geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a triakis tetrahedron (or kistetrahedron) is a

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan so ...

with 12 faces. Each Catalan solid is the dual of an

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...

. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be seen as a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

with a

triangular pyramid In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

added to each face; that is, it is the Kleetope of the tetrahedron. It is very similar to the net for the

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It ...

, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name. The length of the shorter edges is that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area and volume .

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

for the 8 vertices of a triakis tetrahedron centered at the origin, are the points (±5/3, ±5/3, ±5/3) with an even number of minus signs, along with the points (±1, ±1, ±1) with an odd number of minus signs: *(5/3, 5/3, 5/3), (5/3, -5/3, -5/3), (-5/3, 5/3, -5/3), (-5/3, -5/3, 5/3) *(-1, 1, 1), (1, -1, 1), (1, 1, -1), (-1, -1, -1) The length of the shorter edges of this triakis tetrahedron equals

2\sqrt

. The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals

\arccos(-7/18)\approx 112.885\,380\,476\,16^

and the acute ones equal

\arccos(5/6)\approx 33.557\,309\,761\,92^

Tetartoid symmetry

The triakis tetrahedron can be made as a degenerate limit of a tetartoid:

Orthogonal projections

Variations

A triakis tetrahedron with equilateral triangle faces represents a

net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...

of the four-dimensional regular polytope known as the

. :

If the triangles are right-angled isosceles, the faces will be coplanar and form a cubic volume. This can be seen by adding the 6 edges of

inside of a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

. :

Stellations

This chiral figure is one of thirteen

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

s allowed by Miller's rules.

Related polyhedra

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

figures have (*''n''32) reflectional

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

References

* (Section 3-9) * (The thirteen semiregular convex polyhedra and their duals, Page 14, Triakistetrahedron) *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,

(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )

External links

* Catalan solids {{Polyhedron-stub