Tetragonal disphenoid honeycomb

Tetragonal disphenoid tetrahedral honeycomb
Type	convex uniform honeycomb dual
Coxeter-Dynkin diagram
Cell type	Tetragonal disphenoid
Face types	isosceles triangle {3}
Vertex figure	tetrakis hexahedron
Space group	Im3m (229)
Symmetry	[[4, 3, 4]]
Coxeter group	${\displaystyle {\tilde {C}}_{3}}$, [4, 3, 4]
Dual	Bitruncated cubic honeycomb
Properties	cell-transitive, face-transitive, vertex-transitive

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.^[1]

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