Triakis truncated tetrahedron

In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb.[1][2]

Not to be confused with truncated triakis tetrahedron.
Triakis truncated tetrahedron
(Click here for rotating model)
TypePlesiohedron
Conway notationk3tT
Faces4 hexagons
12 isosceles triangles
Edges30
Vertices16
DualOrder-3 truncated triakis tetrahedron
Propertiesconvex, space-filling

The triakis truncated tetrahedron is the shape of the Voronoi cell of the carbon atoms in diamond, which lie on the diamond cubic crystal structure.[3][4] As the Voronoi cell of a symmetric space pattern, it is a plesiohedron.[5]

Construction

For space-filling, the triakis truncated tetrahedron can be constructed as follows:

  1. Truncate a regular tetrahedron such that the big faces are regular hexagons.
  2. Add an extra vertex at the center of each of the four smaller tetrahedra that were removed.

See also

References
  1. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. p. 332. ISBN 978-1568812205.
  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. Föppl, L. (1914). "Der Fundamentalbereich des Diamantgitters". Phys. Z. 15: 191–193.
  4. Conway, John. "Voronoi Polyhedron". geometry.puzzles. Retrieved 20 September 2012.
  5. Grünbaum, Branko; Shephard, G. C. (1980), "Tilings with congruent tiles", Bulletin of the American Mathematical Society, New Series, 3 (3): 951–973, doi:10.1090/S0273-0979-1980-14827-2, MR 0585178.


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