geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

regular polyhedra A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different eq ...

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

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentregular skew apeirohedra.

History

According to

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

, in 1926

John Flinders Petrie John Flinders Petrie (26 April 1907 – 1972) was an English mathematician. Petrie was the great grandson of the explorer and navigator, Matthew Flinders. He met the geometer Harold Scott MacDonald Coxeter as a student, beginning a lifelong fri ...

generalized the concept of regular skew polygons (nonplanar polygons) to ''regular skew polyhedra''. Coxeter offered a modified

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

for these figures, with implying the

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

, -gons around a vertex, and -gonal holes. Their vertex figures are

skew polygon In geometry, a skew polygon is a closed polygonal chain in Euclidean space. It is a figure (geometry), figure similar to a polygon except its Vertex (geometry), vertices are not all coplanarity, coplanar. While a polygon is ordinarily defined a ...

s, zig-zagging between two planes. The regular skew polyhedra, represented by , follow this equation: :

2 \cos \frac \cos \frac = \cos \frac

A first set , repeats the five convex

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s, and one nonconvex Kepler–Poinsot solid: :

Finite regular skew polyhedra

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

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 honeycomb In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells. Twenty-eight such honey ...

s, the finite forms all represent manifold surfaces within the cells of the uniform 4-polytopes. Polyhedra of the form are related to

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 ref ...

symmetry of p,r,q,r) which reduces to the linear ,p,rwhen q is 2. Coxeter gives these symmetry as nowiki/>[(''p'',''r'',''q'',''r'')sup>+">''p'',''r'',''q'',''r'').html" ;"title="nowiki/>[(''p'',''r'',''q'',''r'')">nowiki/>[(''p'',''r'',''q'',''r'')sup>+which he says is isomorphic to his abstract group (2''p'',2''q'', 2,''r''). The related honeycomb has the extended symmetry [[(''p'',''r'',''q'',''r''). is represented by the faces of the Bitruncation, bitruncated

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

, and is represented by square faces of the runcinated . produces a ''n''-''n''

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

, and specifically fits inside of a x

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

.

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 _2''q'', and group G^{''l'',''m'',''q''}.

Higher dimensions

Regular skew polyhedra can also be constructed in dimensions higher than 4 as

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...

s 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.

Notes

References

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

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

, ''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*

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

, ''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. *Garner, C. W. L. ''Regular Skew Polyhedra in Hyperbolic Three-Space.'' Can. J. Math. 19, 1179-1186, 1967. * E. Schulte, J.M. Will
On Coxeter's regular skew polyhedra
Discrete Mathematics, Volume 60, June–July 1986, Pages 253–262 Polyhedra

History

Finite regular skew polyhedra

Higher dimensions

See also

Notes

References