Truncated triakis tetrahedron

It is constructed from a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 mirror-symmetric pentagons.

Triakis tetrahedron

A topologically similar equilateral polyhedron can be constructed by using 12 regular pentagons with 4 equilateral but nonplanar hexagons, each vertex with internal angles alternating between 108 and 132 degrees.

Topologically, as a near-miss Johnson solid, the four hexagons corresponding to the face planes of a tetrahedron are triambi, with equal edges but alternating angles, while the pentagons only have reflection symmetry.

If all of a triakis tetrahedron's vertices, of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a trihexakis truncated tetrahedron.

Truncation of only the 3-valence vertices yields the order-3 truncated triakis tetrahedron, which looks like a tetrahedron with each face raised by a low triangular frustum. The dual to that truncation will be the triakis truncated tetrahedron.

Hexakis truncated tetrahedron rotating

The dual of the order-6 Truncated triakis tetrahedron is called a hexakis truncated tetrahedron. It is constructed by a truncated tetrahedron with hexagonal pyramids augmented. If all of the triangles are made regular, the polyhedron becomes a failed Johnson solid, with coplanar triangles in a truncated tetrahedron volume.

truncated tetrahedron

Hexakis truncated tetrahedron

Net

• Near-miss Johnson solid
• Truncated tetrakis cube
• Truncated triakis octahedron
• Truncated triakis icosahedron

• Johnson Solid Near Misses: Number 22
• George Hart's Polyhedron generator - "t6kT" (Conway polyhedron notation)