In geometry, a regular skew apeirohedron is an infinite

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 ...

, with either

skew Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not ...

regular faces or skew regular vertex figures.

History

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of

regular skew polygon The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrumen ...

s (nonplanar polygons) to finite regular skew polyhedra in 4-dimensions, and infinite regular skew apeirohedra in 3-dimensions (described here). Coxeter identified 3 forms, with planar faces and skew vertex figures, two are complements of each other. They are all named with a modified

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

, where there are ''l''-gonal faces, ''m'' faces around each vertex, with ''holes'' identified as ''n''-gonal missing faces. Coxeter offered a modified

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: * 2 sin(/''l'') · sin(/''m'') = cos(/''n'')

Regular skew apeirohedra of Euclidean 3-space

The three Euclidean solutions in 3-space are , , and .

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...

named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron. #Mucube: : 6 squares about each vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

.) #Muoctahedron: : 4 hexagons about each vertex (related to

bitruncated cubic honeycomb The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of ...

, constructed by

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

with their square faces removed and linking hole pairs of holes together.) #Mutetrahedron: : 6 hexagons about each vertex (related to quarter cubic honeycomb, constructed by truncated tetrahedron cells, removing triangle faces, and linking sets of four around a faceless tetrahedron.) Coxeter gives these regular skew apeirohedra with extended chiral symmetry nowiki/>[(''p'',''q'',''p'',''r'')sup>+">''p'',''q'',''p'',''r'').html" ;"title="nowiki/>[(''p'',''q'',''p'',''r'')">nowiki/>[(''p'',''q'',''p'',''r'')sup>+which he says is isomorphic to his abstract group (2''q'',2''r'', 2,''p''). The related honeycomb has the extended symmetry [[(''p'',''q'',''p'',''r'').

Regular skew apeirohedra in hyperbolic 3-space

In 1967, C. W. L. Garner identified 31 hyperbolic skew apeirohedra with

vertex figures, found in a similar search to the 3 above from Euclidean space.Garner, C. W. L. ''Regular Skew Polyhedra in Hyperbolic Three-Space.'' Can. J. Math. 19, 1179–1186, 1967

Note: His paper says there are 32, but one is self-dual, leaving 31. These represent 14 compact and 17 paracompact regular skew polyhedra in hyperbolic space, constructed from the symmetry of a subset of linear and cyclic Coxeter groups graphs of the form (''p'',''q'',''p'',''r''), These define ''regular skew polyhedra'' and dual . For the special case of linear graph groups ''r'' = 2, this represents the Coxeter group 'p'',''q'',''p'' It generates regular skews and . All of these exist as a subset of faces of the

convex uniform honeycombs in hyperbolic space In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wyt ...

. The skew apeirohedron shares the same antiprism vertex figure with the honeycomb, but only the zig-zag edge faces of the vertex figure are realized, while the other faces make "holes".

References

Petrie–Coxeter Maps Revisited

PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

, Isabel Hubard, Egon Schulte, Asia Ivic Weiss, 2005 * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ,'' * Peter McMullen
''Four-Dimensional Regular Polyhedra''
Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355–387 * Coxeter, ''Regular Polytopes'', Third edition, (1973), Dover edition, *''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", ''Scripta Mathematica'' 6 (1939) 240–244. ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559–591] * Coxeter, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn, 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.) **Coxeter, H. S. M. ''Regular Skew Polyhedra in Three and Four Dimensions.'' Proc. London Math. Soc. 43, 33–62, 1937. Polyhedra

History

Regular skew apeirohedra of Euclidean 3-space

Regular skew apeirohedra in hyperbolic 3-space

See also

References