Tetragonal disphenoid honeycomb

The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an oblate tetrahedrille or shortened to obtetrahedrille.

A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four edges have 6 cells, and 2 edges have 4 cells.

The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb.

Its vertices form the A*
3 / D*
3 lattice, which is also known as the Body-Centered Cubic lattice.

This honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular octahedron. When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of parallelepiped called a trigonal trapezohedron.

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