| Acronym | batch |
| Name |
bitruncated cubic honeycomb, truncated-octahedral honeycomb, Voronoi complex of body-centered cubic (bcc) lattice |
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| Vertex figure |
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| Dual | bichon |
| Confer |
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External links |
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As can be read from the matrices below, at every edge there are 2 hexagons. Thus we get as pseudo cells something with hexagons only. From the vertex incidence we further read off that this pseudo tiling happens to use 4 hexagons per vertex. From the here truely being used cells (toe) it is deduced, that any straight edge sequence of that seeming x6o4o needs to be mod-wrapped to square holes. Therefore those pseudo cells rather are the skew polyhedron x6o4o|4 instead.
The facial subset, obtained when neglecting one tetrahedral subset of hexagons within each toe (e.g. the hexagons x3x . . and . . x3x in the representation x3x3x3x3*a), by itself describes the Wythoffian infinite skew polyhedron x6o6x|6, a modwrap of the hyperbolical tiling x6o6x.
By virtue of an outer symmetry this is a non-quasiregular monotoxal honeycomb, that is all edges belong to the same equivalence class.
Incidence matrix according to Dynkin symbol
o4x3x4o (N → ∞) . . . . | 12N ♦ 2 2 | 1 4 1 | 2 2 --------+-----+---------+----------+---- . x . . | 2 | 12N * | 1 2 0 | 2 1 . . x . | 2 | * 12N | 0 2 1 | 1 2 --------+-----+---------+----------+---- o4x . . | 4 | 4 0 | 3N * * | 2 0 . x3x . | 6 | 3 3 | * 8N * | 1 1 . . x4o | 4 | 0 4 | * * 3N | 0 2 --------+-----+---------+----------+---- o4x3x . ♦ 24 | 24 12 | 6 8 0 | N * . x3x4o ♦ 24 | 12 24 | 0 8 6 | * N
or . . . . | 6N ♦ 4 | 2 4 | 4 -----------+----+-----+-------+-- . x . . & | 2 | 12N | 1 2 | 3 -----------+----+-----+-------+-- o4x . . & | 4 | 4 | 3N * | 2 . x3x . | 6 | 6 | * 4N | 2 -----------+----+-----+-------+-- o4x3x . & ♦ 24 | 36 | 6 8 | N snubbed forms: o4s3s4o
o4x3x4/3o (N → ∞) . . . . | 12N ♦ 2 2 | 1 4 1 | 2 2 ----------+-----+---------+----------+---- . x . . | 2 | 12N * | 1 2 0 | 2 1 . . x . | 2 | * 12N | 0 2 1 | 1 2 ----------+-----+---------+----------+---- o4x . . | 4 | 4 0 | 3N * * | 2 0 . x3x . | 6 | 3 3 | * 8N * | 1 1 . . x4/3o | 4 | 0 4 | * * 3N | 0 2 ----------+-----+---------+----------+---- o4x3x . ♦ 24 | 24 12 | 6 8 0 | N * . x3x4/3o ♦ 24 | 12 24 | 0 8 6 | * N
o4/3x3x4/3o (N → ∞) . . . . | 12N ♦ 2 2 | 1 4 1 | 2 2 ------------+-----+---------+----------+---- . x . . | 2 | 12N * | 1 2 0 | 2 1 . . x . | 2 | * 12N | 0 2 1 | 1 2 ------------+-----+---------+----------+---- o4/3x . . | 4 | 4 0 | 3N * * | 2 0 . x3x . | 6 | 3 3 | * 8N * | 1 1 . . x4/3o | 4 | 0 4 | * * 3N | 0 2 ------------+-----+---------+----------+---- o4/3x3x . ♦ 24 | 24 12 | 6 8 0 | N * . x3x4/3o ♦ 24 | 12 24 | 0 8 6 | * N
or . . . . | 6N ♦ 4 | 2 4 | 4 ---------------+----+-----+-------+-- . x . . & | 2 | 12N | 1 2 | 3 ---------------+----+-----+-------+-- o4/3x . . & | 4 | 4 | 3N * | 2 . x3x . | 6 | 6 | * 4N | 2 ---------------+----+-----+-------+-- o4/3x3x . & ♦ 24 | 36 | 6 8 | N
x3x3x *b4o (N → ∞) . . . . | 24N ♦ 1 2 1 | 2 1 2 1 | 2 1 1 -----------+-----+-------------+-------------+------- x . . . | 2 | 12N * * | 2 1 0 0 | 2 1 0 . x . . | 2 | * 24N * | 1 0 1 1 | 1 1 1 . . x . | 2 | * * 12N | 0 1 2 0 | 2 0 1 -----------+-----+-------------+-------------+------- x3x . . | 6 | 3 3 0 | 8N * * * | 1 1 0 x . x . | 4 | 2 0 2 | * 6N * * | 2 0 0 . x3x . | 6 | 0 3 3 | * * 8N * | 1 0 1 . x . *b4o | 4 | 0 4 0 | * * * 6N | 0 1 1 -----------+-----+-------------+-------------+------- x3x3x . ♦ 24 | 12 12 12 | 4 6 4 0 | 2N * * x3x . *b4o ♦ 24 | 12 24 0 | 8 0 0 6 | * N * . x3x *b4o ♦ 24 | 0 24 12 | 0 0 8 6 | * * N snubbed forms: s3s3s *b4o
x3x3x *b4/3o (N → ∞) . . . . | 24N ♦ 1 2 1 | 2 1 2 1 | 2 1 1 -------------+-----+-------------+-------------+------- x . . . | 2 | 12N * * | 2 1 0 0 | 2 1 0 . x . . | 2 | * 24N * | 1 0 1 1 | 1 1 1 . . x . | 2 | * * 12N | 0 1 2 0 | 2 0 1 -------------+-----+-------------+-------------+------- x3x . . | 6 | 3 3 0 | 8N * * * | 1 1 0 x . x . | 4 | 2 0 2 | * 6N * * | 2 0 0 . x3x . | 6 | 0 3 3 | * * 8N * | 1 0 1 . x . *b4/3o | 4 | 0 4 0 | * * * 6N | 0 1 1 -------------+-----+-------------+-------------+------- x3x3x . ♦ 24 | 12 12 12 | 4 6 4 0 | 2N * * x3x . *b4/3o ♦ 24 | 12 24 0 | 8 0 0 6 | * N * . x3x *b4/3o ♦ 24 | 0 24 12 | 0 0 8 6 | * * N
x3x3x3x3*a (N → ∞) . . . . | 24N ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 -----------+-----+-----------------+-------------------+-------- x . . . | 2 | 12N * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 12N * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 12N * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 12N | 0 0 1 0 1 1 | 0 1 1 1 -----------+-----+-----------------+-------------------+-------- x3x . . | 6 | 3 3 0 0 | 4N * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 6N * * * * | 1 0 1 0 x . . x3*a | 6 | 3 0 0 3 | * * 4N * * * | 0 1 1 0 . x3x . | 6 | 0 3 3 0 | * * * 4N * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 6N * | 0 1 0 1 . . x3x | 6 | 0 0 3 3 | * * * * * 4N | 0 0 1 1 -----------+-----+-----------------+-------------------+-------- x3x3x . ♦ 24 | 12 12 12 0 | 4 6 0 4 0 0 | N * * * x3x . x3*a ♦ 24 | 12 12 0 12 | 4 0 4 0 6 0 | * N * * x . x3x3*a ♦ 24 | 12 0 12 12 | 0 6 4 0 0 4 | * * N * . x3x3x ♦ 24 | 0 12 12 12 | 0 0 0 4 6 4 | * * * N snubbed forms: s3s3s3s3*a
s4x3x4o (N → ∞)
demi( . . . . ) | 24N ♦ 1 2 1 | 2 1 1 2 | 1 2 1
----------------+-----+-------------+-------------+-------
demi( . x . . ) | 2 | 12N * * | 2 0 1 0 | 1 2 0
demi( . . x . ) | 2 | * 24N * | 1 1 0 1 | 1 1 1
sefa( s4x . . ) | 2 | * * 12N | 0 0 1 2 | 0 2 1
----------------+-----+-------------+-------------+-------
demi( . x3x . ) | 6 | 3 3 0 | 8N * * * | 1 1 0
demi( . . x4o ) | 4 | 0 4 0 | * 6N * * | 1 0 1
s4x . . | 4 | 2 0 2 | * * 6N * | 0 2 0
sefa( s4x3x . ) | 6 | 0 3 3 | * * * 8N | 0 1 1
----------------+-----+-------------+-------------+-------
demi( . x3x4o ) ♦ 24 | 12 24 0 | 8 6 0 0 | N * *
s4x3x . ♦ 24 | 12 12 12 | 4 0 6 4 | * 2N *
sefa( s4x3x4o ) ♦ 24 | 0 24 12 | 0 6 0 8 | * * N
starting figure: x4x3x4o
:qooo:4:xuxu:4:ooqo:&##x (N → ∞) → all heights = 1/sqrt(2) = 0.707107 o... 4 o... 4 o... | 4N * * * ♦ 2 1 0 0 0 1 | 1 2 1 2 0 0 | 2 2 .o.. 4 .o.. 4 .o.. | * 2N * * ♦ 0 2 2 0 0 0 | 0 4 1 0 1 0 | 2 2 ..o. 4 ..o. 4 ..o. | * * 4N * ♦ 0 0 1 2 1 0 | 0 2 0 2 1 1 | 2 2 ...o 4 ...o 4 ...o | * * * 2N ♦ 0 0 0 0 2 2 | 0 0 1 4 1 0 | 2 2 -------------------------+-------------+-------------------+-----------------+---- .... x... .... | 2 0 0 0 | 4N * * * * * | 1 1 0 1 0 0 | 1 2 oo.. 4 oo.. 4 oo.. &#x | 1 1 0 0 | * 4N * * * * | 0 2 1 0 0 0 | 2 1 .oo. 4 .oo. 4 .oo. &#x | 0 1 1 0 | * * 4N * * * | 0 2 0 0 1 0 | 1 2 .... ..x. .... | 0 0 2 0 | * * * 4N * * | 0 1 0 1 0 1 | 2 1 ..oo 4 ..oo 4 ..oo &#x | 0 0 1 1 | * * * * 4N * | 0 0 0 2 1 0 | 1 2 :o..o:4:o..o:4:o..o:&#x | 1 0 0 1 | * * * * * 4N | 0 0 1 2 0 0 | 2 1 -------------------------+-------------+-------------------+-----------------+---- .... x... 4 o... | 4 0 0 0 | 4 0 0 0 0 0 | N * * * * * | 0 2 .... xux. .... &#xt | 2 2 2 0 | 1 2 2 1 0 0 | * 4N * * * * | 1 1 :qo.o: .... .... &#xt | 2 1 0 1 | 0 2 0 0 0 2 | * * 2N * * * | 2 0 .... :x.xu: .... &#xt | 2 0 2 2 | 1 0 0 1 2 2 | * * * 4N * * | 1 1 .... .... .oqo &#xt | 0 1 2 1 | 0 0 2 0 2 0 | * * * * 2N * | 0 2 ..o. 4 ..x. .... | 0 0 4 0 | 0 0 0 4 0 0 | * * * * * N | 2 0 -------------------------+-------------+-------------------+-----------------+---- :qooo:4:xuxu: .... &#xt ♦ 8 4 8 4 | 4 8 4 8 4 8 | 0 4 4 4 0 2 | N * .... :xuxu:4:ooqo:&#xt ♦ 8 4 8 4 | 8 4 8 4 8 4 | 2 4 0 4 4 0 | * N
:xxu:3:xux:3:uxx:3*a&##x (N → ∞) → all heights = sqrt(2/3) = 0.816497 o.. 3 o.. 3 o.. 3*a | 6N * * ♦ 1 1 1 0 0 0 0 0 1 | 1 1 1 0 0 0 1 1 1 | 2 1 1 .o. 3 .o. 3 .o. 3*a | * 6N * ♦ 0 0 1 1 1 1 0 0 0 | 0 1 1 1 1 0 1 0 1 | 1 2 1 ..o 3 ..o 3 ..o 3*a | * * 6N ♦ 0 0 0 0 0 1 1 1 1 | 0 0 1 0 1 1 1 1 1 | 1 1 2 -------------------------+----------+----------------------------+-------------------------+------ x.. ... ... | 2 0 0 | 3N * * * * * * * * | 1 1 0 0 0 0 1 0 0 | 2 1 0 ... x.. ... | 2 0 0 | * 3N * * * * * * * | 1 0 1 0 0 0 0 1 0 | 2 0 1 oo. 3 oo. 3 oo. 3*a&#x | 1 1 0 | * * 6N * * * * * * | 0 1 1 0 0 0 0 0 1 | 1 1 1 .x. ... ... | 0 2 0 | * * * 3N * * * * * | 0 1 0 1 0 0 1 0 0 | 1 2 0 ... ... .x. | 0 2 0 | * * * * 3N * * * * | 0 0 0 1 1 0 0 0 1 | 0 2 1 .oo 3 .oo 3 .oo 3*a&#x | 0 1 1 | * * * * * 6N * * * | 0 0 1 0 1 0 1 0 0 | 1 1 1 ... ..x ... | 0 0 2 | * * * * * * 3N * * | 0 0 1 0 0 1 0 1 0 | 1 0 2 ... ... ..x | 0 0 2 | * * * * * * * 3N * | 0 0 0 0 1 1 0 0 1 | 0 1 2 :o.o:3:o.o:3:o.o:3*a&#x | 1 0 1 | * * * * * * * * 6N | 0 0 0 0 0 0 1 1 1 | 1 1 1 -------------------------+----------+----------------------------+-------------------------+------ x.. 3 x.. ... | 6 0 0 | 3 3 0 0 0 0 0 0 0 | N * * * * * * * * | 2 0 0 xx. ... ... &#x | 2 2 0 | 1 0 2 1 0 0 0 0 0 | * 3N * * * * * * * | 1 1 0 ... xux ... &#xt | 2 2 2 | 0 1 2 0 0 2 1 0 0 | * * 3N * * * * * * | 1 0 1 .x. ... .x. 3*a | 0 6 0 | 0 0 0 3 3 0 0 0 0 | * * * N * * * * * | 0 2 0 ... ... .xx &#x | 0 2 2 | 0 0 0 0 1 2 0 1 0 | * * * * 3N * * * * | 0 1 1 ... ..x 3 ..x | 0 0 6 | 0 0 0 0 0 0 3 3 0 | * * * * * N * * * | 0 0 2 :xxu: ... ... &#xt | 2 2 2 | 1 0 0 1 0 2 0 0 2 | * * * * * * 3N * * | 1 1 0 ... :x.x: ... &#x | 2 0 2 | 0 1 0 0 0 0 1 0 2 | * * * * * * * 3N * | 1 0 1 ... ... :uxx: &#xt | 2 2 2 | 0 0 2 0 1 0 0 1 2 | * * * * * * * * 3N | 0 1 1 -------------------------+----------+----------------------------+-------------------------+------ :xxu:3:xux: ... &#xt ♦ 12 6 6 | 6 6 6 3 0 6 3 0 6 | 2 3 3 0 0 0 3 3 0 | N * * :xxu: ... :uxx:3*a&#xt ♦ 6 12 6 | 3 0 6 6 6 6 0 3 6 | 0 3 0 2 3 0 3 0 3 | * N * ... :xux:3:uxx: &#xt ♦ 6 6 12 | 0 3 6 0 3 6 6 6 6 | 0 0 3 0 3 2 0 3 3 | * * N
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