Regular Skew Polyhedron - Infinite Regular Skew Polyhedra

Infinite Regular Skew Polyhedra

There are 3 regular skew polyhedra, the first two being duals:

  1. {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.)
  2. {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.)
  3. {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

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