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.
gollark: MUAHAHAHA, commit.
gollark: ... somehow.
gollark: Oh, right, I must login.
gollark: Aha, "fixed".
gollark: Oh bee oh apiary form. It was intuitive enough with git (due to better editor support, not any intrinsic goodness).

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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