Geometry in Nature

Many things in the natural world have a simple underlying mathematics, and we often miss it unless our eyes are open; but the symmetry is always there.

We are all familiar with the interlocking hexagons of a honeycomb, made from equilateral triangles.











But there is another mathematical trick which Nature uses in some surprising ways. The shape of a snail's shell, the scales on a pine cone, the whorls in a flower, the breeding of rabbits (it's true, see below). Much more, and it all follows a simple mathematical sequence -
The Fibonacci Series.

It goes like this:

You start with nothing, zero
0
along comes a 'first', the number one
0, 1
add them together 0+1=1
0, 1, 1
then add 1+1 to get 2
0, 1, 1, 2
keep adding the last two numbers, so 1+2=3
0, 1, 1, 2, 3
then 2+3=5
0, 1, 1, 2, 3, 5
and so on...
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

If you divide each number in the sequence by the previous one* you get increasingly more accurate approximations to one particular number...

1÷1=1
2÷1=2
3÷2=1.5
5÷3=1.6667
8÷5=1.6
13÷8=1.6250
21÷13=1.6154
34÷21=1.6190
55÷34=1.6176
89÷55=1.6182
144÷89=1.6180...

That number is the Golden Ratio, that pleasing blend of height and width beloved of artists and architects for millennia. But Nature has been using it for far longer:

Take the Golden Ratio and make a triangle where, if the base equals 1 then the long sides are 1.6180. A Golden Triangle, Now let's play by putting them together...










And again you can join the dots to find a snail shell:

Now watch this short animated video by Cristóbal Vila which explains things in a beautiful way.

Nature by Numbers [3:44]


Oh! I promised rabbits, here you go...

Fibonacci with RABBITS!!!! [1:57]

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*Avoiding division by zero (1÷0 is not infinity, it is just not a number, it's undefined and so it's the worst approximation you can possibly get)