Wednesday, 14 April 2010

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]

--------------------------
*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)

5 comments:

Duncanr said...

fascinating post, Andy

P.S. like your revolving blog roll on the sidebar

Andy Holroyd said...

Glad to please Duncan.

The rolling list shows fellow members of the Atheist Blogroll. I got the widgit from there.

jill said...

yes, I agree...very interesting, but over my head! :-)

dave hambidge said...

I need to be fully awake Andy but I think I got the gist of it, maybe! Rolling sidepanel is very neat BTW

Andy Holroyd said...

Jill, Dave,

There seems to be an underlying symmetry in the world around us. Often we ask "why?" when another question is the simpler "how?". Understanding 'how' can sometimes illuminate the 'why', and then it all becomes clearer. Honest :)