User:Tomruen/Six-cube enneahedron

Six cube enneahedron

Faces	9: 6 concave hexagons 3 self-crossing hexagons
Edges	27
Vertices	18
Euler characteristic	0
Genus	2
Symmetry group	[2⁺,6], (2*3), order 12
Rotation group	[2,3]⁺, (233), order 6
Dual polyhedron	?
Properties	Nonorientable, polycube

In geometry, a six-cube enneahedron is a non-orientable enneahedron. It has 9 faces, 27 edges, and 18 vertices. With polyomino faces at right angles, it is a polycube. It has [2⁺,6] symmetry order 12, with three reflection plane, and 2-fold rotation axes.

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 6 of 8 cubes within a 2×2×2 cubic honeycomb. Coplanar neighboring squares are merged into polyominos, resulting in nine total faces: 6 hexagons as 3 squares combined in a tromino V, 3 crossed-hexagons as 2 squared connected in a diagonal domino, .

The enneahedron has 18 vertices, 6, 6, and 6 by planar levels. It has 27 edges.

It has the appearance of a simple toroidal polyhedron with a cyclic volume between the 6 cubes. If two central coinciding vertices are added, it can become an ordinary polyhedron "pinched" on the coinciding vertices.

References

[1]

Construction

See also

References