Acronym sabirch
Name small birhombatocubic honeycomb
Confer
variants:
this case 0 < b:a < 1/sqrt(2) / case b:a = 1/sqrt(2) / case 1/sqrt(2) < b:a < 3/sqrt(2) / case b:a = 3/sqrt(2) / case 3/sqrt(2) < b:a < ∞
general polytopal classes:
isogonal  

This honeycomb is the general form of one of the 3 phases of the tegum sum of 2 inverted general srich variants, ao4ob3bo4oa&#zc, here in fact the realm of 0 < b:a < 1/sqrt(2). The other ones will be mabirch for 1/sqrt(2) < b:a < 3/sqrt(2) resp. gabirch for 3/sqrt(2) < b:a < ∞.

Further there are singular transition cases too. The relevant ones for the current realm are the bordering cases b:a = 0 (bichon) and b:a = 1/sqrt(2) (rich) respectively. While in the latter limit pairs of square antipodia obviously combine into further coes, the tetragonal disphenoids degenerate into flat squares only and the a-edges themselves become pseudo, in the former process of b → 0 one observes the followings: the b-coes clearly vanish by size, the corresponding vertices will coincide by 12; the triangular antiprisms ob3bo&#c then become an likewise 6 fold coincident single edge; but the a-base connected pairs of square antipodia ao4ob&#c, instead of becoming according square pyramids each, rather happen to swap into rosettes of 4 further disphenoids.


Incidence matrix according to Dynkin symbol

ao4ob3bo4oa&#zc   (N → ∞)   → height = 0
                              c = sqrt[3a2-2ab sqrt(2)+2b2]/2
                              0 < b:a < 1/sqrt(2)

o.4o.3o.4o.     & | 12N |   2   4   4 |  1  2  2   6   6 | 1  4  2  2
------------------+-----+-------------+------------------+-----------
a. .. .. ..     & |   2 | 12N   *   * |  1  0  0   2   0 | 0  2  1  0
.. .. b. ..     & |   2 |   * 24N   * |  0  1  1   0   1 | 1  1  0  1
oo4oo3oo4oo&#c    |   2 |   *   * 24N |  0  0  0   2   2 | 0  2  1  1
------------------+-----+-------------+------------------+-----------
a.4o. .. ..     & |   4 |   4   0   0 | 3N  *  *   *   * | 0  2  0  0
.. o.3b. ..     & |   3 |   0   3   0 |  * 8N  *   *   * | 1  0  0  1
.. .. b.4o.     & |   4 |   0   4   0 |  *  * 6N   *   * | 1  1  0  0
ao .. .. ..&#c  & |   3 |   1   0   2 |  *  *  * 24N   * | 0  1  1  0
.. ob .. ..&#c  & |   3 |   0   1   2 |  *  *  *   * 24N | 0  1  0  1
------------------+-----+-------------+------------------+-----------
.. o.3b.4o.     & |  12 |   0  24   0 |  0  8  6   0   0 | N  *  *  * b-co
ao4ob .. ..&#c  & |   8 |   4   4   8 |  1  0  1   4   4 | * 6N  *  * square antipodium
ao .. .. oa&#c    |   4 |   2   0   4 |  0  0  0   4   0 | *  * 6N  * tetragonal disphenoid
.. ob3bo ..&#c    |   6 |   0   6   6 |  0  2  0   0   6 | *  *  * 4N triangular antiprism

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