Truncated triakis tetrahedron

The truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.

Not to be confused with triakis truncated tetrahedron.
Truncated triakis tetrahedron

(Click for rotating model)
Conway notationt6kT = dk6tT
Faces4 hexagons
12 pentagons
Edges42
Vertices28
DualHexakis truncated tetrahedron
Vertex configuration4 (5.5.5)
24 (5.5.6)
Symmetry groupTd
Propertiesconvex

Net

Construction

It is constructed from taking 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.

Full truncation

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
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
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See also


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