# Polytope of full rank

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A polytope is of **full rank** if its dimension is maximal for its rank. Specifically a finite non-degenerate polytope is full rank iff its dimension is equal to its rank, and a non-degenerate apeirotope is full rank iff its dimension is one less than its rank. Classical senses of the word polytope (convex polytope, star polytope etc.) are full rank. The term full rank is used in the context of skew polytopes, since polytopes which are not of full rank are necessarily skew.

The regular and chiral polytopes of full rank have been fully enumerated.

## Examples[edit | edit source]

- The mucube is not of full rank since its rank (3) is equal to its dimension (3), and it has an infinite number of edges.
- The square duocomb is not of full rank since its rank (3) is less than its dimension (4).
- The Petrial tesseract is of full rank since its rank (4) is equal to its dimension (4) and it has a finite number of flags.
- The apeir tetrahedron is full rank since its rank (4) is one greater than its dimension (3) and it has an infinite number of flags.

## Properties[edit | edit source]

- Polytopes of full rank are always pure.

## Bibliography[edit | edit source]

- McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF).
*Discrete Computational Geometry*.

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