This image of a hyperbolic 3 - manifold was created with the mathematical visualisation software Cuved Spaces.
The mathematician who wrote the programme, Jeff Weeks, calls it "a flight simulator for multiconnected universes".
The Klein bottle is a one-sided surface - like the wellknown Möbius band - but is even more fascinating, since it is closed and has no border and neither an enclosed interior nor exterior.
The simplest kaleidocycle is a ring of an even number of tetrahedra. (Tetrahedra are pyramids with equal edges and equilateral triangles as sides). The tetrahedra are connected on perpendicular edges.
What makes it special is that you can continually twist it inwards or outwards while it shows different sides of each tetrahedron.
The Mandelbrot set, that wonderfully intricate little fractal. All that beauty, an infinitely detailed interface between what is in the set and what is excluded, is created by a very simple little algorithm.