The triangular tiling honeycomb is one of 11 paracompact regular space-filling

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 ...

s (or honeycombs) in

hyperbolic 3-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 called ''paracompact'' because it has infinite

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 ...

and

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, with all vertices as

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'' ...

s at infinity. It has

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 ...

, being composed of

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 ...

cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its

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 ...

Symmetry

] It has two lower reflective symmetry constructions, as an Alternation (geometry), alternated

order-6 hexagonal tiling honeycomb In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells with an infinite number of faces. Each cell is ...

, ↔ , and as from , which alternates 3 types (colors) of triangular tilings around every edge. In

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

, the removal of the 3rd and 4th mirrors, ,6,3^*creates a new

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 ...

^[3,3/sup>">,3.html" ;"title="^{[3,3">^{[3,3/sup> , subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: ↔ .

Related Tilings

It is similar to the 2D hyperbolic infinite-order apeirogonal tiling, , with infinite apeirogonal faces, and with all vertices on the ideal surface.
:

Related honeycombs

The triangular tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs.

There are nine uniform honeycombs in the ,6,3Coxeter 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 ...
family, including this regular form as well as the bitruncated form, t_1,2, with all 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.

The honeycomb is also part of a series of polychora and honeycombs with triangular edge 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 triangular tiling honeycomb

The rectified triangular tiling honeycomb, , has 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 mathemat ...
cells, 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.

Symmetry

A lower symmetry of this honeycomb can be constructed as a ''cantic order-6 hexagonal tiling honeycomb'', ↔ . A second lower-index construction is ↔ .

Truncated triangular tiling honeycomb

The truncated triangular tiling honeycomb, , is a lower-symmetry form of the hexagonal tiling honeycomb
In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells composed of an infinite number of faces. Each cell is a ...
, . It contains 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 ...
facets with a 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.

Bitruncated triangular tiling honeycomb

The bitruncated triangular tiling honeycomb, , has 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 a tetragonal disphenoid

In geometry, a disphenoid () is a tetrahedron whose four Face (geometry), faces are Congruence (geometry), congruent acute-angled triangles. It can also be described as a tetrahedron in which every two Edge (geometry), edges that are opposite ea ...
vertex figure.

Cantellated triangular tiling honeycomb

The cantellated triangular tiling honeycomb, , has 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 ...
, 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''. 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.

Symmetry

It can also be constructed as a cantic snub triangular tiling honeycomb, , a half-symmetry form with symmetry ⁺,6,3

Cantitruncated triangular tiling honeycomb

The cantitruncated triangular tiling honeycomb, , has 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 ...
, 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 ...
, 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.

Runcinated triangular tiling honeycomb

The runcinated triangular tiling honeycomb, , has 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''. A unif ...
cells, with a hexagonal antiprism

In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

Antiprisms are similar to prisms except the bases are twisted relative to each oth ...
vertex figure.

Runcitruncated triangular tiling honeycomb

The runcitruncated triangular tiling honeycomb, , has 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 ...
, 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 ...
, 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 ...
, 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 to ...
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 ...
.

Symmetry

It can also be constructed as a runcicantic snub triangular tiling honeycomb, , a half-symmetry form with symmetry ⁺,6,3

Omnitruncated triangular tiling honeycomb

The omnitruncated triangular tiling honeycomb, , has 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 to ...
cells, with a phyllic disphenoid

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.

Runcisnub triangular tiling honeycomb

The runcisnub triangular tiling honeycomb, , has 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. ...
, 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 ...
, 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 ...
, and 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, ...
cells. It is vertex-transitive

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
, but not uniform, since it contains Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
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, ...
cells.

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

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, f ...
'', 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'' {{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

Honeycombs (geometry)
Self-dual tilings
Triangular tilings}}