Shape (EntityClass, 2)

(Redirected from Uniform polytope)

A shape is any geometric entity. It is the most general term; there are many subsets of shapes. Common subsets are summarized below, with the point being the simplest element of each set.

Polytope

Main article: Polytope

Powertope

Main article: Powertope

A powertope is a shape which can be written as a shape raised to the power of another shape. Technically, all shapes would be powertopes since they can all be written as themselves raised to the power of digon, so in order to be labeled a powertope, both the base and exponent shapes must be at least two-dimensional. This of course means that there are no powertopes in prime dimensions.

Rotope

Main article: Rotope

The rotopes are the classic set of shapes enumerated by Alkaline, later extended by others. Unfortunately these extensions have produced a quantity of undesirable "shapes", which pollute the set of rotopes and make it difficult to use. However, it is still to this day the most intuitive construction-based set known of. The set of rotopes includes the hypercube, hypersphere and simplex of each dimension.

Bracketope

Main article: Bracketope

The bracketopes are a set enumerated by Keiji, which was initially an attempt to replace rotopes, but did not fare much better as once again it produced a quantity of undesirable shapes, which in this case were duplicates of legitimate ones. The set of bracketopes includes the hypercube, hypersphere and cross polytope of each dimension.

Convex shape

A convex shape is a shape which is its own convex hull; it has a genus of zero and all digons connecting two points in the shape also lie inside the shape.

Concave shape

A concave shape is a shape which is not convex but also is not self-intersecting.

Glaze

Main article: Glaze

Stellar shape

A stellar shape is a shape which is self-intersecting.