Acronym trat
Name triangular tiling,
Delone complex of hexagonal lattice A2

 regular 2-coloring of triangles `©` 3-coloring of vertices 6-coloring of triangles
Vertex figure [36]
Vertex layers
(first ones only)
 Layer Symmetry Subsymmetries o3o6o o3o . . o6o 1 x3o6o x3o .{3} first . o6overtex first 2 o3u . . x6overtex figure 3 u3x . . o6h 4 x3d . . u6o ... ... ...
(d = u+x = 3x)
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform tiling member: ditatha )
Dual hexat
Confer
uniform relative:
that   hexat   ditatha
related CRF tilings:
pextrat   pacrothat   uBxx3xoAo3xooA3*a&#zx   uBxx3uxBx3xooA3*a&#zx
related rhomb tiling:
rhombic xAoo3xoAo3xooA3*a&#zx
general polytopal classes:
partial Stott expansions   regular   noble polytopes
External

The 3-coloring of the vertices allows for partial Stott expansions running through pextrat and pacrothat finally towards rothat.

This tiling also could be considered as a vertex overlay of hexat plus 3 shifted copies of trats with edge size 3 (see below, resp. the above 6-coloring of triangles). Then this decomposition allows for a further 3-step transformation running through uBxx3xoAo3xooA3*a&#zx and uBxx3uxBx3xooA3*a&#zx towards a pure hexat (A = d = 3x, B = 4x). – So this transformation sequence is not a true partial Stott expansion one because pairs of triangles become transformed into single hexagons. But this could be recovered by combining these pairs into rhombs. The corresponding sequence member here then would be rhomb-xAoo3xoAo3xooA3*a&#zx.

Incidence matrix according to Dynkin symbol

```x3o6o   (N → ∞)

. . . | N |  6 |  6
------+---+----+---
x . . | 2 | 3N |  2
------+---+----+---
x3o . | 3 |  3 | 2N
```

```x3o3o3*a    (N → ∞)

. . .    | N |  6 | 3 3
---------+---+----+----
x . .    | 2 | 3N | 1 1
---------+---+----+----
x3o .    | 3 |  3 | N *
x . o3*a | 3 |  3 | * N
```

```x3o6/5o   (N → ∞)

. .   . | N |  6 |  6
--------+---+----+---
x .   . | 2 | 3N |  2
--------+---+----+---
x3o   . | 3 |  3 | 2N
```

```x3/2o6o   (N → ∞)

.   . . | N |  6 |  6
--------+---+----+---
x   . . | 2 | 3N |  2
--------+---+----+---
x3/2o . | 3 |  3 | 2N
```

```x3/2o6/5o   (N → ∞)

.   .   . | N |  6 |  6
----------+---+----+---
x   .   . | 2 | 3N |  2
----------+---+----+---
x3/2o   . | 3 |  3 | 2N
```

```x3/2o3/2o3*a    (N → ∞)

.   .   .    | N |  6 | 3 3
-------------+---+----+----
x   .   .    | 2 | 3N | 1 1
-------------+---+----+----
x3/2o   .    | 3 |  3 | N *
x   .   o3*a | 3 |  3 | * N
```

```o3/2x3/2o3*a    (N → ∞)

.   .   .    | N |  6 | 3 3
-------------+---+----+----
.   x   .    | 2 | 3N | 1 1
-------------+---+----+----
o3/2x   .    | 3 |  3 | N *
.   x3/2o .  | 3 |  3 | * N
```

```s6o3o    (N → ∞)

demi( . . . ) | N |  6 | 3 3
--------------+---+----+----
sefa( s6o . ) | 2 | 3N | 1 1
--------------+---+----+----
s6o .   ♦ 3 |  3 | N *
sefa( s6o3o ) | 3 |  3 | * N

starting figure: x6o3o
```

```s3s6o    (N → ∞)

demi( . . . ) | 3N |  4  2 |  2 1  3
--------------+----+-------+--------
sefa( s3s . ) |  2 | 6N  * |  1 0  1
sefa( . s6o ) |  2 |  * 3N |  0 1  1
--------------+----+-------+--------
s3s .   ♦  3 |  3  0 | 2N *  *
. s6o   ♦  3 |  0  3 |  * N  *
sefa( s3s6o ) |  3 |  2  1 |  * * 3N

starting figure: x3x6o
```

```s3s3s3*a   (N → ∞)

demi( . . .    ) | 3N |  2  2  2 | 1 1 1  3
-----------------+----+----------+---------
sefa( s3s .    ) |  2 | 3N  *  * | 1 0 0  1
sefa( s . s3*a ) |  2 |  * 3N  * | 0 1 0  1
sefa( . s3s    ) |  2 |  *  * 3N | 0 0 1  1
-----------------+----+----------+---------
s3s .      ♦  3 |  3  0  0 | N * *  *
s . s3*a   ♦  3 |  0  3  0 | * N *  *
. s3s      ♦  3 |  0  0  3 | * * N  *
sefa( s3s3s3*a ) |  3 |  1  1  1 | * * * 3N

starting figure: x3x3x3*a
```

```xdoo3xodo3xood3*a&#zx   (N → ∞, d = 3x)

o...3o...3o...3*a    | 6N * * * |  1  1  1  1  1  1 |  1  1  1  1  1  1
.o..3.o..3.o..3*a    |  * N * * |  0  0  0  6  0  0 |  3  3  0  0  0  0
..o.3..o.3..o.3*a    |  * * N * |  0  0  0  0  6  0 |  0  0  3  3  0  0
...o3...o3...o3*a    |  * * * N |  0  0  0  0  0  6 |  0  0  0  0  3  3
---------------------+----------+-------------------+------------------
x... .... ....       |  2 0 0 0 | 3N  *  *  *  *  * |  0  0  1  0  1  0
.... x... ....       |  2 0 0 0 |  * 3N  *  *  *  * |  1  0  0  0  0  1
.... .... x...       |  2 0 0 0 |  *  * 3N  *  *  * |  0  1  0  1  0  0
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  * |  0  0  1  1  0  0
o..o3o..o3o..o3*a&#x |  1 0 0 1 |  *  *  *  *  * 6N |  0  0  0  0  1  1
---------------------+----------+-------------------+------------------
.... xo.. ....   &#x |  2 1 0 0 |  0  1  0  2  0  0 | 3N  *  *  *  *  *
.... .... xo..   &#x |  2 1 0 0 |  0  0  1  2  0  0 |  * 3N  *  *  *  *
x.o. .... ....   &#x |  2 0 1 0 |  1  0  0  0  2  0 |  *  * 3N  *  *  *
.... .... x.o.   &#x |  2 0 1 0 |  0  0  1  0  2  0 |  *  *  * 3N  *  *
x..o .... ....   &#x |  2 0 0 1 |  1  0  0  0  0  2 |  *  *  *  * 3N  *
.... x..o ....   &#x |  2 0 0 1 |  0  1  0  0  0  2 |  *  *  *  *  * 3N
```

```:xo:∞:ox:&##x   (N → ∞)   → heights = sqrt(3)/2 = 0.866025

o. ∞ o.     | N * | 2  2 0  2 | 2 1 2 1
.o ∞ .o     | * N | 0  2 2  2 | 1 2 1 2
-------------+-----+-----------+--------
x.   ..     | 2 0 | N  * *  * | 1 0 1 0
oo ∞ oo &#x | 1 1 | * 2N *  * | 1 1 0 0
..   .x     | 0 2 | *  * N  * | 0 1 0 1
:oo:∞:oo:&#x | 1 1 | *  * * 2N | 0 0 1 1
-------------+-----+-----------+--------
xo   .. &#x | 2 1 | 1  2 0  0 | N * * *
..   ox &#x | 1 2 | 0  2 1  0 | * N * *
:xo: :..:&#x | 2 1 | 1  0 0  2 | * * N *
:..: :ox:&#x | 1 2 | 0  0 1  2 | * * * N
```