geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

or skew regular

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

History

According to

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, in 1926

John Flinders Petrie In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

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

s, 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

, ''m'' l-gons around a vertex, and ''n''-gonal holes. Their vertex figures are

skew polygon 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 ...

s, 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 ...

named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron. #Mucube: : 6

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

s about each vertex (related to

cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a re ...

, 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

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

s 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 The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is c ...

, constructed by

truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...

cells, removing triangle faces, and linking sets of four around a faceless

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

.) Coxeter gives these regular skew apeirohedra with extended chiral symmetry nowiki/>[(''p'',''q'',''p'',''r'')sup>+.html" ;"title="''p'',''q'',''p'',''r'').html" ;"title="nowiki/>[(''p'',''q'',''p'',''r'')">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

s, 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 group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

s 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 In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

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

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ,'' *

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

''Four-Dimensional Regular Polyhedra''
Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355–387 *

, ''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 ''Scripta Mathematica'' was a quarterly journal published by Yeshiva University devoted to the philosophy, history, and expository treatment of mathematics. It was said to be, at its time, "the only mathematical magazine in the world edited by spe ...

'' 6 (1939) 240–244. ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559–591*

, ''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