Triakis truncated tetrahedron explained

Type:	Plesiohedron
Faces:	4 hexagons 12 isosceles triangles
Edges:	30
Vertices:	16
Dual:	16Order-3 truncated triakis tetrahedron
Properties:	convex

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]}

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

• Quarter cubic honeycomb
• Truncated tetrahedron
• Triakis tetrahedron

Notes and References

1. Book: Conway. John H.. Burgiel. Heidi. Goodman-Strauss. Chaim. The Symmetries of Things. 332. 2008. 978-1568812205.
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. L.. Föppl. 1914. Der Fundamentalbereich des Diamantgitters. Phys. Z.. 15. 191–193.
4. Web site: Conway. John. Voronoi Polyhedron. geometry.puzzles. 20 September 2012.
5. .