By Keith Ball

1. Maths in Nature By Keith Ball

2. 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.

3. 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!

4. 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.

5. 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?

6. 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!

7. 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?

8. 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!

9. 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.

10. 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?

11. 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?

12. 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.

13. Fractal trees

14. Fractal ferns

15. Fractal spiral

16. 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.