User:Tomruen/Two-cube enneahedron

Two cube enneahedron
Showing 6 red square faces, 3 yellow crossed-hexagon races (one in white). There is no central vertex.
Faces	9: 6 squares 3 self-crossing hexagons
Edges	21
Vertices	14
Euler characteristic	2
Convex hull	Stretched rhombic dodecahedron
Genus	0
Symmetry group	D_3d, [2⁺,6], (2*3), order 12
Rotation group	D₃, [2,3]⁺, (233), order 6
Dual polyhedron	Double-stacked octahedron
Properties	polycube

In geometry, a two-cube enneahedron is a nonconvex enneahedron. It has 9 faces, 21 edges, and 24 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.

Its Euler_characteristic is 2, a topological sphere. Its convex form can be seen as a triangular prism with the top and bottom triangles dissected with mid-edges and center into 3 coplanar kite faces. Its dual can be seen as two regular octahedra sharing a common face.

Construction

As a polycube it can be constructed as the union of 2 cubes with one comoon vertex. Coplanar neighboring squares are merged into polyominos, resulting in nine total faces: 6 square and 3 crossed-hexagons as 2 squared connected in a diagonal domino, .

The enneahedron has 14 vertices, 4, 6, and 4 by planar levels. It has 24 edges.

Convex form

The geometry can be adjusted into a polyhedron with 3 regular hexagons in a loop, with 2 polar vertices added to make kite faces on top and bottom. It has D_3h symmetry, order 12. It is similar to an elongated rhombic dodecahedron, which has 4-fold symmetry rather than 3-fold.

References

[1]

Construction

Convex form

See also

References