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

Thetetragonal disphenoid tetrahedral honeycomb is a space-fillingtessellation (orhoneycomb) inEuclidean 3-space made up of identicaltetragonal disphenoidal cells. Cells areface-transitive with 4 identicalisosceles triangle faces.John Horton Conway calls it anoblate tetrahedrille or shortened toobtetrahedrille.^[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 uniformbitruncated cubic honeycomb.

Its vertices form the A^*
₃ / D^*
₃ lattice, which is also known as thebody-centered cubic lattice.

This honeycomb'svertex figure is atetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms arhombic 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 irregularoctahedron. 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 ofparallelepiped called atrigonal trapezohedron.

An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with acubic honeycomb, subdividing it at the planes${\displaystyle x=y}$,${\displaystyle x=z}$, and${\displaystyle y=z}$ (i.e. subdividing each cube intopath-tetrahedra), then squashing it along the main diagonal until the distance between the points (0, 0, 0) and (1, 1, 1) becomes the same as the distance between the points (0, 0, 0) and (0, 0, 1).

Hexakis cubic honeycomb Pyramidille[2]
Type	Dual uniform honeycomb
Coxeter–Dynkin diagrams
Cell	Isoscelessquare pyramid
Faces	Triangle square
Space group Fibrifold notation	Pm3m (221) 4−:2
Coxeter group	${\displaystyle {\tilde{C}}_3}$, [4, 3, 4]
vertex figures	,
Dual	Truncated cubic honeycomb
Properties	Cell-transitive

Thehexakis cubic honeycomb is a uniform space-fillingtessellation (orhoneycomb) in Euclidean 3-space.John Horton Conway calls it apyramidille.^[2]

Cells can be seen in a translational cube, using 4 vertices on one face, and the cube center. Edges are colored by how many cells are around each of them.

It can be seen as acubic honeycomb with each cube subdivided by a center point into 6square pyramid cells.

There are two types of planes of faces: one as asquare tiling, and flattenedtriangular tiling with half of the triangles removed asholes.

Tiling plane
Symmetry	p4m, [4,4] (*442)	pmm, [∞,2,∞] (*2222)

It is dual to thetruncated cubic honeycomb with octahedral and truncated cubic cells:

If the square pyramids of thepyramidille arejoined on their bases, another honeycomb is created with identical vertices and edges, called asquare bipyramidal honeycomb, or the dual of therectified cubic honeycomb.

It is analogous to the 2-dimensionaltetrakis square tiling:

Square bipyramidal honeycomb Oblate octahedrille[2]
Type	Dual uniform honeycomb
Coxeter–Dynkin diagrams
Cell	Square bipyramid
Faces	Triangles
Space group Fibrifold notation	Pm3m (221) 4−:2
Coxeter group	${\displaystyle {\tilde{C}}_3}$, [4,3,4]
vertex figures	,
Dual	Rectified cubic honeycomb
Properties	Cell-transitive,Face-transitive

Thesquare bipyramidal honeycomb is a uniform space-fillingtessellation (orhoneycomb) in Euclidean 3-space.John Horton Conway calls it anoblate octahedrille or shortened tooboctahedrille.^[1]

A cell can be seen positioned within a translational cube, with 4 vertices mid-edge and 2 vertices in opposite faces. Edges are colored and labeled by the number of cells around the edge.

It can be seen as acubic honeycomb with each cube subdivided by a center point into 6square pyramid cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into square bipyramids (octahedra). Its vertex and edge framework is identical to thehexakis cubic honeycomb.

There is one type of plane with faces: a flattenedtriangular tiling with half of the triangles asholes. These cut face-diagonally through the original cubes. There are alsosquare tiling plane that exist as nonfaceholes passing through the centers of the octahedral cells.

Tiling plane	Square tiling "holes"	flattenedtriangular tiling
Symmetry	p4m, [4,4] (*442)	pmm, [∞,2,∞] (*2222)

It is dual to therectified cubic honeycomb with octahedral and cuboctahedral cells:

Phyllic disphenoidal honeycomb Eighth pyramidille[3]
(No image)
Type	Dual uniform honeycomb
Coxeter-Dynkin diagrams
Cell	Phyllic disphenoid
Faces	Rhombus Triangle
Space group Fibrifold notation Coxeter notation	Im3m (229) 8o:2 [[4,3,4]]
Coxeter group	[4,3,4],${\displaystyle {\tilde{C}}_3}$
vertex figures	,
Dual	Omnitruncated cubic honeycomb
Properties	Cell-transitive,face-transitive

Thephyllic disphenoidal honeycomb is a uniform space-fillingtessellation (orhoneycomb) in Euclidean 3-space.John Horton Conway calls this anEighth pyramidille.^[3]

A cell can be seen as 1/48 of a translational cube with vertices positioned: one corner, one edge center, one face center, and the cube center. The edge colors and labels specify how many cells exist around the edge. It is one 1/6 of a smaller cube, with 6 phyllic disphenoidal cells sharing a common diagonal axis.

It is dual to theomnitruncated cubic honeycomb:

• Architectonic and catoptric tessellation
• Cubic honeycomb
• Space frame
• Tetrahedral-octahedral honeycomb
• Triakis truncated tetrahedral honeycomb

• Gibb, William (1990), "Paper patterns: solid shapes from metric paper",Mathematics in School,19 (3):2–4, reprinted inPritchard, Chris, ed. (2003),The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Cambridge University Press, pp. 363–366,ISBN 0-521-53162-4.
• Senechal, Marjorie (1981), "Which tetrahedra fill space?",Mathematics Magazine,54 (5), Mathematical Association of America:227–243,doi:10.2307/2689983,JSTOR 2689983.
• Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "21. Naming Archimedean and Catalan Polyhedra and Tilings".The Symmetries of Things. A K Peters, Ltd. pp. 292–298.ISBN 978-1-56881-220-5.

• 3-honeycombs
• Tetrahedra