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:
“And universal Nature, through her vast
And crowded whole, an infinite paroquet,
Repeats one note.”
—Ralph Waldo Emerson (1803–1882)
“While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.”
—Anton Pavlovich Chekhov (1860–1904)