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Also visit my site - como ganhar dinheiro na internet 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 and volume are:

$A=3\sqrt{7+4\sqrt{2}}$

$V=\frac{1}{2}(3+2\sqrt{2}).$

Orthogonal projections

The triakis octahedron and its dual truncated cube have five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B₂ and A₂ Coxeter planes.

Orthogonal projections

Centered by	Vertex	Edge 3-8	Edge 8-8	Face Octagon	Face Triangle
Truncated cube
Triakis octahedron
Projective symmetry	[2]	[2]	[2]	[4]	[6]

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 (*n32) 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 (*n42) reflectional symmetry. Template:Truncated figure4 table

References

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
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• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)

External links

• Template:Mathworld2
• Triakis Octahedron – Interactive Polyhedron Model
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "dtC"