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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. That is, the right depicted infinite pseudopolyhedron evidently is of the same type as x3o8o. However, it still is not equivalent, simply because it clearly shows up those square holes, which in the hyperbolic tiling are not contained.
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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