Disphenoid tetrahedral honeycomb

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Disphenoid tetrahedral honeycomb

Type	convex uniform honeycomb dual
Cell type	disphenoid tetrahedron
Face types	isosceles triangle {3}
Cells/edge	4 or 6
Cells/vertex	24
Symmetry group	Im3m
Dual	Bitruncated cubic honeycomb
Properties	cell-transitive, face-transitive

The disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical nonregular tetrahedral cells.

The stretched tetrahedron cells are called disphenoids. Cells are face-transitive with 4 identical isosceles triangle faces.

24 disphenoids meet at each vertex of the tessellation. 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.

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

[edit] See also

cubic honeycomb

[edit] References

Gibb, William (1990), “Paper patterns: solid shapes from metric paper”, Mathematics in School 19 (3): 2–4 , reprinted in Pritchard, 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 .