Acronym ...
Name hyperbolic shieldotritetragonal tiling,
hyperbolic tritetratrigonal tiling
 
 ©   
Circumradius sqrt[-(4+9 sqrt(2))/8] = 1.446026 i
Vertex figure [3,6,4,6]
Confer
related hyperbolic polytopes:
x3o4o3*a   pex-x3o4o3*a   pabex-x3o4o3*a   pac-x3o4x3*a  
general polytopal classes:
partial Stott expansions  
External
links
wikipedia  

The naming prefix part shieldo- was made up ad hoc, kind of corresponding to the similar prefix rhombi- for the digonal counterpart of the trigonal one in this case. It also corresponds to the fact that semiregular polygons with equal edges but alternating angles are commonly called rhombs, shields, etc. Nonetheless, the here used hexagons are regular ones, just as the tetragons in the related cases are true squares. It is only that both are positioned within a lower surrounding symmetry.

This tiling allows for a consistent 4-coloring either of the triangle class or the hexagon class. Any such choice then implies a similar 4-coloring of the other one: provided by the complement (as a subset) of adjacent ones. The squares thereby would get 6-colored.


Incidence matrix according to Dynkin symbol

x3x3o4*a   (N → ∞)

. . .    | 12N |   2   2 |  2  1  1
---------+-----+---------+---------
x . .    |   2 | 12N   * |  1  1  0
. x .    |   2 |   * 12N |  1  0  1
---------+-----+---------+---------
x3x .    |   6 |   3   3 | 4N  *  *
x . o4*a |   4 |   4   0 |  * 3N  *
. x3o    |   3 |   0   3 |  *  * 4N

x3o8s   (N → ∞)

demi( . . . ) | 12N |   2   2 |  1  1  2
--------------+-----+---------+---------
demi( x . . ) |   2 | 12N   * |  1  0  1
sefa( . o8s ) |   2 |   * 12N |  0  1  1
--------------+-----+---------+---------
demi( x3o . ) |   3 |   3   0 | 4N  *  *
      . o8s      4 |   0   4 |  * 3N  *
sefa( x3o8s ) |   6 |   3   3 |  *  * 4N

starting figure: x3o8x

(as 4-coloring either of the triangle class or of the hexagon class)   (N → ∞)

2N  *  *  *  *  *  *  *  *  *  *  * |  1  1  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0 |  1  1  0  0  0  1  0  0 1 0 0 0 0 0  rg/-b
 * 2N  *  *  *  *  *  *  *  *  *  * |  1  1  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  0 |  1  1  0  0  1  0  0  0 1 0 0 0 0 0  rg/-s
 *  * 2N  *  *  *  *  *  *  *  *  * |  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0 |  1  0  1  0  0  0  1  0 0 1 0 0 0 0  rb/-g
 *  *  * 2N  *  *  *  *  *  *  *  * |  0  0  1  1  0  0  0  0  0  0  0  0  1  0  0  0  0  1  0  0  0  0  0  0 |  1  0  1  0  1  0  0  0 0 1 0 0 0 0  rb/-s
 *  *  *  * 2N  *  *  *  *  *  *  * |  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  1  0  0  0  1  0  0  0 |  1  0  0  1  0  0  1  0 0 0 1 0 0 0  rs/-g
 *  *  *  *  * 2N  *  *  *  *  *  * |  0  0  0  0  1  1  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0 |  1  0  0  1  0  1  0  0 0 0 1 0 0 0  rs/-b
 *  *  *  *  *  * 2N  *  *  *  *  * |  0  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0 |  0  1  1  0  0  0  0  1 0 0 0 1 0 0  gb/-r
 *  *  *  *  *  *  * 2N  *  *  *  * |  0  0  0  0  0  0  1  1  0  0  0  0  0  0  1  0  0  1  0  0  0  0  0  0 |  0  1  1  0  1  0  0  0 0 0 0 1 0 0  gb/-s
 *  *  *  *  *  *  *  * 2N  *  *  * |  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  0  1 |  0  1  0  1  0  0  0  1 0 0 0 0 1 0  gs/-r
 *  *  *  *  *  *  *  *  * 2N  *  * |  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  1  0  0  0  1  0  0  0  0 |  0  1  0  1  0  1  0  0 0 0 0 0 1 0  gs/-b
 *  *  *  *  *  *  *  *  *  * 2N  * |  0  0  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  1  1 |  0  0  1  1  0  0  0  1 0 0 0 0 0 1  bs/-r
 *  *  *  *  *  *  *  *  *  *  * 2N |  0  0  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0  1  0  1  0  0  0 |  0  0  1  1  0  0  1  0 0 0 0 0 0 1  bs/-g
------------------------------------+-------------------------------------------------------------------------+------------------------------------
 1  1  0  0  0  0  0  0  0  0  0  0 | 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  0  0  0  0  0  0  0 1 0 0 0 0 0  rg:r
 1  1  0  0  0  0  0  0  0  0  0  0 |  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  1  0  0  0  0  0  0 1 0 0 0 0 0  rg:g
 0  0  1  1  0  0  0  0  0  0  0  0 |  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  0  0  0  0  0  0  0 0 1 0 0 0 0  rb:r
 0  0  1  1  0  0  0  0  0  0  0  0 |  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  0  0  0  0  0 0 1 0 0 0 0  rb:b
 0  0  0  0  1  1  0  0  0  0  0  0 |  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  0  0  0  0  0  0  0 0 0 1 0 0 0  rs:r
 0  0  0  0  1  1  0  0  0  0  0  0 |  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  0  1  0  0  0  0 0 0 1 0 0 0  rs:s
 0  0  0  0  0  0  1  1  0  0  0  0 |  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  1  0  0  0  0  0  0 0 0 0 1 0 0  gb:g
 0  0  0  0  0  0  1  1  0  0  0  0 |  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  0  0  0  0  0 0 0 0 1 0 0  gb:b
 0  0  0  0  0  0  0  0  1  1  0  0 |  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  1  0  0  0  0  0  0 0 0 0 0 1 0  gs:g
 0  0  0  0  0  0  0  0  1  1  0  0 |  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  0  1  0  0  0  0 0 0 0 0 1 0  gs:s
 0  0  0  0  0  0  0  0  0  0  1  1 |  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  0  0  0  0  0 0 0 0 0 0 1  bs:b
 0  0  0  0  0  0  0  0  0  0  1  1 |  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  0  1  0  0  0  0 0 0 0 0 0 1  bs:s
 0  1  0  1  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  *  * |  1  0  0  0  1  0  0  0 0 0 0 0 0 0
 1  0  0  0  0  1  0  0  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  *  * |  1  0  0  0  0  1  0  0 0 0 0 0 0 0
 0  1  0  0  0  0  0  1  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  *  * |  0  1  0  0  1  0  0  0 0 0 0 0 0 0
 1  0  0  0  0  0  0  0  0  1  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  *  * |  0  1  0  0  0  1  0  0 0 0 0 0 0 0
 0  0  1  0  1  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  *  * |  1  0  0  0  0  0  1  0 0 0 0 0 0 0
 0  0  0  1  0  0  0  1  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  *  * |  0  0  1  0  1  0  0  0 0 0 0 0 0 0
 0  0  1  0  0  0  0  0  0  0  0  1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  *  * |  0  0  1  0  0  0  1  0 0 0 0 0 0 0
 0  0  0  0  0  1  0  0  0  1  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  *  * |  0  0  0  1  0  1  0  0 0 0 0 0 0 0
 0  0  0  0  1  0  0  0  0  0  0  1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  *  * |  0  0  0  1  0  0  1  0 0 0 0 0 0 0
 0  0  0  0  0  0  1  0  1  0  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  *  * |  0  1  0  0  0  0  0  1 0 0 0 0 0 0
 0  0  0  0  0  0  1  0  0  0  1  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N  * |  0  0  1  0  0  0  0  1 0 0 0 0 0 0
 0  0  0  0  0  0  0  0  1  0  1  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 2N |  0  0  0  1  0  0  0  1 0 0 0 0 0 0
------------------------------------+-------------------------------------------------------------------------+------------------------------------
 1  1  1  1  1  1  0  0  0  0  0  0 |  1  0  1  0  1  0  0  0  0  0  0  0  1  1  0  0  1  0  0  0  0  0  0  0 | 2N  *  *  *  *  *  *  * * * * * * *  r
 1  1  0  0  0  0  1  1  1  1  0  0 |  0  1  0  0  0  0  1  0  1  0  0  0  0  0  1  1  0  0  0  0  0  1  0  0 |  * 2N  *  *  *  *  *  * * * * * * *  g
 0  0  1  1  0  0  1  1  0  0  1  1 |  0  0  0  1  0  0  0  1  0  0  1  0  0  0  0  0  0  1  1  0  0  0  1  0 |  *  * 2N  *  *  *  *  * * * * * * *  b
 0  0  0  0  1  1  0  0  1  1  1  1 |  0  0  0  0  0  1  0  0  0  1  0  1  0  0  0  0  0  0  0  1  1  0  0  1 |  *  *  * 2N  *  *  *  * * * * * * *  s
 0  1  0  1  0  0  0  1  0  0  0  0 |  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  1  0  0  0  0  0  0 |  *  *  *  * 2N  *  *  * * * * * * *  -s
 1  0  0  0  0  1  0  0  0  1  0  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  1  0  0  0  0 |  *  *  *  *  * 2N  *  * * * * * * *  -b
 0  0  1  0  1  0  0  0  0  0  0  1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  1  0  0  0 |  *  *  *  *  *  * 2N  * * * * * * *  -g
 0  0  0  0  0  0  1  0  1  0  1  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1 |  *  *  *  *  *  *  * 2N * * * * * *  -r
 2  2  0  0  0  0  0  0  0  0  0  0 |  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * N * * * * *
 0  0  2  2  0  0  0  0  0  0  0  0 |  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * * N * * * *
 0  0  0  0  2  2  0  0  0  0  0  0 |  0  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * * * N * * *
 0  0  0  0  0  0  2  2  0  0  0  0 |  0  0  0  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * * * * N * *
 0  0  0  0  0  0  0  0  2  2  0  0 |  0  0  0  0  0  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * * * * * N *
 0  0  0  0  0  0  0  0  0  0  2  2 |  0  0  0  0  0  0  0  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  *  *  *  *  * * * * * * N

© 2004-2019
top of page