In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h, or , is a semiregular tessellation with

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 ...

and

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 arranged in an

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 e ...

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 named after its construction, as an

alteration Alteration(s) may refer to: * Alteration (music), the use of a neighboring pitch in the chromatic scale in place of its diatonic neighbor. ** Alteration, in the mensural notation used by renaissance music, the lengthening of a breve, semibreve or ...

of a hexagonal tiling honeycomb.

Symmetry constructions

It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: ,3,3 ,6,3 ,3,6 ,3^[3/sup>">.html" ;"title=",3^{[3">,3^{[3/sup>and [3^{[3,3">">,3<sup>[3<_a>_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>">,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 honeycombs

The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, ; the runcic hexagonal tiling honeycomb, ; and the runcicantic hexagonal tiling honeycomb
In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, a ...
, .

Cantic hexagonal tiling honeycomb

The cantic hexagonal tiling honeycomb, h₂, or , is composed of 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 e ...
, 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 tetrahedr ...
, 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 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 ...
.

Runcic hexagonal tiling honeycomb

The runcic hexagonal tiling honeycomb, h₃, or , 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 ...
, 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''. ...
, 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 ...
, and 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 ...
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 ...
.

Runcicantic hexagonal tiling honeycomb

The runcicantic hexagonal tiling honeycomb, h_2,3, or , 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 tetrahedr ...
, 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''. ...
, 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 hexagons and 6 squares), 36 ...
, 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 rectangular

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
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
* Semiregular honeycomb
* Hexagonal tiling honeycomb

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'' {{ISBN, 0-8247-0709-5 (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

Hexagonal tilings
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