## Spirals

The starting square has side 1. Another side 1 square appears, and then a side 2 square is added across the top of them. Then a square of side 3 appears to the left, and a sqare of side 5 appears underneath.

The sequence of the sides of the squares is like this…
1, 1, 2, 3, 5,...
and that is the Fibonacci series! The next term is obtained by adding the two previous terms.

If you reproduce the construction on some graph paper you can make a spiral curve through the squares – the kind of spiral you get is called the equiangular spiral or the logarithmic spiral.

The equiangular bit comes from the property that at any point on the spiral, a line drawn from the point to the centre of the spiral will make the same angle to the path of the spiral. In the case of the spiral generated from the square construction above, that angle will by 90¬?. If you used a rectangle as the starting shape, you could change the angle… How would the ratio of the sides of the rectangle change the angle of the spiral?

Fibonacci Numbers and the Golden Section by Ron Knott
A very full and rewarding set of pages. The Fibonacci numbers and the golden section do seem to pop up in a wide variety of places in the natural world. I supose that this is because things grow in stages and the extra growth depends on the current size of the organism.
Equiangular spiral from St Andrews
St Andrews University has a large project around the history of Maths. A small part of this huge resource includes pages about common curves and graphs. The Equiangular Spiral page gives you the basics including the polar equation for the spiral and provides links to more information, including…
Equiangular Spiral by Xah Lee
Full page with some mathematical definitions, illustrations of equiangular spirals that have angles different from 90° and pictures of spirals in nature including a very nice looking kind of cauliflower. Xah also has a page on the Archimedian spiral – much more common in architecture. This page is part of Xah’s ambitious looking site called A Visual Dictionary of Famous Plane Curves.