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:
“The unique eludes us; yet we remain faithful to the ideal of it; and in spite of sense and of our merely abstract thinking, it becomes for us the most real thing in the actual world, although for us it is the elusive goal of an infinite quest.”
—Josiah Royce (1855–1916)
“They were regular in being gay, they learned little things that are things in being gay, they learned many little things that are things in being gay, they were gay every day, they were regular, they were gay, they were gay the same length of time every day, they were gay, they were quite regularly gay.”
—Gertrude Stein (1874–1946)