Жартылай текше ұялы тор (Жартылай реттелген кеңістікті толтырушы мозаика)

Quasiregular space filling tesselation
Alternated cubic honeycomb  TypeUniform honeycombFamilyAlternated hypercubic honeycombSimplectic honeycombIndexingJ21,31,51, A2W9, G1Schläfli symbolsh{4,3,4}{3[4]}ht0,3{4,3,4}h{4,4}h{∞}ht0,2{4,4}h{∞}h{∞}h{∞}h{∞}s{∞}s{∞}s{∞}Coxeter diagrams = = = = Cells{3,3} {3,4} Facestriangle {3}Edge figure[{3,3}. {3,4}]2(rectangle)Vertex figure(cuboctahedron)Symmetry groupFmm (225)Coxeter group, [4,31,1]DualDodecahedrillerhombic dodecahedral honeycombCell: Propertiesvertex transitive, edge transitive, quasiregular honeycomb
The tetrahedral octahedral honeycomb, alternated cubic honeycomb is a quasiregular space filling tessellation (or honeycomb) in Euclidean 3 space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual a dodecahedrille. R. Buckminster Fuller combines the two words octahedron and tetrahedron into octet truss, a rhombohedron consisting of one octahedron (or two square pyramids) and two opposite tetrahedra. It is vertex transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge transitive with 2 tetrahedra and 2 octahedra alternating on each edge. It is part of an infinite family of uniform honeycombs called alternated hypercubic honeycombs, formed as an alternation of a hypercubic honeycomb and being composed of demihypercube and cross polytope facets. It is also part of another infinite family of uniform honeycombs called simplectic honeycombs. In this case of 3 space, the cubic honeycomb is alternated, reducing the cubic cells to tetrahedra, and the deleted vertices create octahedral voids. As such it can be represented by an extended Schläfli symbol h{4,3,4} as containing half the vertices of the {4,3,4} cubic honeycomb. There is a similar honeycomb called gyrated tetrahedral octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra. The tetrahedral octahedral honeycomb can have its symmetry doubled by placing tetrahedra on the octahedral cells, creating a nonuniform honeycomb consisting of tetrahedra and octahedra (as triangular antiprisms). Its vertex figure is an order 3 truncated triakis tetrahedron. This honeycomb is the dual of the triakis truncated tetrahedral honeycomb, with triakis truncated tetrahedral cells.

The alternated cubic honeycomb can be sliced into sections, where new square faces are created from inside of the octahedron. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges. A second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells.

The [4,3,4], , Coxeter group generates 15 permutations of uniform honeycombs, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.

The [4,31,1], , Coxeter group generates 9 permutations of uniform honeycombs, 4 with distinct geometry including the alternated cubic honeycomb.

This honeycomb is one of five distinct uniform honeycombs constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

The dual of a runcic cubic honeycomb is called a quarter cubille, with Coxeter diagram , with faces in 2 of 4 hyperplanes of the , [4,31,1] symmetry fundamental domain. Cells can be seen as 1/4 of dissected cube, using 4 vertices and the center. Four cells exist around 6 edges, and 3 cells around 3 edges.

The dual to the runcitruncated cubic honeycomb is called a half pyramidille, with Coxeter diagram Faces exist in 3 of 4 hyperplanes of the [4,31,1], Coxeter group. Cells are irregular pyramids and can be seen as 1/12 of a cube, or 1/24 of a rhombic dodecahedron, each defined with three corner and the cube center.

A related uniform skew apeirohedron exists with the same vertex arrangement, but triangles and square removed. It can be seen as truncated tetrahedra and truncated cubes augmented together.

The geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb. The hexagonal prism cells become octahedra and the voids create triangular bipyramids which can be divided into pairs of tetrahedra of this honeycomb. This honeycomb with bipyramids is called a ditetrahedral octahedral honeycomb. There are 3 Coxeter Dynkin diagrams, which can be seen as 1, 2, or 3 colors of octahedra:

Gyroelongated alternated cubic honeycombTypeUniform honeycombSchläfli symbolh{4,3,4}:ge{3,6}h1{∞}Coxeter diagramCells{3,3} {3,4} (3.4.4) Facestriangle {3}square {4}Vertex figureSpace groupP63/mmc (194)[3,6,2+,∞]Propertiesvertex transitive
The gyroelongated alternated cubic honeycomb or elongated triangular antiprismatic cellulation is a space filling tessellation (or honeycomb) in Euclidean 3 space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2. It is vertex transitive with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex. It is one of 28 convex uniform honeycombs. The elongated alternated cubic honeycomb has the same arrangement of cells at each vertex, but the overall arrangement differs. In the elongated form, each prism meets a tetrahedron at one of its triangular faces and an octahedron at the other; in the gyroelongated form, the prism meets the same kind of deltahedron at each end.

Elongated alternated cubic honeycombTypeUniform honeycombSchläfli symbolh{4,3,4}:e{3,6}g1{∞}Cells{3,3} {3,4} (3.4.4) Facestriangle {3}square {4}Vertex figuretriangular cupola joined to isosceles hexagonal pyramidSymmetry group [6,(3,2+,∞,2+)] ?Propertiesvertex transitive
The elongated alternated cubic honeycomb or elongated triangular gyroprismatic cellulation is a space filling tessellation (or honeycomb) in Euclidean 3 space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2. It is vertex transitive with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex. Each prism meets an octahedron at one end and a tetrahedron at the other. It is one of 28 convex uniform honeycombs. It has a gyrated form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.