In geometry, an infinite skew polyhedron is an extension of the idea of a polyhedron, consisting of regular polygon faces with nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
Infinite skew polyhedra have also been called polyhedral sponges, and also hyperbolic tessellations because they can be seen as related to hyperbolic space tessellations which also have negative angle defects. They are examples of the more general class of infinite polyhedra, or apeirohedra.
Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically, an infinite skew polyhedron divides space into two halves. If one half is thought of as solid the figure is sometimes called a partial honeycomb.
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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:
Coxeter and Petrie found three of these that filled 3-space:
| Regular skew polyhedra (partial) | ||
|---|---|---|
{4,6|4} |
{6,4|4} |
{6,6|3} |
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.
J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular pseudopolyhedrons, including the three from Coxeter and four new ones.
Gott relaxed the definition of regularity to allow his new figures. Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent. Thus, Gott's new examples are not regular by Coxeter and Petrie's definition.
Gott called the full set of regular polyhedra, regular tilings, and regular pseudopolyhedrons as regular generalized polyhedra, representable by a {p,q} Schläfli symbol, with by p-gonal faces, q around each vertex.
A.F. Wells also published a list of pseudopolyhedra in the 1960s, including different forms with the same symbol: {4,5}, {3,7}, {3,8}, {3,10}, {3,12}.
However neither the term "pseudopolyhedron" nor Gott's definition of regularity have achieved wide usage.
Prismatic form: {4,5} |
There are two prismatic forms:
{3,10} is also formed from parallel planes of triangular tilings, with alternating octahedral holes going both ways.
{5,5} is composed of 3 coplanar pentagons around a vertex and two perpendicular pentagons filling the gap.
Gott also acknowledged that there are other periodic forms of the regular planar tessellations. Both the square tiling {4,4} and triangular tiling {3,6} can be curved into approximating infinite cylinders in 3-space.
He wrote some theorems:
There are many other semiregular (vertex-transitive) infinite skew polyhedra. Wachmann, Burt and Kleinmann (1974) discovered many examples but it is not known whether their list is complete.
Three are illustrated here:
A prismatic semiregular skew polyhedron with vertex configuration 4.4.4.6. |
A (partial) semiregular skew polyhedron with vertex configuration 4.8.4.8. Related to the omnitruncated cubic honeycomb. |
A (partial) semiregular skew polyhedron with vertex configuration 3.4.4.4.4. Related to the Runcitruncated cubic honeycomb. |