AirToob     From the page:

In "Nature by Numbers," filmmaker Cristóbal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci sequence. Then it moves on to the Golden and Angle Ratios, the Delaunay Triangulation and Voronoi Tessellations. This would be math-class gold, and it's awfully sweet even if math class is years behind you.

If you are fascinated by this brilliant animation (which I found on Ian's pages), here is a quick guide to finding out more (should you need or want it):

The Fibonacci Sequence is formed by starting with 0,1 and adding the two numbers together to get a new number (giving 0,1,1). If you keep doing this using the last 2 numbers in the sequence then you get 0,1,1,2,3,5,8,13,21... Fibonacci thought up this sequence (in the year 1202) as described in Fibonacci's Rabbits, which is well worth reading, as is all the other information that you will find there. Two numbers "a" and "b" are said to be in the Golden Ratio if the ratio of "a" to "b" is the same as the ratio of "a+b" to "a". You can't write this ratio exactly as a decimal number, but it is approximately 1.618.

The Golden Ratio, referred to as phi (greek letter), is approximated ever more closely by the ratio of any two successive numbers in the Fibonacci Sequence, as the sequence is taken further and further. From the sequence above you can see that 13/8 is an approximation of the ratio, but 21/13 is a better approximation (and so it goes on).

Spookily, you can start a sequence with any two numbers, e.g. (235,1) and extend it using the same rules as for the Fibonacci Sequence, and the ratio between the last two numbers will still get closer and closer to the Golden Ratio as the sequence extends. In other words, it is the rules, rather than the starting numbers, that matter.

The Golden Section is a line segment divided according to the Golden Ratio.

The Golden Angle is what you get if you divide a circle's circumference according to the Golden Section, draw two lines from the centre of the circle to where the circumference is split, and take the smaller angle between those two lines. It is about 137.5 degrees. You will see it appearing in the animation.

I can't tell you much about Delaunay Triangulations and Voronoi Tessellations (not having met them before), except that they are reflected in the way in which Nature packs things together. This amazing video has certainly inspired me to try to find out more.

If you like this...

[A great post on the Fibonacci series from laydgray]
[... and you might like the mind-stretchers tag at the top of this post] 