where:
    B		A = x3o3o3*a (trat)
A    		B = o3x3o3*a (gyro trat)
    C		C = o3o3x3*a (alt. gyro trat)

This euclidean honeycomb uses trat (for cells) in the sense of an infinite horohedron. Even honeycomb slabs, like tratap, are being used herein.

Note however, as a gyrotrigonism, within position space of the lace city display it just happens to remain of finite reach. Thence it more looks like an infintely long triangular needle. – If, on the other hand, these triangular needles would get arranged along an orthogonal trat, the according blend "trat gytrat" much better is known as hext.

Incidence matrix according to Dynkin symbol

xoo3oxo3oox3*a &#x   (N → ∞)   → pw. heights = sqrt(2/3) = 0.816497

o..3o..3o..3*a     & | 3N ♦  6  6 |  3  3  9  9  6 | 1  6  4  4 12 | 2  7
---------------------+----+-------+----------------+---------------+-----
x.. ... ...        & |  2 | 9N  * |  1  1  1  1  0 | 1  2  1  1  1 | 2  2
oo.3oo.3oo.3*a &#x & |  2 |  * 9N |  0  0  2  2  2 | 0  2  1  1  5 | 1  4
---------------------+----+-------+----------------+---------------+-----
x..3o.. ...        & |  3 |  3  0 | 3N  *  *  *  * | 1  1  0  1  0 | 2  1
x.. ... o..3*a     & |  3 |  3  0 |  * 3N  *  *  * | 1  1  1  0  0 | 2  1
xo. ... ...    &#x & |  3 |  1  2 |  *  * 9N  *  * | 0  1  1  0  1 | 1  2
... ox. ...    &#x & |  3 |  1  2 |  *  *  * 9N  * | 0  1  0  1  1 | 1  2
ooo3ooo3ooo3*a &#x   |  3 |  0  3 |  *  *  *  * 6N | 0  0  0  0  3 | 0  3
---------------------+----+-------+----------------+---------------+-----
x..3o..3o..3*a     & ♦  N | 3N  0 |  N  N  0  0  0 | 3  *  *  *  * | 2  0
xo.3ox. ...    &#x & ♦  6 |  6  6 |  1  1  3  3  0 | * 3N  *  *  * | 1  1
xo. ... oo.3*a &#x & ♦  4 |  3  3 |  0  1  3  0  0 | *  * 3N  *  * | 1  1
... ox.3oo.    &#x & ♦  4 |  3  3 |  1  0  0  3  0 | *  *  * 3N  * | 1  1
xoo ... ...    &#x & ♦  4 |  1  5 |  0  0  1  1  2 | *  *  *  * 9N | 0  2
---------------------+----+-------+----------------+---------------+-----
xo.3ox.3oo.3*a &#x & ♦ 2N | 6N 3N | 2N 2N 3N 3N  0 | 2  N  N  N  0 | 3  *
xoo3oxo ...    &#x & ♦  7 |  6 12 |  1  1  6  6  6 | 0  1  1  1  6 | * 3N

or
o..3o..3o..3*a     & | 3N ♦  6  6 |  6  18  6 | 1  6  8 12 | 2  7
---------------------+----+-------+-----------+------------+-----
x.. ... ...        & |  2 | 9N  * |  2   2  0 | 1  2  2  1 | 2  2
oo.3oo.3oo.3*a &#x & |  2 |  * 9N |  0   4  2 | 0  2  2  5 | 1  4
---------------------+----+-------+-----------+------------+-----
x..3o.. ...        & |  3 |  3  0 | 6N   *  * | 1  1  1  0 | 2  1
xo. ... ...    &#x & |  3 |  1  2 |  * 18N  * | 0  1  1  1 | 1  2
ooo3ooo3ooo3*a &#x   |  3 |  0  3 |  *   * 6N | 0  0  0  3 | 0  3
---------------------+----+-------+-----------+------------+-----
x..3o..3o..3*a     & ♦  N | 3N  0 | 2N   0  0 | 3  *  *  * | 2  0
xo.3ox. ...    &#x & ♦  6 |  6  6 |  2   6  0 | * 3N  *  * | 1  1
xo. ... oo.3*a &#x & ♦  4 |  3  3 |  1   3  0 | *  * 6N  * | 1  1
xoo ... ...    &#x & ♦  4 |  1  5 |  0   2  2 | *  *  * 9N | 0  2
---------------------+----+-------+-----------+------------+-----
xo.3ox.3oo.3*a &#x & ♦ 2N | 6N 3N | 4N  6N  0 | 2  N 2N  0 | 3  *
xoo3oxo ...    &#x & ♦  7 |  6 12 |  2  12  6 | 0  1  2  6 | * 3N