hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

, a uniform honeycomb in hyperbolic space is a uniform

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

of uniform polyhedral cells. In 3-dimensional

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...

there are nine

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 ...

families of compact

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...

s, generated as

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

s, and represented by permutations of

rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

of the

Coxeter diagram 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 t ...

s for each family.

Hyperbolic uniform honeycomb families

Honeycombs are divided between compact and paracompact forms defined by

s, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups.

Compact uniform honeycomb families

The nine compact

s are listed here with their

s, in order of the relative volumes of their fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. Two known examples are cited with the family below. Only two families are related as a mirror-removal halving: ,3^1,1↔ ,3,4,1⁺ There are just two radical subgroups with non-simplicial domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is 4,3,4,3^*) represented by Coxeter diagrams an index 6 subgroup with a

trigonal trapezohedron In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...

fundamental domain ↔ , which can be extended by restoring one mirror as . The other is ,(3,5)^* index 120 with a

dodecahedral In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

fundamental domain.

Paracompact hyperbolic uniform honeycombs

There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

, including ideal vertices at infinity. Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these

triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...

fundamental domains (double tetrahedra) as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, with the constraint that at least one node must be ringed across infinite order branches.

,5,3family

There are 9 forms, generated by ring permutations of the

: ,5,3or One related

non-wythoffian In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

form is constructed from the vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a

tetrahedrally diminished dodecahedron In geometry, a tetrahedrally diminished dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical ...

.Wendy Y. Krieger, Walls and bridges: The view from six dimensions, ''Symmetry: Culture and Science'' Volume 16, Number 2, pages 171–192 (2005

/ref> Another is constructed with 2 antipodal vertices removed. The bitruncated and runcinated forms (5 and 6) contain the faces of two

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 ...

s: and .

,3,4family

There are 15 forms, generated by ring permutations of the

: ,3,4or . This family is related to the group ,3^1,1by a half symmetry ,3,4,1⁺ or ↔ , when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔ .

,3,5family

There are 9 forms, generated by ring permutations of the

: ,3,5or The bitruncated and runcinated forms (29 and 30) contain the faces of two

s: and .

,3^1,1family

There are 11 forms (and only 4 not shared with ,3,4family), generated by ring permutations of the

: ,3^1,1or . If the branch ring states match, an extended symmetry can double into the ,3,4family, ↔ .

4,3,3,3)family

There are 9 forms, generated by ring permutations of the

: The bitruncated and runcinated forms (41 and 42) contain the faces of two

s: and .

5,3,3,3)family

There are 9 forms, generated by ring permutations of the

: The bitruncated and runcinated forms (50 and 51) contain the faces of two

s: and .

4,3,4,3)family

There are 6 forms, generated by ring permutations of the

: . There are 4 extended symmetries possible based on the symmetry of the rings: , , , and . This symmetry family is also related to a radical subgroup, index 6, ↔ , constructed by 4,3,4,3^*) and represents a

fundamental domain. The truncated forms (57 and 58) contain the faces of two

s: and .

4,3,5,3)family

There are 9 forms, generated by ring permutations of the

: The truncated forms (65 and 66) contain the faces of two

s: and .

5,3,5,3)family

There are 6 forms, generated by ring permutations of the

: . There are 4 extended symmetries possible based on the symmetry of the rings: , , , and . The truncated forms (72 and 73) contain the faces of two

s: and .

Other non-Wythoffians

There are several other known non-Wythoffian uniform compact hyperbolic honeycombs, and it is not known how many are left to be discovered. Two have been listed above as diminishings of the icosahedral honeycomb . In 1997 Wendy Krieger discovered an infinite series of uniform hyperbolic honeycombs with pseudoicosahedral vertex figures, made from 8 cubes and 12 ''p''-gonal prisms at a vertex for any integer ''p''. In the case ''p'' = 4, all cells are cubes and the result is the order-5 cubic honeycomb. Another two known ones are related to ''noncompact'' families. The tessellation consists of

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

s and infinite order-8 triangular tilings . However the latter intersect the sphere at infinity orthogonally, having exactly the same curvature as the hyperbolic space, and can be replaced by mirror images of the remainder of the tessellation, resulting in a ''compact'' uniform honeycomb consisting only of the truncated cubes. (So they are analogous to the hemi-faces of spherical

hemipolyhedra In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other p ...

.) Something similar can be done with the tessellation consisting of small rhombicuboctahedra , infinite order-8 triangular tilings , and infinite

order-8 square tiling In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around eve ...

s . The order-8 square tilings already intersect the sphere at infinity orthogonally, and if the order-8 triangular tilings are augmented with a set of

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...

s, the surface passing through their centre points also intersects the sphere at infinity orthogonally. After replacing with mirror images, the result is a compact honeycomb containing the small rhombicuboctahedra and the triangular prisms. Another non-Wythoffian was discovered in 2021. It has as vertex figure a snub cube with 8 vertices removed and contains two

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...

and eight

snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...

s at each vertex. Subsequently Krieger found a non-Wythoffian with a snub cube as the vertex figure, containing 32 tetrahedra and 6 octahedra at each vertex, and that the truncated and rectified versions of this honeycomb are still uniform. In 2022, Richard Klitzing generalised this construction to use any snub as vertex figure: the result is compact for p=4 or 5, paracompact for p=6, and hypercompact for p>6.

Summary enumeration of compact uniform honeycombs

This is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform.

Notes

References

* James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990) * ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, , (Chapter 10
Regular Honeycombs in Hyperbolic Space
*

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 ...

, ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) * Jeffrey R. Weeks ''The Shape of Space, 2nd edition'' (Chapters 16–17: Geometries on Three-manifolds I,II

Coxeter Decompositions of Hyperbolic Tetrahedra

arXiv arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...

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. ...

, A. Felikson, December 2002 * C. W. L. Garner, ''Regular Skew Polyhedra in Hyperbolic Three-Space'' Can. J. Math. 19, 1179–1186, 1967.

br>
* Norman Johnson (mathematician), Norman Johnson, ''Geometries and Transformations'' (2018), Chapters 11,12,13 *N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups 1999, Volume 4, Issue 4, pp 329–35

* N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, ''Commensurability classes of hyperbolic Coxeter groups'' H³: p130

* {{KlitzingPolytopes, hyperbolic.htm#3D-compact, Hyperbolic honeycombs, H3 compact Honeycombs (geometry)

Hyperbolic uniform honeycomb families

Compact uniform honeycomb families

Paracompact hyperbolic uniform honeycombs

,5,3family

,3,4family

,3,5family

,31,1family

4,3,3,3)family

5,3,3,3)family

4,3,4,3)family

4,3,5,3)family

5,3,5,3)family

Other non-Wythoffians

Summary enumeration of compact uniform honeycombs

See also

Notes

References

,3^1,1family