World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0025925754
Reproduction Date:

Title: Thah  
Author: World Heritage Encyclopedia
Language: English
Subject: Tripuri cuisine
Publisher: World Heritage Encyclopedia


Type Uniform star polyhedron
Elements F = 7, E = 12
V = 6 (χ = 1)
Faces by sides 4{3}+3{4}
Wythoff symbol(s) 3/2 3 | 2 (double-covering)
Symmetry group Td, [3,3], *332
Index references U04, C36, W67
Bowers acronym Thah
Vertex figure) dual polyhedron)

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. Its Coxeter-Dynkin diagram is .

It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but actually that represents a double covering of the tetrahemihexahedron with 8 triangles and 6 squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)

It is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.

The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.

It is the three-dimensional demicross polytope.

Related polyhedra

It has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.


The dual figure is the tetrahemihexacron.

It is 2-covered by the cuboctahedron,[1] which accordingly has the same abstract vertex figure (2 triangles and two squares: and twice the vertices, edges, and faces.


Related surfaces

It is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a projective polyhedron, yielding a representation of the real projective plane[1] very similar to the Roman surface.

Roman surface


External links

  • MathWorld
  • Uniform polyhedra and duals
  • Paper model
  • Great Stella: software used to create main image on this page
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.