In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a triakis tetrahedron is an Archimedean dual

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

 solid, or a Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

. Its dual is the truncated tetrahedron

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

.

It can be seen as 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 triangular pyramids

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...

 added to each face; that is, it is the Kleetope

Kleetope

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...

 of the tetrahedron. This interpretation is expressed in the name.

If the triakis tetrahedron has shorter edge lengths 1, it has area and volume .

Variations

A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.

Stellations

This chiral figure is one of thirteen stellation

Stellation

Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

s allowed by Miller's rules.