Truncated triakis tetrahedron
Type	Conway polyhedron Conway polyhedron notation Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...
Faces	4 hexagons 12 pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and... s
Edges	42
Vertices	28
Dual	Hexakis truncated tetrahedron
Vertex configuration Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...	4 (5.5.5) 24 (5.5.6)
Symmetry group	Td
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

The truncated triakis tetrahedron is a convex polyhedron with 16 faces: 4 sets of 3 pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

s arranged in a tetrahedral

Tetrahedral symmetry

150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...

 arrangement, with 4 hexagons in the gaps. It is constructed from taking a triakis tetrahedron

Triakis tetrahedron

In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron...

 by truncating

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

 the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 irregular pentagons.

A topologically similar equilateral

Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

 polyhedron can be constructed by using 12 regular

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 pentagons with 4 equilateral

Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

 but nonplanar hexagons, each vertex with internal angle

Internal angle

In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon...

s alternating between 108 and 132 degrees.

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 simpler vertices yields what looks like a tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 with each face raised by a low triangular frustum

Frustum

In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

. The dual to that truncation will be the triakis truncated tetrahedron.

External links

• Johnson Solid Near Misses: Number 23
• George Hart's Polyhedron generator - "t6dtT" (Conway polyhedron notation
  Conway polyhedron notation
  Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...
  
  )