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In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
These finite regular skew polyhedra in 4-space can be seen as a subset of the faces of uniform 4-polytopes. They have planar regular polygon faces, but regular skew polygon vertex figures . Two dual solutions are related to the 5-cell , two dual solutions are related to the 24-cell , and an infinite set of self-dual duoprisms generate regular ...
Gott called the full set of regular polyhedra, regular tilings, and regular pseudopolyhedra as regular generalized polyhedra, representable by a {p,q} Schläfli symbol, with by p-gonal faces, q around each vertex. However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage.
In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra.While apeirohedra are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure.
Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures. For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes.
In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges.
The Petrie polygon of a regular polyhedron can be defined as the skew polygon (whose vertices do not all lie in the same plane) such that every two consecutive sides (but not three) belong to one of the faces of the polyhedron. Each finite regular polyhedron can be orthogonally projected onto a plane so that the Petrie polygon becomes a regular ...
The regular skew polyhedron, {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices.