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' ...
, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs 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. ...
. It is ''paracompact'' because it has 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 ...
with an infinite number of faces. Each cell is a hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
whose vertices lie on a horosphere
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of ...
: a flat plane in hyperbolic space that approaches a single ideal point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left- limiting parallels to ''l'' through ''P' ...
at infinity.
The 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 mor ...
of the hexagonal tiling honeycomb is . Since that of the hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
of the plane is , this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
is , the 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 lines ...
of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.
Related tilings
The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolicinfinite-order apeirogonal tiling
In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of , which means it has countably infinitely many apeirogons around all its ideal vertices.
Symmetry
This tiling represen ...
, , with infinite apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to ...
al faces, and with all vertices on the ideal surface.
: 
It contains and that tile 2- hypercycle surfaces, which are similar to the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol , having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.
The order-2 a ...
, respectively):
: 
Symmetry
The order-6 hexagonal tiling honeycomb has a half-symmetry construction: .
It also has an index-6 subgroup, ,3*,6 with a non-simplex fundamental domain. This subgroup corresponds to a 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 ...
with six order-3 branches and three infinite-order branches in the shape of a triangular prism: .
Related polytopes and honeycombs
The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space. There are nine uniform honeycombs in the ,3,6Coxeter 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 ...
family, including this regular form.
This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: ↔ .
The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces ( polygons), ...
and honeycombs with triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
vertex figures:
It is also part of a sequence of regular polychora
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces ( polygons), ...
and honeycombs with hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
cells:
It is also part of a sequence of regular polychora
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces ( polygons), ...
and honeycombs with regular deltahedral 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 lines ...
s:
Rectified order-6 hexagonal tiling honeycomb
The rectified order-6 hexagonal tiling honeycomb, t1, hastriangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
and trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
facets, with a hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
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 lines ...
.
it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q, ↔ .
It is analogous to 2D hyperbolic order-4 apeirogonal tiling
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of .
Symmetry
This tiling represents the mirror lines of *2∞ symmetry. It dual to this tiling represents the fundamental domains o ...
, r with infinite apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to ...
al faces, and with all vertices on the ideal surface.
: 
Related honeycombs
The order-6 hexagonal tiling honeycomb is part of a series of honeycombs withhexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
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 lines ...
s:
It is also part of a matrix of 3-dimensional quarter honeycombs: q
Truncated order-6 hexagonal tiling honeycomb
The truncated order-6 hexagonal tiling honeycomb, t0,1, hastriangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
and truncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applies to ...
facets, with a hexagonal pyramid 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 lines ...
.Rotation around 3 fold axis
Bitruncated order-6 hexagonal tiling honeycomb
The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . It containshexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
facets, with a 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 ...
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 lines ...
.
Cantellated order-6 hexagonal tiling honeycomb
The cantellated order-6 hexagonal tiling honeycomb, t0,2, hastrihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
, rhombitrihexagonal tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr.
John Conway calls it a rhombihexadeltille.Conway, 2008, ...
, and hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
cells, with a wedge
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
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 lines ...
.
Cantitruncated order-6 hexagonal tiling honeycomb
The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2, hashexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
, truncated trihexagonal tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''.
Names
Uniform colorings
The ...
, and hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
cells, with a mirrored sphenoid 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 lines ...
.
Runcinated order-6 hexagonal tiling honeycomb
The runcinated order-6 hexagonal tiling honeycomb, t0,3, hashexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
and hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
cells, with a triangular antiprism
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 at ea ...
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 lines ...
.
It is analogous to the 2D hyperbolic rhombihexahexagonal tiling
In geometry, the tetrahexagonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol r.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal ...
, rr, with square and hexagonal faces:
: 
Runcitruncated order-6 hexagonal tiling honeycomb
The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3, hastruncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applies to ...
, rhombitrihexagonal tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr.
John Conway calls it a rhombihexadeltille.Conway, 2008, ...
, hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used t ...
, and dodecagonal prism
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.
If faces are all regular, it is a uniform polyhedron.
Use
It is used in the construction of two prismatic u ...
cells, with an isosceles-trapezoidal pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
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 lines ...
.
Omnitruncated order-6 hexagonal tiling honeycomb
The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3, hastruncated trihexagonal tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''.
Names
Uniform colorings
The ...
and dodecagonal prism
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.
If faces are all regular, it is a uniform polyhedron.
Use
It is used in the construction of two prismatic u ...
cells, with a phyllic disphenoid
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...
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 lines ...
.
Alternated order-6 hexagonal tiling honeycomb
The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, ↔ . It containstriangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
facets in a hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
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 lines ...
.
Cantic order-6 hexagonal tiling honeycomb
The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of therectified triangular tiling honeycomb
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycomb (geometry), honeycombs) in Hyperbolic space, hyperbolic 3-space. It is called ''paracompact'' because it has infinite Cell (geometry), cells ...
, ↔ , with trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
and hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
facets in a 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''. ...
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 lines ...
.
Runcic order-6 hexagonal tiling honeycomb
The runcic hexagonal tiling honeycomb, h3, , or , hashexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
, rhombitrihexagonal tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr.
John Conway calls it a rhombihexadeltille.Conway, 2008, ...
, triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
, and 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''. ...
facets, with a triangular cupola
In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron.
Formulae
The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, wi ...
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 lines ...
.
Runicantic order-6 hexagonal tiling honeycomb
The runcicantic order-6 hexagonal tiling honeycomb, h2,3, , or , containstruncated trihexagonal tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''.
Names
Uniform colorings
The ...
, truncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applies to ...
, trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
, and 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''. ...
facets, with a rectangular pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
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 lines ...
.
See also
*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 ...
* Regular tessellations of hyperbolic 3-space
* Paracompact uniform honeycomb
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron Cell (geometry), cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of Coxeter diagram#Paracompact (Koszul simplex groups), ...
s
References
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)
* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, , (Chapter 10Regular Honeycombs in Hyperbolic Space
Table III * Jeffrey R. Weeks ''The Shape of Space, 2nd edition'' {{isbn, 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II) * Norman Johnson ''Uniform Polytopes'', Manuscript ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 ** N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 13: Hyperbolic Coxeter groups Hexagonal tilings Honeycombs (geometry) Self-dual tilings