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 ...

, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

. It can be seen as an

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

with

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 ...

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

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. Kleetopes are named after Victor Klee. Exam ...

of the octahedron. It is also sometimes called a ''trisoctahedron'', or, more fully, ''trigonal trisoctahedron''. Both names reflect that it has three triangular faces for every face of an octahedron. The ''tetragonal trisoctahedron'' is another name for the

deltoidal icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...

, a different polyhedron with three quadrilateral faces for every face of an octahedron. This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center. If its shorter edges have length 1, its surface area and volume are: :

\begin A &= 3\sqrt \\ V &= \frac \end

Cartesian coordinates

Put

\alpha=\sqrt-1

, then the 14 points

(\pm\alpha, \pm\alpha, \pm\alpha)

and

(\pm 1, 0, 0)

(0, \pm 1, 0)

and

(0, 0, \pm 1)

are the vertices of a triakis octahedron centered at the origin. The length of the long edges equals

\sqrt

, and that of the short edges

2\sqrt-2

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

\arccos(\frac-\frac\sqrt)\approx 117.200\,570\,380\,16^

and the acute ones equal

\arccos(\frac+\frac\sqrt)\approx 31.399\,714\,809\,92^

Orthogonal projections

The ''triakis octahedron'' has three symmetry positions, two located on vertices, and one mid-edge:

Cultural references

* A triakis octahedron is a vital element in the plot of cult author Hugh Cook's novel '' The Wishstone and the Wonderworkers''.

Related polyhedra

The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. The triakis octahedron 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.

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

figures have (*''n''42) reflectional symmetry.

References

* (Section 3-9) * (The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron) * ''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 octahedron)

External links

*
Triakis Octahedron
– Interactive Polyhedron Model

www.georgehart.com: The Encyclopedia of Polyhedra **

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...

br>model
*

Try: "dtC" {{Polyhedron-stub Catalan solids