User:Tomruen/Eight-cube decahedron

Eight cube decahedron
Seen here with 2 yellow concave hexagons, 4 red concave hexagons, 2 green squares, and 2 blue central intersecting concave octagons.
Faces	10: 2 concave octagons 4+2 concave hexagons 2 squares
Edges	30
Vertices	20
Euler characteristic	0
Genus	2
Symmetry group	[2⁺,2], (2*2), order 4
Rotation group	[2]⁺, (22), order 2
Dual polyhedron	?
Properties	Nonorientable, polycube

In geometry, an eight-cube decahedron is a non-orientable decahedron. It has 10 faces, 30 edges, and 20 vertices. With polyomino faces, it is a polycube. It has [2⁺,2] symmetry order 4, with one reflection plane, and one 2-fold rotation axis.

It was constructed as a simple example of a non-orientable polyhedron. Its Euler_characteristic is zero, and its genus is 2. It can be seen as a connected sum decomposition of two real projective planes.

Construction

As a polycube it can be constructed as the union of 8 of 12 cubes within a 3×2×2 cubic honeycomb. Coplanar neighboring squares are merged into polyominos, resulting in ten total faces: two as squares, 2 hexagons as 3 squares combined in a tromino V, 4 hexagons as 4 squares combined in a tetromino L/J, and 2 intersecting octagons as 4 squares combined in a tetromino S/Z.

The decahedron has 20 vertices, 6, 8, and 6 by planar levels. It has 30 edges, but missing an edge in the center. The two S and Z faces intersect each other orthogonally through the center. If two coinciding central edges were added and the Z/S faces divided, it could be a toroidal polyhedron with a degenerate digonal hole.

It has the appearance of a simple toroidal polyhedron with a cyclic volume between the 8 cubes, but the central crossing faces swap the interior and exterior, so it becomes a one-sided polyhedron.

References

[1]

Construction

See also

References