| 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 edit
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.