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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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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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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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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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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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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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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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"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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