In
hyperbolic geometry, 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
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
. In 3-dimensional
hyperbolic space there are nine
Coxeter group families of compact
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, generated as
Wythoff constructions, and represented by
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of
rings of the
Coxeter diagrams for each family.
Hyperbolic uniform honeycomb families
Honeycombs are divided between compact and paracompact forms defined by
Coxeter groups, 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
Coxeter groups are listed here with their
Coxeter diagrams,
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:
,31,1↔
,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
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
↔ , which can be extended by restoring one mirror as . The other is
,(3,5)* index 120 with a
dodecahedral 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 or
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 ...
, 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 domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s (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
Coxeter group:
,5,3or
One related
non-wythoffian 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.
[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 Coxeter group: ,3,4or .
This family is related to the group ,31,1by 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 Coxeter group: ,3,5or
The bitruncated and runcinated forms (29 and 30) 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 .
,31,1family
There are 11 forms (and only 4 not shared with ,3,4family), generated by ring permutations of the Coxeter group: ,31,1or . 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 Coxeter group:
The bitruncated and runcinated forms (41 and 42) 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 .
5,3,3,3)family
There are 9 forms, generated by ring permutations of the Coxeter group:
The bitruncated and runcinated forms (50 and 51) 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 .
4,3,4,3)family
There are 6 forms, generated by ring permutations of the Coxeter group: . 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 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
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
.
The truncated forms (57 and 58) 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 .
4,3,5,3)family
There are 9 forms, generated by ring permutations of the Coxeter group:
The truncated forms (65 and 66) 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 .
5,3,5,3)family
There are 6 forms, generated by ring permutations of the Coxeter group: . 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 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 .
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 .[
Another two known ones are related to ''noncompact'' families. The tessellation consists of truncated cubes and infinite ]order-8 triangular tiling
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of ', having eight regular triangles around each vertex.
Uniform colorings
The half symmetry +,8,3= 4,3,3)can be show ...
s . 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 ...
.)order-8 triangular tiling
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of ', having eight regular triangles around each vertex.
Uniform colorings
The half symmetry +,8,3= 4,3,3)can be show ...
s , 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 unif ...
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.[ 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.
See also
* Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
* List of regular polytopes#Tessellations of hyperbolic 3-space
Notes
References
* James E. Humphreys
James Edward Humphreys (December 10, 1939 – August 27, 2020) was an American mathematician, who worked in algebraic groups, Lie groups, and Lie algebras and applications of these mathematical structures. He is known as the author of several m ...
, ''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, '' Regular Polytopes'', 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/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. 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. ...
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)