Acronym | sabirch |
Name | small birhombatocubic honeycomb |
Confer |
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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