The order-4 octahedral honeycomb is a regular paracompact honeycomb 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 ''paracompact'' because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by

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 ...

, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III

Symmetry

A half symmetry construction, ,4,4,1⁺ exists as , with two alternating types (colors) of octahedral cells: ↔ . A second half symmetry is ,4,1⁺,4 ↔ . A higher index sub-symmetry, ,4,4^* which is index 8, exists with a pyramidal fundamental domain, (3,∞,3)),((3,∞,3)) . This honeycomb contains and that tile 2- hypercycle surfaces, which are similar to the paracompact

infinite-order triangular tiling In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . All vertices are ''ideal'', located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection. ...

s and , respectively:
: H2chess 23ib

Related polytopes and honeycombs

The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs. There are fifteen uniform honeycombs in the ,4,4 Coxeter group family, including this regular form. It is a part of a sequence of honeycombs with a square tiling vertex figure: It a part of a sequence of regular polychora and honeycombs with octahedral

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 ...

Rectified order-4 octahedral honeycomb

The rectified order-4 octahedral honeycomb, t₁, has cuboctahedron and square tiling facets, with a square prism vertex figure. H3 344 CC center 0100

Truncated order-4 octahedral honeycomb

The truncated order-4 octahedral honeycomb, t_0,1, 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 square 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.

Bitruncated order-4 octahedral honeycomb

The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.

Cantellated order-4 octahedral honeycomb

The cantellated order-4 octahedral honeycomb, t_0,2, has rhombicuboctahedron,

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 square tiling facets, with a wedge vertex figure.

Cantitruncated order-4 octahedral honeycomb

The cantitruncated order-4 octahedral honeycomb, t_0,1,2, 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 ...

, and truncated square tiling facets, with a mirrored sphenoid vertex figure.

Runcinated order-4 octahedral honeycomb

The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.

Runcitruncated order-4 octahedral honeycomb

The runcitruncated order-4 octahedral honeycomb, t_0,1,3, has

, hexagonal prism, and square tiling facets, with a

vertex figure.

Runcicantellated order-4 octahedral honeycomb

The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.

Omnitruncated order-4 octahedral honeycomb

The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.

Snub order-4 octahedral honeycomb

The snub order-4 octahedral honeycomb, s, has Coxeter diagram . It is a

scaliform honeycomb In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...

, with

, square tiling, and

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

facets.

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'', (2015) Chapter 13: Hyperbolic Coxeter groups ** Norman W. Johnson and Asia Ivic Weis
Quadratic Integers and Coxeter Groups

PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 Honeycombs (geometry)