Triakis truncated tetrahedron

Not to be confused with truncated triakis 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]

Triakis truncated tetrahedron
Type	Plesiohedron
Faces	4 hexagons 12 isosceles triangles
Edges	30
Vertices	16
Conway notation	k3tT
Dual polyhedron	16|Order-3 truncated triakis tetrahedron
Properties	convex

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

Triakis truncated tetrahedral honeycomb

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