Triakis octahedron
Template:Short description Template:Semireg dual polyhedra db In geometry, a triakis octahedron (or trigonal trisoctahedron<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> or kisoctahedron<ref>Conway, Symmetries of things, p. 284</ref>) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope 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, 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 at different relative distances from the center.
If its shorter edges have length of 1, its surface area and volume are:
- <math>\begin{align} A &= 3\sqrt{7+4\sqrt{2}} \\ V &= \frac{3+2\sqrt{2}}{2} \end{align}</math>
Cartesian coordinatesEdit
Let Template:Nowrap, then the 14 points Template:Nowrap and Template:Nowrap, Template:Nowrap and Template:Nowrap are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals Template:Sqrt, and that of the short edges Template:Nowrap.
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals Template:Nowrap ≈ Template:Val° and the acute ones equal Template:Nowrap ≈ Template:Val°.
Orthogonal projectionsEdit
The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:
| Projective symmetry |
[2] | [4] | [6] |
|---|---|---|---|
| Triakis octahedron |
File:Dual truncated cube t01 e88.png | File:Dual truncated cube t01 B2.png | File:Dual truncated cube t01.png |
| Truncated cube |
File:Cube t01 e88.png | File:3-cube t01 B2.svg | File:3-cube t01.svg |
Cultural referencesEdit
- A triakis octahedron is a vital element in the plot of cult author Hugh Cook's novel The Wishstone and the Wonderworkers.
Related polyhedraEdit
The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
Template:Octahedral truncations
The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
Template:Truncated figure1 table
The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n42) reflectional symmetry. Template:Truncated figure4 table
ReferencesEdit
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
- Template:Citation (The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron)
- The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:Isbn [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)
External linksEdit
- Template:Mathworld2
- Triakis Octahedron – Interactive Polyhedron Model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
Template:Catalan solids Template:Polyhedron navigator Template:Polyhedron-stub