Tetrahemihexahedron
Tetrahemihexahedron  

Type  Uniform star polyhedron 
Elements  F = 7, E = 12 V = 6 (χ = 1) 
Faces by sides  4{3}+3{4} 
Coxeter diagram 
(doublecovering) 
Wythoff symbol  3/2 3  2 (doublecovering) 
Symmetry group  T_{d}, [3,3], *332 
Index references  U_{04}, C_{36}, W_{67} 
Dual polyhedron  Tetrahemihexacron

Vertex figure  3.4.3/2.4 
Bowers acronym  Thah 
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U_{4}. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.^{[1]} Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).
The tetrahemihexahedron is the only nonprismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3  2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)
The tetrahemihexahedron 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 "halfasmany" 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.
Related surfaces
The tetrahemihexahedron is a nonorientable 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^{[2]} very similar to the Roman surface.
Roman surface 
Related polyhedra
The tetrahemihexahedron 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 of the tetrahemihexahedron is the
The tetrahemihexahedron is 2covered by the cuboctahedron,^{[2]} which accordingly has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the abstract polyhedron hemicuboctahedron.
Cuboctahedron 
Tetrahemihexahedron 
The tetrahemihexahedron may also be constructed as a crossed triangular cuploid. All cuploids and their duals are topologically projective planes.^{[3]}
3  5  7  n⁄d 

{3/2} Crossed triangular cuploid (upside down) 
Pentagrammic cuploid

{7/2} Heptagrammic cuploid 
2 
—  Crossed pentagonal cuploid (upside down) 
{7/4} Crossed heptagrammic cuploid 
4 
Tetrahemihexacron
Tetrahemihexacron  

Type  Star polyhedron 
Face  — 
Elements  F = 6, E = 12 V = 7 (χ = 1) 
Symmetry group  T_{d}, [3,3], *332 
Index references  DU_{04} 
dual polyhedron  Tetrahemihexahedron 
The tetrahemihexacron is the
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.^{[4]} In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
Topologically, the tetrahemihexacron is considered to contain seven vertices. The three vertices considered at infinity (the
References
 ^ Maeder, Roman. "04: tetrahemihexahedron". MathConsult.
 ^ ^{a} ^{b} (Richter)
 ^ Polyhedral Models of the Projective Plane, Paul Gailiunas, Bridges 2018 Conference Proceedings
 ^ (Wenninger 2003, p. 101)
 Richter, David A., Two Models of the Real Projective Plane
 MR 0730208(Page 101, Duals of the (nine) hemipolyhedra)