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An octahedron with a square pseudo-face highlighted

A pseudo-element of a polytope P  is a polytope whose lower-dimensional elements are all elements of P , but which is not an element of P . For example, the octahedron has three square pseudo-faces, while the tetrahemihexahedron (which has the same edges) has the same squares as faces. However, four of the octahedron's faces are pseudo-faces of the tetrahemihexahedron.

Terminology[edit | edit source]

The term pseudo-element refers to polytopes of non-specific rank. For referring to psuedo-elements of a specific rank the same terminology used for elements is used, but with the prefix of "pseudo-". For example rank 2 pseudo-elements can be called pseudo-faces, and rank n-1  pseudo-elements can be called pseudo-facets.

Pseudo-elements of a particular polytope may also be referred to by the polytope name prefixed with "pseudo-". For example a pseudo-face that is a square may be called a "pseudo-square", or a pseudo-cell that is a tetrahemihexahedron may be called a "pseudo-tetrahemihexahedron".

Special classes of Pseudo-elements[edit | edit source]

There are several specific classes of psuedo-elements of note.

Holes[edit | edit source]

Am icosahedron with a pentagonal hole highlighted

Holes, and more generally k -holes, are a particular type of pseudo-face. On a polyhedron, a hole is a sequence of edges obtained by starting at a vertex following the second-leftmost edge at every vertex until returning to the start.

Holes are not strictly pseudo-faces, since in some cases they can be true faces. For example the holes of the cube are its faces. For a regular polyhedron, k -holes are pseudo-faces iff where d  is the vertex degree of the polyhedron.

Holes are relevant in discussions of abstract regular polytopes and regular skew polytopes. The can be used in the construction of polytopes such as the mucube and muoctahedron, and they are related to the facetting operation.

Petrie polygons[edit | edit source]

Petrie polygons are another type of psuedo-face relevant to regular abstract and skew polytopes. The Petrie polygons of a regular polyhedron are walks of its edges such that every pair of two consecutive edges are are mutually incident on a single face, but no three consecutive edges are mutually incident on a single face.

Unlike holes, Petrie polygons are planar only if the polyhedron is skew. This means that in context where planarity is mandated Petrie polygons are not likely to be relevant.