Infinite Regular Skew Polyhedra
There are 3 regular skew polyhedra, the first two being duals:
- {4,6|4}: 6 squares on a vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless cube.)
- {6,4|4}: 4 hexagons on a vertex (related to bitruncated cubic honeycomb, constructed by truncated octahedron with their square faces removed and linking hole pairs of holes together.)
- {6,6|3}: 6 hexagons on a vertex (related to quarter cubic honeycomb, constructed by truncated tetrahedron cells, removing triangle faces, and linking sets of four around a faceless tetrahedron.)
Also solutions to the equation above are the Euclidean regular tilings {3,6}, {6,3}, {4,4}, represented as {3,6|6}, {6,3|6}, and {4,4|∞}.
Here are some partial representations, vertical projected views of their skew vertex figures, and partial corresponding uniform honeycombs.
| Partial polyhedra | ||
|---|---|---|
{4,6|4} |
{6,4|4} |
{6,6|3} |
| Vertex figures | ||
{4,6} |
{6,4} |
{6,6} |
| Related convex uniform honeycombs | ||
Runcinated cubic honeycomb t0,3{4,3,4} |
Bitruncated cubic t1,2{4,3,4} |
quarter cubic honeycomb t0,1{3} |
Read more about this topic: Regular Skew Polyhedron
Famous quotes containing the words infinite and/or regular:
“Let the will embrace the highest ideals freely and with infinite strength, but let action first take hold of what lies closest.”
—Franz Grillparzer (1791–1872)
“A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.”
—Henry David Thoreau (1817–1862)