Regular skew polyhedron


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.
Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra.

History

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons to regular skew polyhedra.
Coxeter offered a modified Schläfli symbol for these figures, with implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by, follow this equation:
A first set, repeats the five convex Platonic solids, and one nonconvex Kepler-Poinsot solid:
FacesEdgesVerticespPolyhedronSymmetry
order
= 4640Tetrahedron12
= 81260Octahedron24
= 61280Cube24
= 2030120Icosahedron60
= 1230200Dodecahedron60
= 1230124Great dodecahedron60

Finite regular skew polyhedra of 4-space

also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".
Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform polychora.
Polyhedra of the form are related to Coxeter group symmetry of , which reduces to the linear when q is 2. Coxeter gives these symmetry as +] which he says is isomorphic to his abstract group. The related honeycomb has the extended symmetry +]93-3 duoprism1632161D4xD44,2,4]+]164-4 duoprism or tesseract2550251D5xD55,2,5]+]255-5 duoprism3672361D6xD66,2,6]+]366-6 duoprismn22n2n21DnxDnn,2,n]+]n2n-n duoprism3060206S53,3,3]+]60Runcinated 5-cell2060306S53,3,3]+]60Bitruncated 5-cell288576144733,4,3]+]576Runcinated 24-cell144576288733,4,3]+]576Bitruncated 24-cell
FacesEdgesVerticespStructureSymmetryOrderRelated uniform polychora
901807210A65/2,5,5/2]+]360Runcinated grand stellated 120-cell
721809010A65/2,5,5/2]+]360Bitruncated grand stellated 120-cell

FacesEdgesVerticespStructureOrder
4080325?160
3280405?160
42842410LF168
24844210LF168
721807219A6360
182546156105LF1092
156546182105LF1092
156546156118LF1092
6121224272171LF2448
2721224612171LF2448
1536537613441249?10752
1344537615361249?10752

A final set is based on Coxeter's further extended form or with q2 unspecified:. These can also be represented a regular finite map or 2q, and group Gl,m,q.
or 2qFacesEdgesVerticespStructureOrderNotes
= 2q2q23q2q21G3,6,2q2q2
= 62q23q23q*/2G3,6,2q2q2
= 85684243LF168
= 8112168428PGL336Related to complex polyhedron 4,
= 6841685615PGL336Related to complex polyhedron ,
= 1236454615614LF1092
= 1436454615614LF1092
= 10720108027046G3,8,102160Related to complex polyhedron 5,
= 8720108021673G3,8,102160Related to complex polyhedron 4,
= 4122483S4×S248
= 42460209A5×S2120
= 820243036552231LF6072
= 16358453761536129G3,7,1710752
= 10121801827040601016LF×A336540

Higher dimensions

Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew icosahedron by H. S. M. Coxeter. The dodecahedron can be similarly embedded into the 10-demicube.