacteria can survive for long periods of time in
a non-growing state. This particular phase of growth is interesting
for research into a number of pathogens as it is thought to best
represent the state in which bacteria survive during a number of
diseases.
The formula for the growth of population can summarized in this way:
Let's see an example.
In 1950, the world's human population was 2,555,982,611. With a growth rate of
approximately 1.68%, what was the population in 1955?
First, let's figure out what everything is:

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We can ignore the decimal part since it's not a full person.
So, our guess is that the world's population in 1955 was 2,779,960,539.
The actual population was 2,780,296,616 so we were pretty close.

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Snowflakes
Snowflakes’ patterns can be
incredibly complex. Since snowflakes
can branch differently down to
individual water molecules, the
number of possibilities is extremely
large. Without a restrictive theory
that constrains snowflakes to a
limited number of shapes, it seems
probable that no two snowflakes are
alike.

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Fractals aren't just something we learn about in math class. They are also a gorgeous
part of the natural world. Here are some of the most stunning examples of these
repeating patterns.
Romanesco broccoli is a particularly symmetrical fractal.
The fern is one of many flora that are fractal; it’s an especially good example.
Each part is the roughly the same as the
whole. When we break a leaf off of the
original and it looks like the original – break
a leaf off of that leaf and that looks like the
original also.
The delicate Queen Anne’s Lace, which
is really just wild carrot, is a beautiful
example of a floral fractal. Each blossom
produces smaller iterative blooms. This
particular image was shot from
underneath to demonstrate the fractal
nature of the plant

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IIIIII.. FFIIBBOONNAACCCCII SSEEQQUUEENNCCEE
Leonardo Fibonacci was an Italian
mathematician. Fibonacci was regarded as the first
great European mathematician of the Middle Ages
and he’s responsible for the creation of the
Fibonacci Sequence:
The first two terms are F0= 0 and F1= 1. This sequence has a simple law: every element,
after the third, is obtained by adding the previous two. See: 1 +1 = 2, 2 +1 = 3, 3 +2 = 5,
and so on.
Fibonacci and a population of rabbits (example)
Fibonacci considered the growth of a population idealized (not biologically
realistic) of rabbits. The numbers describe the number of couples in the population of
rabbits after n months if we deduce that:
● on the first month only a couple is born;
● couples are only expected to be sexually mature (and reproduce) after the
second month of life;
● there are no problems in genetic inbreeding;
● every month, every fertile couple gives birth to a new couple;
● rabbits never die.
● The rabbits born in January 1. will be fertile after 2 months. Therefore, on March
1 they will have descendents
● On April 1, the initial couple are still fertile so they will have another couple of
descendents.
● If we reason similarly, we can deduce that, on June 1, there will be 8 pairs of
rabbits; on July 1, 13 couples; on August 1, 21 couples and so on.
● After a year, that is, January 1st of the following year, we’re expected to have 144
pairs of rabbits.

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Fibonacci numbers in nature
The Fibonacci numbers appear everywhere in nature, from the leaf arrangement
in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales
of a pineapple.
Probably most of us have never taken the time to examine very carefully the
number or arrangement of petals on a flower. If we were to do so, we would find that the
number of petals on a flower for many flowers is a Fibonacci number.
3 petals lily, iris
5 petals buttercup, wild rose, larkspur, columbine
8 petals delphiniums, clematis
13 petals ragwort, corn marigold, cineraria
21 petals aster, black-eyed susan, chicory
34 petals plantain, pyrethrum
55, 89 petals michelmas daisies, the asteraceae family

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“Logarithmic Spiral” of a common shell.
In nature, we have a lot of examples of the golden ratio
and the Fibonacci sequence. We can find them in all
natural world. The Fibonacci spiral appears not only in
the perfect nautilus shell but also in pinecone, pineapple,
in hurricanes, ram's horns. The Fibonacci numbers
increase at a ratio that is revealed in objects and spirals.
The Chambered Nautilus if cut in half reveals a series of
chambers. Each chamber increases in size as the mollusk
grows. They also grow in a spiral shape.

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This same spiral and ratio is present in a great many products of nature; the pinecone,
the pineapple.
If we look at the bottom of a pinecone. We can found same kinds of spirals. They don’t
go around and around in a circle – they go out like fireworks. Look at the pictures above,
to see what that looks like.
Another examples of spiral:

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SSEEEEDD PPAATTTTEERRNNSS OOFF
SSUUNNFFLLOOWWEERRSS
All the sunflowers in the world
show a number of spirals that
are within the Fibonacci
Sequence.
Look at the following images of a sunflower:
By observing closely the seeds configuration you will see how appears a kind of spiral
patterns. In the top left picture we have highlighted three of the spirals typologies that
could be found on almost any sunflower.
Well, if you look at one of the typologies, for example the one in green, and you go to the
illustration above right you can check that there is a certain number of spirals like this,
specifically 55 spirals.

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We have more examples in the two upper panels, cyan and orange, they are also
arranged following values that are within the sequence: 34 and 21 spirals.
A lot of people love honey made by tiny bees.
These insects use so much mathematical
strategy throughout their daily lives. Just their
hives use angles, shape, tessellation and
addition.
Wasps and bees exhibit polygons in
their nests. Hexagons create nests
that require less material and work
to build. It is an efficient way of
partitioning that also saves energy.
Hexagonal cell requires minimum
amount of wax for construction
while it stores maximum amount of
honey

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IIVV.. TTEESSSSEELLAATTIIOONNSS
In nature we can see samples of tessellations. This phenomenon is really beautiful and
incredible. Here you can see some examples :
a) The Giant's Causeway, located in Ireland, is an fascinating formation found in
nature. It is a collection of hexagons tesselating the ground - even in 3D at some
points.

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b) Rock formation in "White Pocket", Vermillion Cliffs National Monument,
c) Veins in a leaf.
d) Dragonfly
e) Cracked dried

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VV.. TTHHEE GGAAMMEE OOFF LLIIFFEE
"The mathematician’s patterns, like the
painter’s or the poet’s must be
beautiful; the ideas like the colours or
the words, must fit together in a
harmonious way. Beauty is the first
test: there is no permanent place in the
world for ugly mathematics."
GH Hardy (1877 – 1947)
Introduction
Everybody can easily understand that a population of living beings can evolve
depending on the environmental conditions around it. Under their premises, for more
than one century scientists and mathematicians in particular have been studying and
developing different models to better understand the future evolution of a given
population under a set of given environmental conditions. This research field was called
“Theoretical Biology”: traditional population
genetic models are frequently stochastic. For
instance around 1925 two mathematicians,
the Italian Vito Volterra and the American
Alfred James Lotka, developed
independently a model, later named Lotka-
Volterra prey-predator model, to describe
the dynamics of biological systems in which
two species interact, one as a predator and
the other as prey.
In recent years, mathematical modeling of developmental processes has earned new
respect. Not only mathematical models have been used to validate hypotheses made
from experimental data, but designing and testing these models has led to testable

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experimental predictions. There are now many different impressive cases in which
mathematical models have given a substantial new contribution to a better knowledge of
biological systems by suggesting how connections between local interactions among
system components relate to their wider biological effects.
In evolutionary game theory, biologists work together with mathematicians to obtain a
deterministic mathematical form, with selection acting directly on inherited genetic
types.
The game
The Game of Life is a simulation game that reproduces the real-life processes of the
living organisms: it is not a game in the conventional sense as there are no players and
nobody wins or loses; once the "pieces" are placed in the starting position, the rules
determine everything that happens later. Nevertheless, the game is full of surprises: in
most cases, it is impossible to look at a starting position (pattern) and see what will
happen in the future. The only way to find out is to follow the rules of the game.
It was invented by the British mathematician
John Conway (Liverpool 1937) who was teaching
at Cambridge University.
In 1970 “Scientific American” published an article
in which is mentioned for the first time this game.
To play the game is necessary to have a
chessboard extended infinitely in all directions:
the basic idea is to start with a simple
configuration of organisms, one to a cell, then
observe how it changes as we apply
Conway's 'genetic laws' for births, deaths
and survivals.
Conway's genetic laws are easy:
• each cell can have eight neighbors.
• each cell can evolve according to one of these 3 rules:
• Survival: every cell with 2 or 3 neighbors cells survives for the next
generation
• Death:
• each cell with 4 or more neighbors dies for overpopulation
• every cell with one neighbor or none dies for isolation.

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• Births: each empty cell adjacent to exactly 3 neighbors is a birth cell.
Conway choose these rules quite carefully after trying many other possibilities: some
caused the cells to die too fast while others caused too many cells to be born. The rules
he eventually choose balance these tendencies, making it hard to tell whether a pattern
will die out completely, form a stable population, or grow forever.
Conway conjectures that no pattern can grow without limit. He has offered a prize of
$50 to the first person who can prove or disprove this conjecture.
We can observe that:
• there should be no initial pattern for which there is a simple proof that the
population can grow without limit;
• there should be simple initial patterns that apparently grow without limit;
• there should be simple initial patterns that grow and change for a considerable
period of time before going to end in three possible ways: fading away completely
(from overcrowding or becoming too sparse), setting into a stable configuration that
remains unchanged, or entering in oscillating phase in which they repeat an endless
cycle of two or more periods.
From a random initial pattern of living cells on the grid, observers will find the
population constantly changing as the generations tick by.
The patterns that emerge from the simple rules may be considered a form of beauty.
Small isolated subpatterns with no initial symmetry tend to become symmetrical.
Once this happens, the symmetry may increase in richness, but it cannot be lost unless a
nearby subpattern comes close enough to disturb it. In a very few cases the society
eventually dies out, with all living cells vanishing, though this may not happen for a
great many generations. Most initial patterns eventually "burn out", producing either
stable figures or patterns that oscillate forever between two or more states; many also
produce one or more gliders or spaceships that travel indefinitely away from the initial
location.
The game and reality
Life is just one example of a cellular automaton, which is any system in which rules are
applied to cells and their neighbors in a regular grid.
There has been much recent interest in cellular automata, a field of mathematical
research. Life is one of the simplest cellular automata to have been studied, but many
others have been invented, often to simulate systems in the real world.
The Game of Life is one of the simplest examples of "emergent complexity" or "self-
organizing systems" : this topic captured the attention of scientists and mathematicians
in diverse fields who are studying of how elaborate patterns and behaviors can emerge
from very simple rules. For instance, it helps us understand how the petals on a rose or
the stripes on a zebra can arise from a tissue of living cells growing together or it how

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the diversity of life that evolved on earth along the ages.
The game and computer programming
Early patterns with unknown futures led computer programmers across the world to
write programs to track the evolution of Life patterns. Most of the early algorithms were
similar; they represented Life patterns as two-dimensional arrays in computer memory.
Typically two arrays are used, one to hold the current generation, and one in which to
calculate its successor. Often 0 and 1 represent dead and live cells respectively. A nested
for-loop considers each element of the current array in turn, counting the live
neighbours of each cell to decide whether the corresponding element of the successor
array should be 0 or 1. The successor array is displayed. For the next iteration the arrays
swap roles so that the successor array in the last iteration becomes the current array in
the next iteration.
The rules can be rearranged from an egocentric approach of the inner field regarding its
neighbors to a scientific observers point: if the sum of all nine fields is 3, the inner field
state for the next generation will be life if the all-field sum is 4, the inner field retains its
current state and every other sum sets the inner field to death.
The game and its inventor
This is an interview to Conway talking about the game:
http://a.parsons.edu/~joseph/k2/gameoflife/
Let's play the game... using this link:
http://www.kongregate.com/games/shaman4d/conways-game-of-life
To see more - here is the LINK to the presentation prepared by Italian students.

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VVII.. CCHHAAOOSS TTHHEEOORRYY AANNDD MMAATTHHSS
“Chaos was the law of nature: Order was the dream of man”
-Henry Adams-
INTRODUCTION: When we talk about chaos we normally mean a sense of disorder and
randomness. But It’s very important to understand that, in the new science conception,
chaos is seen, on the contrary, as a very complex order, so complex to escape to the
human’s perception and comprehension. According to this, chaos is the real essence of
order.
THE HISTORY OF CHAOS: Until the 1960’ the world of science was relatively simple:
everything could be explained with simple formulas and everything behaved in a
predictable way. The theory of chaos was worked out when the
classical science could explain anymore the irregular and
inconstant aspects of nature. In 1961 Edward Lorenz, an
American meteorologist and one of the pioneer of the theory of
chaos, was working on his weather forecasting machine and he
decided he wanted to examine a previous day sequence in more
detail. He typed the numbers from the previous day’s print out
into the computers and went to get a coffee. When he returned
he couldn’t believe at his eyes. The new weather was nothing like
the original. Then he realized what happened: he had made a
minimal error that normally would have been considered
insignificant. Repeating the simulation with slightly different values the results was
completely different. Lorenz realized that weather is a chaotic system, that is to say an
extreme sensible system dependent upon initial conditions.
BUTTERFLY EFFECT: This simply means that even the most minute and almost
imperceptible change in the starting conditions could well generate unpredictable
results in the final outcome. Lorenz explained this theory with the simple example of the
Butterfly Effects: a flap of a butterfly’s wings in China has the potential to cause a
hurricane in North America. In conclusion it will never be possible to predict the
behavior of systems that shows extreme sensitivity. That’ s why we can’ t predict
whether for more than a week in advance.
FRACTALS: Fractals are strictly connected with chaos. Gaston Julia is considered the
father of fractals but the study of this structures really evolved with the use of
computers. The French professor Benoit Mandelbrot discovered them studying nature:
he noted that nature has a peculiar tendency to repeat itself, often in strange and
unpredictable patterns. Fractals describe some chaotic behavior and they can be
expressed by complex equations. Fractals are for example the never-ending spirals on a
head of broccoli or the rhythmic scales of snakes.

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INTRODUCTION OF BIOLOGY: Thanks to Mandelbrot scientist started to think that
fractals can be applied in the field of biology. In fact he proposed that lungs show signs
of fractal geometry.
Thanks to fractals and chaos we can accurately describe
and understand various parts of the human body. Some
scientists support that such inefficiency exists in the world
of medicine because the human body is a chaotic system In
Biology, Chaotic systems can be used to show the rhythms
of heartbeats, walking strides, and even the biological
changes of aging. Fractals can be used to model the
structures of nerve networks, circulatory systems, lungs,
and even DNA.
You could find it strange but, it was proved that the absence
of chaos is a sign of disease. According to this theory a group of scientists found that
some subjects with congestive heart failure actually experienced periods of time where
there was no chaos in their heartbeat. When humans walk with variation in their step
they are normal. With the onset of a disease, such as Parkinson’s disease, the human
stride is more constant. An absence of chaos can also be a sign of aging, as our bodies
and the systems within them lose their chaotic characteristics as we grow older.
Brain is another example of chaotic system that can only be modeled using fractal
geometry. In addition to being able to model electrical signals of nerves and the brain,
chaos theory may help solve neurological diseases and progress the invention of
artificial intelligence.
The most significant contribution that chaos theory could make in the field of medicine
would have to do with the creation of chaotic models that would be able to predict
progression of aging or diseases within the body. Applying chaotic models to other
systems of the body such as the immune system could help us in understanding how it
ages. Some scientists support that the knowledge we have gained in fractals and chaos
theory could be helpful to cure diseases such as cancer and cystic fibrosis instead stem
cell, radiation treatment or any other uncertain cures. In conclusion we can affirm that
a human body is healthy when there is a balance between order and chaos.
To see more - here is the LINK to the presentation prepared by Italian students.

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BIBLIOGRAPHY
http://www.demilked.com/amazing-nature-patterns/
http://www.scienceinschool.org/print/40
http://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in-nature/
http://www.microbiologyonline.org.uk/about-microbiology/introducing-
microbes/bacteria
http://textbookofbacteriology.net/growth_3.html
http://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in-nature/
http://farm3.static.flickr.com/2513/3916933802_f61190a991.jpg
http://en.wikipedia.org/wiki/Patterns_in_nature
http://www.pinterest.com/tammyklick/snowflakes-breathtaking-macro-photography/
http://www.scienzainrete.it/en/content/article/nature-numbers
http://nutters.edublogs.org/maths-in-nature/
http://germzoo.blogspot.it/2012/01/death-by-chicken.html
http://www.cellsalive.com/ecoli.htm
http://amrita.vlab.co.in/?sub=3&brch=73&sim=1105&cnt=1
http://www.math.com/students/wonders/life/life.html

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