Triakis truncated tetrahedral honeycomb

Cell type | Triakis truncated tetrahedron
Face types | hexagon
isosceles triangle
Coxeter group | Ã3×2, 3[4] (double)
Space group | Fd3m (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]

## Contents

* 1 Voronoi tessellation
* 2 Relation to quarter cubic honeycomb
* 3 See also
* 4 References

## 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

## 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. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183547682.
3. ↑ Conway, John. "Voronoi Polyhedron". geometry.puzzles. https://groups.google.com/forum/?fromgroups=#!msg/geometry.puzzles/pkL3avbWPoc/ABSaqdQaqu4J. 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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