Example 1: The Möbius Strip

The Möbius strip is the simplest geometric shape which has only one surface and only one edge. It can be created by taking a strip of paper, giving it a half twist along its long axis, and then joining the two narrow ends together. The Möbius strip in 3 dimensions can be represented parametrically f(s,t) as follows:


Paul Bourke


Paul Bourke


M.C. Escher


M.C. Escher


M.C. Escher



Example 2: The Klein Bottle

Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is not able to be constructed in 3 dimensions without intersecting surfaces. It can be realized in 4 dimensions. The classical representation is shown below. The Klein Bottle can be represented parametrically f(u, v) as follows:


Paul Bourke


Paul Bourke



Example 3: Apple

The Apple Surface can be represented parametrically f(u,v) as follows:


Paul Bourke


Paul Bourke



Example 4: The Triaxial Tritorus

The Triaxial Tritorus is defined parametrically as follows:


Paul Bourke


Paul Bourke


Paul Bourke