Acronym trat
Name triangular tiling,
Delone complex of hexagonal lattice
 
       
regular 2-coloring of triangles
©
3-coloring of vertices 6-coloring of triangles
Vertex figure [36]
Vertex layers
(first ones only)
LayerSymmetrySubsymmetries
 o3o6oo3o .. o6o
1x3o6ox3o .
{3} first
. o6o
vertex first
2o3u .. x6o
vertex figure
3u3x .. o6h
4x3A .. u6o
.........
(A=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
links
wikipedia

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 = 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

xAoo3xoAo3xooA3*a&#zx   (N → ∞, A = 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

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