In the field of

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 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. Th ...

. 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 ...

composed of 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 mathemat ...

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 surface 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 more ...

of the hexagonal tiling honeycomb is . Since that of the

is , this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the

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 o ...

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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III

Images

Viewed in perspective outside of a

Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk th ...

, the image above shows one

cell within the honeycomb, and its mid-radius

(the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the

order-3 apeirogonal tiling In geometry, the order-3 apeirogonal tiling is a regular hyperbolic tiling, regular tiling of the Hyperbolic geometry, hyperbolic plane. It is represented by the Schläfli symbol , having three regular Apeirogon#Hyperbolic geometry, apeirogons arou ...

, of H², with

horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...

s circumscribing vertices of

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 the ...

al faces.

Symmetry constructions

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: ,3,3 ,6,3 ,3,6 ,3^[3/sup>.html"_;"title=".html"_;"title=",3^{[3">,3^{[3/sup>">.html"_;"title=",3^{[3">,3^{[3/sup>and_[3^{[3,3.html" ;"title="">,3^{[3/sup>.html" ;"title=".html" ;"title=",3^{[3">,3^{[3/sup>">.html" ;"title=",3^{[3">,3^{[3/sup>and [3^{[3,3">">,3^{[3/sup>.html" ;"title=".html" ;"title=",3^{[3">,3^{[3/sup>">.html" ;"title=",3^{[3">,3^{[3/sup>and [3^{[3,3/sup>] , having 1, 4, 6, 12 and 24 times Paracompact uniform honeycomb#Enumeration, larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^*] (remove 3 mirrors, index 24 subgroup); ,6,3^*or ^*,6,3(remove 2 mirrors, index 6 subgroup); ⁺,6,3,6,1⁺(remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to ^[3,3/sup>.html" ;"title=",3.html" ;"title="^{[3,3">^{[3,3/sup>">,3.html" ;"title="^{[3,3">^{[3,3/sup> The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.

Related polytopes and honeycombs

The hexagonal tiling honeycomb is a List of regular polytopes#Tessellations of hyperbolic 3-space, regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

It is one of 15 uniform paracompact honeycombs in the ,3,3Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.

It is part of a sequence of regular polychora

In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

There are six convex and ten star regu ...
, which include the 5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
, tesseract

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
, and 120-cell

In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral

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 o ...
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 ...
s.

It is also part of a sequence of regular honeycombs of the form , which are each composed of 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 mathemat ...
cells:

Rectified hexagonal tiling honeycomb

The rectified hexagonal tiling honeycomb, t₁, has tetrahedral

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 o ...
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 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 ...
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 ...
. The half-symmetry construction alternates two types of tetrahedra.

Truncated hexagonal tiling honeycomb

The truncated hexagonal tiling honeycomb, t_0,1, has tetrahedral

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 o ...
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 a he ...
facets, with a triangular pyramid

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 o ...
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 ...
.

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t with apeirogonal and triangle faces:
:

Bitruncated hexagonal tiling honeycomb

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,2, has truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...
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 mathemat ...
cells, with a digonal disphenoid 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 ...
.

Cantellated hexagonal tiling honeycomb

The cantellated hexagonal tiling honeycomb, t_0,2, has octahedron

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 ...
, 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, 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''. A unif ...
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
.

Cantitruncated hexagonal tiling honeycomb

The cantitruncated hexagonal tiling honeycomb, t_0,1,2, has truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...
, 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 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 ...
cells, with a mirrored sphenoid

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 o ...
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 ...
.

Runcinated hexagonal tiling honeycomb

The runcinated hexagonal tiling honeycomb, t_0,3, has 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 o ...
, 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 mathemat ...
, 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 to ...
, 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''. A unif ...
cells, with an irregular triangular antiprism 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 ...
.

Runcitruncated hexagonal tiling honeycomb

The runcitruncated hexagonal tiling honeycomb, t_0,1,3, has cuboctahedron

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, 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 ...
, 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 uni ...
, 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 a he ...
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, quadrilat ...
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 ...
.

Runcicantellated hexagonal tiling honeycomb

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,3, has truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...
, 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 to ...
, and 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, 2 ...
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, quadrilat ...
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 ...
.

Omnitruncated hexagonal tiling honeycomb

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,3, has truncated octahedron

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 hexagon, hexagons and 6 Squa ...
, 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 to ...
, 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 uni ...
, and 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 ...
cells, with an irregular 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 o ...
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 ...
.

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 Wyt ...

* 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
* Alternated hexagonal tiling honeycomb
In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h, or , is a semiregular polytope, semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named afte ...

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 10
Regular Honeycombs in Hyperbolic Space
) Table III
* Jeffrey R. Weeks ''The Shape of Space, 2nd edition'' (Chapters 16–17: Geometries on Three-manifolds I,II)
*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'', (2002) H³: p130

External links

*John Baez

John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
, ''Visual Insight''
Honeycomb
(2014/03/15)
*John Baez

John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
, ''Visual Insight''
Honeycomb in Upper Half Space
(2013/09/15)
*John Baez

John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
, ''Visual Insight''
Truncated {6,3,3} Honeycomb
(2016/12/01)

Hexagonal tilings
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