From Surf Wiki (app.surf) — the open knowledge base
Triakis octahedron
Catalan solid with 24 faces
Catalan solid with 24 faces
In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedron) 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: :\begin{align} A &= 3\sqrt{7+4\sqrt{2}} \ V &= \frac{3+2\sqrt{2}}{2} \end{align}
Cartesian coordinates
Let , then the 14 points (±α, ±α, ±α) and (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1) are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals , and that of the short edges 2 − 2.
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals arccos( − ) ≈ ° and the acute ones equal arccos( + ) ≈ °.
Orthogonal projections
The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:
| Projective | |||||
|---|---|---|---|---|---|
| symmetry | Triakis | ||||
| octahedron | Truncated | ||||
| cube | |||||
| [2] | [4] | [6] | |||
| [[File:Dual truncated cube t01 e88.png | 100px]] | [[File:Dual truncated cube t01 B2.svg | 120px]] | [[File:Dual truncated cube t01.svg | 120px]] |
| [[File:Cube t01 e88.svg | 120px]] | [[File:3-cube t01 B2.svg | 120px]] | [[File:3-cube t01.svg | 120px]] |
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 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 face-transitive figures have (**n*42) reflectional symmetry.
References
References
- "Clipart tagged: 'forms'". etc.usf.edu.
- Conway, Symmetries of things, p. 284
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Triakis octahedron — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report