Triakis octahedron
Encyclopedia
In geometry
, a triakis octahedron 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 the fact 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 in different relative distances from the center.
It is also called the small triakis octahedron, so as to differentiate it from the great triakis octahedron
, the dual of the stellated truncated hexahedron.
If its shorter edges have length 1, its surface area is
and its volume is
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 octahedron 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 cube
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....
.
It can be seen as an octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
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 octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect the fact that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron
Deltoidal icositetrahedron
In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....
, 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.
It is also called the small triakis octahedron, so as to differentiate it from the great triakis octahedron
Great triakis octahedron
In geometry, the great triakis octahedron is the dual of the stellated truncated hexahedron . It has 24 intersecting isosceles triangle faces....
, the dual of the stellated truncated hexahedron.
If its shorter edges have length 1, its surface area is
and its volume isCultural references
- A triakis octahedron is a vital element in the plot of cult author Hugh CookHugh Cook (science fiction author)Hugh Cook was a cult author whose works blend fantasy and science fiction. He is best-known for his epic series The Chronicles of an Age of Darkness.-Biography:...
's novel The Wishstone and the Wonderworkers.
External links
- Triakis Octahedron -- Interactive Polyhedron Model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRMLVRMLVRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
model - Conway Notation for Polyhedra Try: "dtC"
- VRML