Triakis truncated tetrahedral honeycomb

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]

Triakis truncated tetrahedral honeycomb
Cell typeTriakis truncated tetrahedron
Face typeshexagon
isosceles triangle
Coxeter groupÃ3×2, [[3[4]]] (double)
Space groupFd3m (227)
PropertiesCell-transitive

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 tetrahedal 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

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


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