Acronym hexat
Name hexagonal tiling,
Voronoi complex of hexagonal lattice A2

```   ©
o3o6x                        x3x6o                        x3x3x3*a                        uBxx3uxBx3uxxB3*a&#zx
. o6x : red                  x3x . : yellow,              x3x .    : blue,                .... .... uxx.&#xt : green, .... .x..3.x..    : yellow
. x6o : red                  x . x3*a : red,                 .... ux.x ....&#xt : blue,  ..x. .... ..x.3*a : teal
. x3x    : yellow               u.xx .... ....&#xt : red,   ...x3...x ....    : rose```
Vertex figure [63]
Snub derivation
```+---+---+---+      .   +   .   +
|   |   |   |          |       |
+---+---+---+      .   +   .   +
|   |   |   |  =>    /   \   /
+---+---+---+      +   .   +   .
|   |   |   |      |       |
+---+---+---+      +   .   +   .
```
Vertex layers
(first ones only)
 Layer Symmetry Subsymmetries o6o3o o6o . . o3o 1 x6o3o x6o .{6} first . o3overtex first 2 u6o . . h3overtex figure 3 x6h . . h3h 4 u6h . . 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

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
```

```ho3oo3oh3*a&#zx   (N → ∞)

o.3o.3o.3*a     | N * |  3 | 3
.o3.o3.o3*a     | * N |  3 | 3
----------------+-----+----+--
oo3oo3oo3*a&#x  | 1 1 | 3N | 2
----------------+-----+----+--
ho .. oh3*a&#zx | 3 3 |  6 | N
```
```or
o.3o.3o.3*a     & | 2N |  3 | 3
------------------+----+----+--
oo3oo3oo3*a&#x    |  2 | 3N | 2
------------------+----+----+--
ho .. oh3*a&#zx   |  6 |  6 | N
```

```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
```