By
Keith Ball
Does maths really appear in
nature?
In a word, yes.
However, unless you know what to look out for, it
isn’t very easy to spot.
For example, would you have thought that the
breeding of rabbits could be modelled on a simple
number sequence?
But that this same sequence can be used to construct
the spiral shape that we see on a sea shell?
In this presentation I aim to show some examples of
the many different cases where you can find maths
in the real world.
A pretty face?
It is quite obvious that the human
face is symmetrical about a
vertical axis down the nose.
However, studies have shown that
the symmetry of another
persons face is a large factor in
determining whether or not we
find them attractive.
The better the quality of the
symmetry, the more
mathematically perfect it is and
the more aesthetically pleasing
we consider it to be.
In short, the better the symmetry of
someone's face, the more
attractive you should find them!
Beehive basics
A beehive is made up of
many hexagons packed
together.
Why hexagons? Not squares
or triangles?
Hexagons fit most closely
together without any gaps,
so they are an ideal shape
to maximise the available
space.
Rabbit multiplication
The breeding of rabbits is a very effective
way of demonstrating the Fibonacci
sequence.
The Fibonacci sequence is a sequence of
numbers formed by adding together the
2 previous numbers.
The Fibonacci sequence starts as0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
So how is this relevant to rabbits breeding?
Lets suppose a newly-born pair of rabbits,
one male and one female, are put in a
field. Rabbits are able to mate at the age
of one month, so that at the end of its
second month of life a female can
produce another pair of rabbits.
Suppose that our rabbits never die and
that the female always produces one
new pair (one male and one female)
every month.
What would happen?
So what happens?
1.
We start off with 1 pair of rabbits in
the field.
2.
At the end of the first month the
original pair mate but there is still
one only 1 pair.
3.
At the end of the second month the
female produces a new pair, so now
there are 2 pairs of rabbits in the
field.
4.
At the end of the third month, the
original female produces a second
pair, making 3 pairs in all in the field.
5.
At the end of the fourth month, the
original female has produced yet
another new pair and the female born
two months ago produces her first
pair, making 5 pairs.
Noticed the sequence yet?
Over the course of 5 months the number of
pairs in the field has gone 1, 1, 2, 3,
5. The Fibonacci sequence!
More Fibonacci
The Fibonacci sequence can also be used in another,
more visual, way.
This is the process of creating Fibonacci rectangles.
1.
Start with two small squares of size 1 next to
each other. On top of both of these draw a
square of size 2
2.
We can now draw a new square - touching both
a unit square and the latest square of side 2 - so
having sides 3 units long
3.
Then another touching both the 2-square and
the 3-square (which has sides of 5 units).
4.
We can continue adding squares around the
picture, each new square having a side which is
as long as the sum of the last two square's sides.
This set of rectangles whose sides are two successive
Fibonacci numbers in length and which are
composed of squares with sides which are
Fibonacci numbers, we call the Fibonacci
Rectangles.
This is only the first 7 numbers in the Fibonacci
sequence.
What would happen if we carried on a lot longer?
So what happens?
As we go on, the squares
begin to form a certain
pattern. If we draw a
line through the corner
of each square we start
to get a spiral shape.
The same spiral shape
that we can see on this
sea shell!
Fibonacci flowers?
The Fibonacci sequence previously
mentioned appears in other cases.
The ratio of consecutive numbers in
the Fibonacci sequence approaches
a number known as the golden
ratio, or phi (1.618033989). Phi is
often found in nature. A Golden
Spiral formed in a manner similar
to the Fibonacci spiral can be found
by tracing the seeds of a sunflower
from the centre outwards.
More Fibonacci flowers?
On many plants, the number of petals is a
Fibonacci number:
3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur
8 petals: delphiniums
13 petals: ragwort, corn marigold,
cineraria, some daisies
21 petals: aster, black-eyed susan,
chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the
asteraceae family.
Ever wondered why its so difficult to
find a 4 leaf clover? Or even a plant of
any kind with 4 petals? Few plants have
4 petals, and 4 is not a Fibonacci
number.
Coincidence?
Natural fractals?
Fractals don’t appear in nature as such, but
they are another clear example of the
way maths and nature can be linked
together.
A fractal is a geometric pattern that is
repeated at every scale and so cannot be
represented by classical geometry.
A famous example is the Koch curve
(shown on the right).
Stage 1 is to draw a straight line.
All stages afterwards are constructed by
rubbing out the middle of a line and
drawing 2 more diagonal lines in its
place (resembling a triangle).
So what would happen if we carried this
fractal on for many more stages?
So what would happen?
We would end up with the
famous Koch snowflake.
By just repeating a simple
pattern, you can create a
snowflake, yet another
example of how maths and
nature can share a
connection.
Are there any more fractals
that create natural images?
Yes! Over the next few slides
are some of the most
impressive natural fractals
that have been discovered.
Fractal trees
Fractal ferns
Fractal spiral
I have covered some of the more well known
examples of the relationship between maths
and nature.
But there are many more out there.
You’d be surprised at what you can find if you
only look hard enough.