Example 1: The Mobius Strip

The Möbius stip 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 parameterically f(s,t) as follows:

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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 realised in 4 dimensions. The classical representation is shown below. The Klein Bottle can be represented parameterically f(u, v) as follows:

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Paul Bourke
Paul Bourke

Example 3: Apple

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

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Paul Bourke
Paul Bourke

Example 4: The Triaxial Tritorus

The Triaxial Tritorus is defined parameterically as follows:

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Paul Bourke
Paul Bourke
Paul Bourke