This honeycomb as a total can not be made uniform: The mere alternated faceting (here starting at grothaph) e.g. would use edges of 3 different sizes: |sefa(s3s)| = x(6,2) = h = sqrt(3) = 1.732051, |s2s| = x(4,2) = q = sqrt(2) = 1.414214 and |sefa(s6s)| = x(12,2) = t = sqrt[2+sqrt(3)] = 1.931852, besides the remaining unit edges (refering to elements of s∞x2s3s6s here).

Incidence matrix according to Dynkin symbol

s∞x2s3s6s   (N → ∞)

demi( .     . . . . ) | 12N |  1  1  1  1  1   2   2 |  1  1  1   3   3   3  2  2 |  1  1 1  1 1  4
----------------------+-----+------------------------+----------------------------+----------------
demi( .     x . . . ) |   2 | 6N  *  *  *  *   *   * |  1  0  0   0   0   0  2  2 |  0  0 0  1 1  3  x
      s     2 s . .   |   2 |  * 6N  *  *  *   *   * |  0  0  0   2   2   0  0  0 |  1  1 0  0 0  2  q
      s     2 . s .   |   2 |  *  * 6N  *  *   *   * |  0  0  0   2   0   2  0  0 |  1  0 1  0 0  2  q
      s     2 . . s   |   2 |  *  *  * 6N  *   *   * |  0  0  0   0   2   2  0  0 |  0  1 1  0 0  2  q
            . s 2 s   |   2 |  *  *  *  * 6N   *   * |  1  0  0   0   2   0  0  0 |  0  1 0  0 0  2  q
sefa( .     . s3s . ) |   2 |  *  *  *  *  * 12N   * |  0  1  0   1   0   0  1  0 |  1  0 0  1 0  1  h
sefa( .     . . s6s ) |   2 |  *  *  *  *  *   * 12N |  0  0  1   0   0   1  0  1 |  0  0 1  0 1  1  t
----------------------+-----+------------------------+----------------------------+----------------
      .     x2s 2 s   |   4 |  2  0  0  0  2   0   0 | 3N  *  *   *   *   *  *  * |  0  0 0  0 0  2  x2q
      .     . s3s .   |   3 |  0  0  0  0  0   3   0 |  * 4N  *   *   *   *  *  * |  1  0 0  1 0  0  h3o
      .     . . s6s   |   6 |  0  0  0  0  0   0   6 |  *  * 2N   *   *   *  *  * |  0  0 1  0 1  0  t6o
sefa( s     2 s3s . ) |   3 |  0  1  1  0  0   1   0 |  *  *  * 12N   *   *  *  * |  1  0 0  0 0  1  oh&#q
sefa( s     2 s 2 s ) |   3 |  0  1  0  1  1   0   0 |  *  *  *   * 12N   *  *  * |  0  1 0  0 0  1  q3o
sefa( s     2 . s6s ) |   3 |  0  0  1  1  0   0   1 |  *  *  *   *   * 12N  *  * |  0  0 1  0 0  1  ot&#q
sefa( .     x2s3s . ) |   4 |  2  0  0  0  0   2   0 |  *  *  *   *   *   * 6N  * |  0  0 0  1 0  1  x2h
sefa( .     x 2 s6s ) |   4 |  2  0  0  0  0   0   2 |  *  *  *   *   *   *  * 6N |  0  0 0  0 1  1  x2t
----------------------+-----+------------------------+----------------------------+----------------
      s     2 s3s .   |   6 |  0  3  3  0  0   6   0 |  0  2  0   6   0   0  0  0 | 2N  * *  * *  *  ho3oh&#q oct variant
      s     2 s 2 s   |   4 |  0  2  0  2  2   0   0 |  0  0  0   0   4   0  0  0 |  * 3N *  * *  *  q-tet
      s     2 . s6s   |  12 |  0  0  6  6  0   0  12 |  0  0  2   0   0  12  0  0 |  *  * N  * *  *  to6ot&#q hap variant
      .     x2s3s .   |   6 |  3  0  0  0  0   6   0 |  0  2  0   0   0   0  3  0 |  *  * * 2N *  *  x h3o trip variant
      .     x 2 s6s   |  12 |  6  0  0  0  0   0  12 |  0  0  2   0   0   0  0  6 |  *  * *  * N  *  x t6o hip variant
sefa( s-inf-x2s3s6s ) |   8 |  3  2  2  2  2   2   2 |  1  0  0   2   2   2  1  1 |  *  * *  * * 6N  etidpy variant

starting figure: x∞x x3x6x