Triakis truncated tetrahedral honeycomb explained

bgcolor=#e7dcc3 colspan=2	Triakis truncated tetrahedral honeycomb
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Cell type	Triakis truncated tetrahedron
Face types	hexagon isosceles triangle
Coxeter group	Ã3×2, 3^[4] (double)
Space group	Fdm (227)
Properties	Cell-transitive

The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914.^{[1] [2]}

Voronoi tessellation

It is the Voronoi tessellation of the carbon atoms in diamond,^{[3] [4]} which lie in the diamond cubic crystal structure.

Being composed entirely of triakis truncated tetrahedra, it is cell-transitive.

Relation to quarter cubic honeycomb

It can be seen as the uniform quarter cubic honeycomb where its tetrahedral cells are subdivided by the center point into 4 shorter tetrahedra, and each adjoined to the adjacent truncated tetrahedral cells.

See also

• Disphenoid tetrahedral honeycomb

Notes and References

1. L.. Föppl. 1914. Der Fundamentalbereich des Diamantgitters. Phys. Z.. 15. 191–193.
2. Grünbaum. B.. Shephard. G. C.. Tilings with Congruent Tiles. Bull. Amer. Math. Soc.. 3. 3. 951–973. 1980. 10.1090/s0273-0979-1980-14827-2. free.
3. Web site: Conway. John. Voronoi Polyhedron. geometry.puzzles. 20 September 2012.
4. Book: Conway. John H.. Burgiel. Heidi. Goodman-Strauss. Chaim. The Symmetries of Things. 332. 2008. 978-1568812205.