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A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered ...

A heptahedron is a polyhedron with seven faces. Because there are 34 heptahedral graphs, there are 34 topologically distinct convex heptahedra. There are three semiregular ...

Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron (Ball and Coxeter ...

There exist polyhedra which can be plaited (braided). Examples include a plaited cube and plaited icosahedron illustrated above (Pargeter 1959, Wells 1991). In the above ...

Sprague (1963) considered the problem of "rolling" five cubes, each which an upright letter "A" on its top, on a chessboard. Here "rolling" means the cubes are moved from ...

A hexagonal antiprism is an antiprism having hexagons as its bases. A stretched hexagonal antiprism is the convex hull of the tetrahedron 3-compound. The dual polyhedron of ...

A cube 13-compound can be constructed as the dual of the octahedron 13-compound. It will be implemented in a future version of the Wolfram Language as ...

A cube 9-compound can be constructed from the vertices of the first dodecahedron 2-compound. It will be implemented in a future version of the Wolfram Language as ...

There are a number of attractive polyhedron compounds consisting of ten octahedra. These compounds will be implemented in a future version of the Wolfram Language as ...

There are a number of attractive polyhedron compounds consisting of 12 octahedra. This compound will be implemented in a future version of the Wolfram Language as ...