(A partial solid honeycomb)
Type	Uniform honeycomb Convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
Schläfli symbol	t0,1{3[4]}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
Cell types	{3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...   (3.6.6) Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
Face types	{3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .... , {6}
Vertex figure	(elongated triangular antiprism)
Cells/vertex	(3.6.6)6+{3,3}2
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	, [3[4]]
Dual	Trigonal trapezohedral honeycomb
Properties	vertex-transitive Vertex-transitive In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same... , edge-transitive

The quarter cubic honeycomb (or bitruncated alternated cubic honeycomb) is a space-filling tessellation

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

 (or honeycomb

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

) in Euclidean 3-space. It is composed of tetrahedra

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 and truncated tetrahedra

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

 in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – consists of four such units of the cubic honeycomb

Cubic honeycomb

The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron....

.

It is vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

 with 6 truncated tetrahedra

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

 and 2 tetrahedra

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 around each vertex.

It is one of 28 convex uniform honeycomb

Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...

s.

The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling

Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex...

.

Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 is an isosceles antiprism

Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

: two equilateral triangles joined by six isosceles triangles.


Edge framework