In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic 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 ...

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: a flat plane in hyperbolic space that approaches a single ideal point 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 order-4 hexagonal tiling honeycomb is . Since that of the

is , this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is , the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III

Images

Symmetry

The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions. The half-symmetry uniform construction has two types (colors) of hexagonal tilings, with Coxeter diagram ↔ . A quarter-symmetry construction also exists, with four colors of hexagonal tilings: . An additional two reflective symmetries exist with non-simplectic fundamental domains: ,3^*,4 which is index 6, with Coxeter diagram ; and ,(3,4)^* which is index 48. The latter has a

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

fundamental domain, and an octahedral Coxeter diagram with three axial infinite branches: . It can be seen as using eight colors to color the hexagonal tilings of the honeycomb. The order-4 hexagonal tiling honeycomb contains , which tile 2- hypercycle surfaces and are similar to the

truncated infinite-order triangular tiling In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t. Symmetry The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror rem ...

, : :

Related polytopes and honeycombs

The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. There are fifteen uniform honeycombs in the ,3,4 Coxeter group family, including this regular form, and its

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

, the

order-6 cubic honeycomb The order-6 cubic honeycomb is a paracompact regular space-filling tessellation (or honeycomb) in hyperbolic 3-space. It is ''paracompact'' because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points a ...

. The order-4 hexagonal tiling honeycomb has a related alternated honeycomb, ↔ , with triangular tiling and octahedron cells. It is a part of sequence of regular honeycombs of the form , all of which are composed of

cells: This honeycomb is also related to the

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures. The aforementioned honeycombs are also quasiregular:

Rectified order-4 hexagonal tiling honeycomb

The rectified order-4 hexagonal tiling honeycomb, t₁, has octahedral and trihexagonal tiling facets, with a square prism vertex figure. It is similar to the 2D hyperbolic

tetraapeirogonal tiling In geometry, the tetraapeirogonal tiling is a uniform tilings in hyperbolic plane, uniform tiling of the hyperbolic geometry, hyperbolic plane with a Schläfli symbol of r. Uniform constructions There are 3 lower symmetry uniform construction, one ...

, r, which alternates apeirogonal and square faces: :

Truncated order-4 hexagonal tiling honeycomb

The truncated order-4 hexagonal tiling honeycomb, t_0,1, has octahedron and truncated hexagonal tiling facets, with a

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...

vertex figure. It is similar to the 2D hyperbolic

truncated order-4 apeirogonal tiling In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t. Uniform colorings A half symmetry coloring is tr, has two types of apeirogons, shown red and yellow here. If the a ...

, t, with apeirogonal and square faces: :

Bitruncated order-4 hexagonal tiling honeycomb

The bitruncated order-4 hexagonal tiling honeycomb, t_1,2, 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 ...

and

cells, with a

digonal 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.

Cantellated order-4 hexagonal tiling honeycomb

The cantellated order-4 hexagonal tiling honeycomb, t_0,2, has cuboctahedron,

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

, and rhombitrihexagonal tiling cells, with a wedge vertex figure.

Cantitruncated order-4 hexagonal tiling honeycomb

The cantitruncated order-4 hexagonal tiling honeycomb, t_0,1,2, has

, and truncated trihexagonal tiling cells, with a mirrored sphenoid vertex figure.

Runcinated order-4 hexagonal tiling honeycomb

The runcinated order-4 hexagonal tiling honeycomb, t_0,3, has

and hexagonal prism cells, with an irregular

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. It contains the 2D hyperbolic

rhombitetrahexagonal tiling In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetrahexagonal tiling, r, as well as an expanded order-4 hexagonal tiling or exp ...

, rr, with square and hexagonal faces. The tiling also has a half symmetry construction .

Runcitruncated order-4 hexagonal tiling honeycomb

The runcitruncated order-4 hexagonal tiling honeycomb, t_0,1,3, has rhombicuboctahedron,

, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Runcicantellated order-4 hexagonal tiling honeycomb

The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.

Omnitruncated order-4 hexagonal tiling honeycomb

The omnitruncated order-4 hexagonal tiling honeycomb, t_0,1,2,3, has

truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...

, truncated trihexagonal tiling, dodecagonal prism, and octagonal prism cells, with an irregular tetrahedron vertex figure.

Alternated order-4 hexagonal tiling honeycomb

The alternated order-4 hexagonal tiling honeycomb, ↔ , is composed of triangular tiling and octahedron cells, in a

vertex figure.

Cantic order-4 hexagonal tiling honeycomb

The cantic order-4 hexagonal tiling honeycomb, ↔ , is composed of trihexagonal tiling,

, and cuboctahedron cells, with a wedge vertex figure.

Runcic order-4 hexagonal tiling honeycomb

The runcic order-4 hexagonal tiling honeycomb, ↔ , is composed of triangular tiling, rhombicuboctahedron,

, and triangular prism cells, with a triangular cupola vertex figure.

Runcicantic order-4 hexagonal tiling honeycomb

The runcicantic order-4 hexagonal tiling honeycomb, ↔ , is composed of trihexagonal tiling,

, truncated cube, and triangular prism cells, with a rectangular pyramid vertex figure.

Quarter order-4 hexagonal tiling honeycomb

The quarter order-4 hexagonal tiling honeycomb, q, or , is composed of triangular tiling, trihexagonal tiling, tetrahedron, and truncated tetrahedron cells, with a triangular cupola vertex figure.

References

* Coxeter, '' Regular Polytopes'', 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 Hexagonal tilings Honeycombs (geometry)