Acronym ...
Name hyperbolic order 8 triangular tiling
 
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Circumradius sqrt[-sqrt(2)]/2 = 0.594604 i
Vertex figure [38]
Dual o3o8x
Confer
related hyperbolic polytopes:
pex-x3o4o3*a   pabex-x3o4o3*a   pac-x3o4x3*a   x3o4x3*a  
general polytopal classes:
partial Stott expansions   regular   noble polytopes  
External
links
wikipedia   wikipedia  

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.

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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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