Uniform Honeycombs In Hyperbolic Space
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In
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
, a uniform honeycomb in hyperbolic space is a uniform
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
of uniform polyhedral
cells Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a d ...
. 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 symme ...
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 ref ...
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 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
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s of rings of the
Coxeter diagram 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 ...
s for each family.


Hyperbolic uniform honeycomb families

Honeycombs are divided between compact and paracompact forms defined by
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 ...
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
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 ...
s are listed here with their
Coxeter diagram 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 ...
s, in order of the relative volumes of their fundamental simplex domains.Felikson, 2002 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 polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the ''trig ...
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 ...
↔ , which can be extended by restoring one mirror as . The other is ,(3,5)* index 120 with a
dodecahedral In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular s ...
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 general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
, including ideal vertices at infinity. Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these
triangular bipyramid A triangular bipyramid is a hexahedron with six triangular faces constructed by attaching two tetrahedra face-to-face. The same shape is also known as a triangular dipyramid or trigonal bipyramid. If these tetrahedra are regular, all faces of a t ...
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 ...
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 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 ...
: ,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 a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are arccos(−) = 109.4712206...° ≈ 109.5° when all four substituents are ...
.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 polyhedr ...
s: and .


,3,4family

There are 15 forms, generated by ring permutations of the
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 ...
: ,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 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 ...
: ,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 polyhedr ...
s: and .


,31,1family

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


5,3,3,3)family

There are 9 forms, generated by ring permutations of the
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 ...
: 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 polyhedr ...
s: and .


4,3,4,3)family

There are 6 forms, generated by ring permutations of the
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 ...
: . 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 polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the ''trig ...
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 ...
. 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 polyhedr ...
s: and .


4,3,5,3)family

There are 9 forms, generated by ring permutations of the
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 ...
: 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 polyhedr ...
s: and .


5,3,5,3)family

There are 6 forms, generated by ring permutations of the
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 ...
: . 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 polyhedr ...
s: and .


Other non-Wythoffians

There are infinitely many known non-Wythoffian uniform compact hyperbolic honeycombs, and there may be more undiscovered ones. 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. The case ''p'' = 2 degenerates to the Euclidean
cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...
. Another four 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 triangle (geometry), triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triak ...
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 tilings . 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 or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...
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. Two more such constructions were discovered in 2023. The first one arises from the fact that and have the same circumradius; the former has
truncated octahedra 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 hexagons and 6 squares), 36 ...
and order-6 square tilings , while the latter has cuboctahedra and order-6 square tilings . A compact uniform honeycomb is taken by discarding the order-6 square tilings they have in common, using only the truncated octahedra and cuboctahedra. The second one arises from a similar construction involving (which has small rhombicosidodecahedra ,
octahedra In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
, and
order-4 pentagonal tiling In geometry, the order-4 pentagonal tiling is a List_of_regular_polytopes#Hyperbolic_tilings, regular tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored q ...
s ) and (which is the prism of the order-4 pentagonal tiling, having
pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...
s and order-4 pentagonal tilings ). These two likewise have the same circumradius, and a compact uniform honeycomb is taken by using only the finite cells of both, discarding the order-4 pentagonal tilings they have in common. 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 (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
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. Kepler first named it in Latin as ''cubus simus'' in 1619 in his Harmonices Mundi. ...
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 (with a snub cube or snub dodecahedral vertex figure respectively), paracompact for p=6 (with a snub trihexagonal tiling as the vertex figure), and hypercompact for p>6. Again, the truncated and rectified versions of these honeycombs are still uniform. There are also other forms based on
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equiva ...
domains. Two known forms generalise the
cubic-octahedral honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb (geometry), honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has ...
, having distorted small rhombicuboctahedral vertex figures. One form has small rhombicuboctahedra, cuboctahedra, and cubes; another has small rhombicosidodecahedra, icosidodecahedra, and cubes. (The version with tetrahedral-symmetry polyhedra is the cubic-octahedral honeycomb, using cuboctahedra, octahedra, and cubes).


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 Face (geometry), faces and is vertex-transitive (Tran ...
* List of regular polytopes#Tessellations of hyperbolic 3-space *
Paracompact uniform honeycombs In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, ...


Notes


References

* J. Humphreys (1990), ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 *
H.S.M. 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 ...
(1954)
"Regular Honeycombs in Hyperbolic Space"
''Proceedings of the International Congress of Mathematicians'', vol. 3, North-Holland, pp. 155–169. Reprinted as Ch. 10 in Coxeter (1999), ''The Beauty of Geometry: Twelve Essays'', Dover, *
H.S.M. 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 ...
(1973), ''
Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) * J. Weeks ''The Shape of Space'', 2nd ed. , Chapters 16–17: Geometries on Three-manifolds I, II * A. Felikson (2002)
"Coxeter Decompositions of Hyperbolic Tetrahedra"
(preprint) * 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 Inc., Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, computer hardware, ...
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* Norman Johnson (mathematician), N. W. Johnson (2018), ''Geometries and Transformations'', Chapters 11–13 * N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz (1999), ''The size of a hyperbolic Coxeter simplex'', Transformation Groups, 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)