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- related hyperbolic polytopes:
- pex-x3o4o3*a pabex-x3o4o3*a pac-x3o4x3*a x3o4x3*a
- general polytopal classes:
- partial Stott expansions regular noble polytopes
links
| Acronym | otrat |
| Name | hyperbolic order 8 triangular tiling |
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| Circumradius | sqrt[-1/sqrt(8)] = 0.594604 i |
| Vertex figure | [38] |
| Dual | o3o8x |
| Confer |
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External links |
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This tiling allows for a consistent 4-coloring of either of the 2 triangle classes (alternate ones). Any such choice then implies a similar 4-coloring of the other triangle class: provided by the complement (as a subset) of adjacent ones. This leads in total to a 8-coloring (if "red" and "no red neighbours" etc. would be distinguished). But these pairs of triangle classes well could be unified in turn, providing thus a coresponding 4-coloring as well.
Seen as an infinite abstract polytope, it further allows for a realization as infinite (pseudo)regular skew polyhedron too, described by snub cubes attached in a cubical lattice.
Incidence matrix according to Dynkin symbol
x3o8o (N → ∞) . . . | 3N | 8 | 8 ------+----+-----+--- x . . | 2 | 12N | 2 ------+----+-----+--- x3o . | 3 | 3 | 8N
o3x3o4*a (N → ∞) . . . | 3N | 8 | 4 4 ---------+----+-----+------ . x . | 2 | 12N | 1 1 ---------+----+-----+------ o3x . | 3 | 3 | 4N * . x3o | 3 | 3 | * 4N
(as 4-coloring of either of the previous triangle classes) (N → ∞) N * * * * * | 2 2 2 2 0 0 0 0 0 0 0 0 | 2 2 0 0 2 2 0 0 rg/-b-s * N * * * * | 2 0 0 0 2 2 2 0 0 0 0 0 | 2 0 2 0 2 0 2 0 rb/-g-s * * N * * * | 0 2 0 0 2 0 0 2 2 0 0 0 | 2 0 0 2 0 2 2 0 rs/-g-b * * * N * * | 0 0 2 0 0 2 0 0 0 2 2 0 | 0 2 2 0 2 0 0 2 gb/-r-s * * * * N * | 0 0 0 2 0 0 0 2 0 2 0 2 | 0 2 0 2 0 2 0 2 gs/-r-b * * * * * N | 0 0 0 0 0 0 2 0 2 0 2 2 | 0 0 2 2 0 0 2 2 bs/-r-g ------------+-------------------------------------+------------------------ 1 1 0 0 0 0 | 2N * * * * * * * * * * * | 1 0 0 0 1 0 0 0 1 0 1 0 0 0 | * 2N * * * * * * * * * * | 1 0 0 0 0 1 0 0 1 0 0 1 0 0 | * * 2N * * * * * * * * * | 0 1 0 0 1 0 0 0 1 0 0 0 1 0 | * * * 2N * * * * * * * * | 0 1 0 0 0 1 0 0 0 1 1 0 0 0 | * * * * 2N * * * * * * * | 1 0 0 0 0 0 1 0 0 1 0 1 0 0 | * * * * * 2N * * * * * * | 0 0 1 0 1 0 0 0 0 1 0 0 0 1 | * * * * * * 2N * * * * * | 0 0 1 0 0 0 1 0 0 0 1 0 1 0 | * * * * * * * 2N * * * * | 0 0 0 1 0 1 0 0 0 0 1 0 0 1 | * * * * * * * * 2N * * * | 0 0 0 1 0 0 1 0 0 0 0 1 1 0 | * * * * * * * * * 2N * * | 0 1 0 0 0 0 0 1 0 0 0 1 0 1 | * * * * * * * * * * 2N * | 0 0 1 0 0 0 0 1 0 0 0 0 1 1 | * * * * * * * * * * * 2N | 0 0 0 1 0 0 0 1 ------------+-------------------------------------+------------------------ 1 1 1 0 0 0 | 1 1 0 0 1 0 0 0 0 0 0 0 | 2N * * * * * * * r 1 0 0 1 1 0 | 0 0 1 1 0 0 0 0 0 1 0 0 | * 2N * * * * * * g 0 1 0 1 0 1 | 0 0 0 0 0 1 1 0 0 0 1 0 | * * 2N * * * * * b 0 0 1 0 1 1 | 0 0 0 0 0 0 0 1 1 0 0 1 | * * * 2N * * * * s 1 1 0 1 0 0 | 1 0 1 0 0 1 0 0 0 0 0 0 | * * * * 2N * * * -s 1 0 1 0 1 0 | 0 1 0 1 0 0 0 1 0 0 0 0 | * * * * * 2N * * -b 0 1 1 0 0 1 | 0 0 0 0 1 0 1 0 1 0 0 0 | * * * * * * 2N * -g 0 0 0 1 1 1 | 0 0 0 0 0 0 0 0 0 1 1 1 | * * * * * * * 2N -r or (by unifying of r and -r etc.) N * * | 0 2 2 2 2 0 | 2 2 2 2 rg/bs * N * | 2 0 2 2 0 2 | 2 2 2 2 rb/gs * * N | 2 2 0 0 2 2 | 2 2 2 2 rs/gb ------+-------------------+------------ 0 1 1 | 2N * * * * * | 1 1 0 0 rg 1 0 1 | * 2N * * * * | 1 0 1 0 rb 1 1 0 | * * 2N * * * | 1 0 0 1 rs 1 1 0 | * * * 2N * * | 0 1 1 0 gb 1 0 1 | * * * * 2N * | 0 1 0 1 gs 0 1 1 | * * * * * 2N | 0 0 1 1 bs ------+-------------------+------------ 1 1 1 | 1 1 1 0 0 0 | 2N * * * r 1 1 1 | 1 0 0 1 1 0 | * 2N * * g 1 1 1 | 0 1 0 1 0 1 | * * 2N * b 1 1 1 | 0 0 1 0 1 1 | * * * 2N s
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