The Mandelbrot Set of complex numbers is defined by the recurrence relation zn+1=zn²+z0 (the set contains all complex numbers z0 that don't cause the the orbit of zn to diverge). This is a close-up of the fractal boundary (one of the most famous fractals ever) of the set in the complex plane.

Approximations of Fractals can be found everywhere in the natural world, particularly in the growth of plants (plants like ferns, etc., often grow recursively, but can't achieve infinite complexity). This is the Romanesco Broccoli, whose top is self-similar- as you zoom in, you see scaled-down versions of the original broccoli. As well as this, its spirals are logarithmic and linked to the Fibonacci numbers. 

The Koch snowflake was one of the first fractals to be mathematically described and is defined in a simple recursive manner, starting with an equilateral triangle. At each step, its perimeter increases by a factor of 4/3, meaning the final snowflake has infinite perimeter. It also implies that the shape is log3(4) = 1.2619-dimensional!

A Roulette is the path traced out by a point on a curve as that curve slides along a second fixed curve. The concept sub-divides into a number of categories like cycloids and trochoids, each of which uses certain curves in certain ways. On the left is an epicycloid, where a small circle rolls around the circumference of a larger. A Glissette is the path of a point, or envelope of a curve, attached to another curve that slides along two other fixed objects.

The Lissajous Curves are the family of curves defined by the parametric equations x = Asin(at+δ), y = Bcos(bt). These curves can be extremely complex in appearance and are very beautiful (see the pictures on the left). They are sometimes used as the centrepieces of logos, as in the Australian Broadcasting Corporation (ABC) logo.

A Rose (or Rhodonea Curve) is a sinusoidal function plotted in polar co-ordinates in 2-D, with general equation r = Acos(kθ).  The appearance of the graph depends on the value of k, and is closed if k is rational, but open and so infinitely long if not. Roses are closely related to Roulettes.

Parametric Surfaces are just the 3-D analogue of parametric curves: here, the x-, y- and z-coordinates of the surface are functions of two parameters. This can produce interesting results, like hyperbolic paraboloids, pseudospheres and cross-caps. To the left is the figure-8 immersion of the Klein bottle in 3-D.

If a Spidron is cut out of paper and "bunched up" in the right way, a 3-D shape like the one on the left will result (it doesn't have to have a hexagonal outline- any regular polygon works). This can easily be done at home- click here for Spidron templates, or simply draw one yourself.

"Spidronising" a polyhedron simply means putting a Spidron on each of its flat surfaces. Obviously, the shape of the Spidron must match the face to which it's being attached- the "Dodeca-Spidroball" on the left is based on a dodecahedron (12 pentagonal faces), and so each Spidron has five arms. Spidronised solids aren't of much practical use, but look really nice.

A Sphidron is like a "smooth" Spidron.