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Thah

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Thah

Tetrahemihexahedron
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.

Octahedron
Tetrahemihexahedron

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: 3.4.3.4) and twice the vertices, edges, and faces.

Cuboctahedron
Tetrahemihexahedron

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

References

External links

  • MathWorld
  • Uniform polyhedra and duals
  • Paper model
  • Great Stella: software used to create main image on this page
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