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A simplex is the X-dimensional form of the line segment - triangle - tetrahedron series, which is formed by beginning with a point and having the next shape in the sequence be a pyramid with the base of the previous shape. For example, a tetrahedron is a triangle-based pyramid.

The schläfli symbol of an X-simplex is $ \{ { 3 }^{ X-1 }\} $ - for example, the schläfli symbol of a pentachoron (4-simplex) is $ \{ { 3 },3,3\} $.

All simplices are self-dual.

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