Keith Ball

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Maths in Nature

• By
• Keith Ball

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Does maths really appear in nature?

• In a word, yes.
• However, unless you know what to look out for, it
  isnt 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.

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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!

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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.

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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 as-
• 0, 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?

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So what happens?

• We start off with 1 pair of rabbits in the field.
• At the end of the first month the original pair
  mate but there is still one only 1 pair.
• At the end of the second month the female
  produces a new pair, so now there are 2 pairs of
  rabbits in the field.
• At the end of the third month, the original
  female produces a second pair, making 3 pairs in
  all in the field.
• 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!

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More Fibonacci

• The Fibonacci sequence can also be used in
  another, more visual, way.
• This is the process of creating Fibonacci
  rectangles.
• Start with two small squares of size 1 next to
  each other. On top of both of these draw a square
  of size 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
• Then another touching both the 2-square and the
  3-square (which has sides of 5 units).
• 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?

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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!

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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.

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More Fibonacci flowers?

• On many plants, the number of petals is a
  Fibonacci number3 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?

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Natural fractals?

• Fractals dont 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?

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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.

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Fractal trees
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Fractal ferns
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Fractal spiral
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• I have covered some of the more well known
  examples of the relationship between maths and
  nature.
• But there are many more out there.
• Youd be surprised at what you can find if you
  only look hard enough.