Acronym hexat
Name hexagonal tiling,
Voronoi complex of hexagonal lattice
 
   ©   
Vertex figure [63]
Snub derivation
+---+---+---+      .   +   .   +
|   |   |   |          |       |
+---+---+---+      .   +   .   +
|   |   |   |  =>    /   \   /  
+---+---+---+      +   .   +   .
|   |   |   |      |       |    
+---+---+---+      +   .   +   .
Vertex layers
(first ones only)
LayerSymmetrySubsymmetries
 o6o3oo6o .. o3o
1x6o3ox6o .
{6} first
. o3o
vertex first
2u6o .. h3o
vertex figure
3x6h .. h3h
4u6h .. o3H
.........
(H=2h)
Dual trat
Confer
Grünbaumian relatives:
2hexat (?)
related CRF tilings:
uBxx3xoAo3xooA3*a&#zx   uBxx3uxBx3xooA3*a&#zx  
related rhomb tilings:
rhombic xAoo3xoAo3xooA3*a&#zx   rhombic uBxx3xoAo3xooA3*a&#zx   rhombic uBxx3uxBx3xooA3*a&#zx  
general polytopal classes:
partial Stott expansions   regular   noble polytopes  
External
links
wikipedia

Both, trat and the tri-hexagonal rhombic tiling (derived from the former by combining 2 triangles each) could be considered as a vertex overlay of hexat plus 3 shifted copies of trats with edge size 3 (see there). Then this decomposition allows for a 3-step partial Stott expansion of the latter running through rhombic uBxx3xoAo3xooA3*a&#zx and rhombic uBxx3uxBx3xooA3*a&#zx towards a pure (6-colored) hexat (A = 3x, B = 4x).


Incidence matrix according to Dynkin symbol

o3o6x   (N → ∞)

. . . | 2N |  3 | 3
------+----+----+--
. . x |  2 | 3N | 2
------+----+----+--
. o6x |  6 |  6 | N

snubbed forms: o3o6s

x3x6o   (N → ∞)

. . . | 6N |  1  2 |  2 1
------+----+-------+-----
x . . |  2 | 3N  * |  2 0
. x . |  2 |  * 6N |  1 1
------+----+-------+-----
x3x . |  6 |  3  3 | 2N *
. x6o |  6 |  0  6 |  * N

snubbed forms: β3x6o, s3s6o

x3x3x3*a   (N → ∞)

. . .    | 6N |  1  1  1 | 1 1 1
---------+----+----------+------
x . .    |  2 | 3N  *  * | 1 1 0
. x .    |  2 |  * 3N  * | 1 0 1
. . x    |  2 |  *  * 3N | 0 1 1
---------+----+----------+------
x3x .    |  6 |  3  3  0 | N * *
x . x3*a |  6 |  3  0  3 | * N *
. x3x    |  6 |  0  3  3 | * * N

snubbed forms: s3s3s3*a

o3o6/5x   (N → ∞)

. .   . | 2N |  3 | 3
--------+----+----+--
. .   x |  2 | 3N | 2
--------+----+----+--
. o6/5x |  6 |  6 | N

o3/2o6x   (N → ∞)

.   . . | 2N |  3 | 3
--------+----+----+--
.   . x |  2 | 3N | 2
--------+----+----+--
.   o6x |  6 |  6 | N

o3/2o6/5x   (N → ∞)

.   .   . | 2N |  3 | 3
----------+----+----+--
.   .   x |  2 | 3N | 2
----------+----+----+--
.   o6/5x |  6 |  6 | N

x3x6/5o   (N → ∞)

. .   . | 6N |  1  2 |  2 1
--------+----+-------+-----
x .   . |  2 | 3N  * |  2 0
. x   . |  2 |  * 6N |  1 1
--------+----+-------+-----
x3x   . |  6 |  3  3 | 2N *
. x6/5o |  6 |  0  6 |  * N

s6x3x   (N → ∞)

demi( . . . ) | 6N |  1  1  1 | 1 1 1
--------------+----+----------+------
demi( . x . ) |  2 | 3N  *  * | 1 1 0
demi( . . x ) |  2 |  * 3N  * | 0 1 1
sefa( s6x . ) |  2 |  *  * 3N | 1 0 1
--------------+----+----------+------
      s6x .     6 |  3  0  3 | N * *
demi( . x3x ) |  6 |  3  3  0 | * N *
sefa( s6x3x ) |  6 |  0  3  3 | * * N

starting figure: x6x3x

x∞s2s∞o   (N → ∞)

demi( . . . . ) | 2N | 1  2 | 3
----------------+----+------+--
demi( x . . . ) |  2 | N  * | 2
      . s2s .   |  2 | * 2N | 2
----------------+----+------+--
sefa( x∞s2s∞o ) |  6 | 2  4 | N

starting figure: x∞x2x∞o

uBxx3uxBx3uxxB3*a&#zx   (N → ∞, B = 4x)

o...3o...3o...3*a     | 6N  *  *  * |  1  1  1  0  0  0  0  0  0 |  1  1  1 0 0 0
.o..3.o..3.o..3*a     |  * 6N  *  * |  1  0  0  1  1  0  0  0  0 |  1  1  0 1 0 0
..o.3..o.3..o.3*a     |  *  * 6N  * |  0  1  0  0  0  1  1  0  0 |  1  0  1 0 1 0
...o3...o3...o3*a     |  *  *  * 6N |  0  0  1  0  0  0  0  1  1 |  0  1  1 0 0 1
----------------------+-------------+----------------------------+---------------
oo..3oo..3oo..3*a&#x  |  1  1  0  0 | 6N  *  *  *  *  *  *  *  * |  1  1  0 0 0 0
o.o.3o.o.3o.o.3*a&#x  |  1  0  1  0 |  * 6N  *  *  *  *  *  *  * |  1  0  1 0 0 0
o..o3o..o3o..o3*a&#x  |  1  0  0  1 |  *  * 6N  *  *  *  *  *  * |  0  1  1 0 0 0
.... .x.. ....        |  0  2  0  0 |  *  *  * 3N  *  *  *  *  * |  0  1  0 1 0 0
.... .... .x..        |  0  2  0  0 |  *  *  *  * 3N  *  *  *  * |  1  0  0 1 0 0
..x. .... ....        |  0  0  2  0 |  *  *  *  *  * 3N  *  *  * |  0  0  1 0 1 0
.... .... ..x.        |  0  0  2  0 |  *  *  *  *  *  * 3N  *  * |  1  0  0 0 1 0
...x .... ....        |  0  0  0  2 |  *  *  *  *  *  *  * 3N  * |  0  0  1 0 0 1
.... ...x ....        |  0  0  0  2 |  *  *  *  *  *  *  *  * 3N |  0  1  0 0 0 1
----------------------+-------------+----------------------------+---------------
.... .... uxx.   &#xt |  2  2  2  0 |  2  2  0  0  1  0  1  0  0 | 3N  *  * * * *
.... ux.x ....   &#xt |  2  2  0  2 |  2  0  2  1  0  0  0  0  1 |  * 3N  * * * *
u.xx .... ....   &#xt |  2  0  2  2 |  0  2  2  0  0  1  0  1  0 |  *  * 3N * * *
.... .x..3.x..        |  0  6  0  0 |  0  0  0  3  3  0  0  0  0 |  *  *  * N * *
..x. .... ..x.3*a     |  0  0  6  0 |  0  0  0  0  0  3  3  0  0 |  *  *  * * N *
...x3...x ....        |  0  0  0  6 |  0  0  0  0  0  0  0  3  3 |  *  *  * * * N

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