According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra.
Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, reresented by {l,m|n}, follow this equation:
- 2*sin(π/l)*sin(π/m)=cos(π/n)
Coxeter and Petrie found three of these that filled 3-space:
| Regular skew polyhedra (partial) | ||
|---|---|---|
{4,6|4} |
{6,4|4} |
{6,6|3} |
There also exist chiral skew polyhedra of types {4,6}, {6,4}, and {6,6}. These skew polyhedra are vertex-transitive, edge-transitive, and face-transitive, but not mirror symmetric (Schulte 2004).
Beyond Euclidean 3-space, C. W. L. Garner determined a set of 32 regular skew polyhedra in hyperbolic 3-space, derived from the 4 regular hyperbolic honeycombs.
Read more about this topic: Infinite Skew Polyhedron
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