hyperbolic honeycomb with seed point in center of right-angled octahedron domain
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- more general:
- o4x4oPo o4x4o2Po
- general polytopal classes:
- regular noble polytopes
links
| Acronym | squah |
| Name |
hyperbolic order 3 square-tiling honeycomb, hyperbolic honeycomb with seed point in center of right-angled octahedron domain |
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| Circumradius | 1/sqrt(-2) = 0.707107 i |
| Dual | x3o4o4o |
| Confer |
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This regular, non-compact hyperbolic tesselation uses squats in the sense of an infinite horohedron as its single cell type.
This regular hyperbolic honeycomb is the single paracompact member of the general family o4x4o2Qo, which would be generally hypercompact for Q>2 (and no longer regular then).
Incidence matrix according to Dynkin symbol
o3o4o4x (N,M → ∞) . . . . | NM ♦ 8 | 12 | 6 --------+----+-----+-----+--- . . . x | 2 | 4NM | 3 | 3 --------+----+-----+-----+--- . . o4x | 4 | 4 | 3NM | 2 --------+----+-----+-----+--- . o4o4x ♦ M | 2M | M | 6N snubbed forms: o3o4o4s
o4x4o4o (N,M,K → ∞) . . . . | NMK ♦ 8 | 4 8 | 4 2 --------+-----+------+----------+-------- . x . . | 2 | 4NMK | 1 2 | 2 1 --------+-----+------+----------+-------- o4x . . | 4 | 4 | NMK * | 2 0 . x4o . | 4 | 4 | * 2NMK | 1 1 --------+-----+------+----------+-------- o4x4o . ♦ 2M | 4M | M M | 2NK * . x4o4o ♦ K | 2K | 0 K | * 2NM snubbed forms: o4s4o4o
o4x4o *b4o (N,M,K,L → ∞) . . . . | NMKL ♦ 8 | 4 4 4 | 2 2 2 -----------+------+-------+----------------+------------ . x . . | 2 | 4NMKL | 1 1 1 | 1 1 1 -----------+------+-------+----------------+------------ o4x . . | 4 | 4 | NMKL * * | 1 1 0 . x4o . | 4 | 4 | * NMKL * | 1 0 1 . x . *b4o | 4 | 4 | * * NMKL | 0 1 1 -----------+------+-------+----------------+------------ o4x4o . ♦ 2M | 4M | M M 0 | NKL * * o4x . *b4o ♦ 2K | 4K | K 0 K | * NML * . x4o *b4o ♦ 2L | 4L | 0 L L | * * NMK snubbed forms: o4s4o *b4o
x4o4x *b4o (N,M,K,L → ∞) . . . . | NMKL ♦ 4 4 | 4 4 4 | 4 1 1 -----------+------+-------------+----------------+------------ x . . . | 2 | 2NMKL * | 2 1 0 | 2 1 0 . . x . | 2 | * 2NMKL | 0 1 2 | 2 0 1 -----------+------+-------------+----------------+------------ x4o . . | 4 | 4 0 | NMKL * * | 1 1 0 x . x . | 4 | 2 2 | * NMKL * | 2 0 0 . o4x . | 4 | 0 4 | * * NMKL | 1 0 1 -----------+------+-------------+----------------+------------ x4o4x . ♦ 4M | 4M 4M | M 2M M | NKL * * x4o . *b4o ♦ K | 2K 0 | K 0 0 | * NML * . o4x *b4o ♦ L | 0 2L | 0 0 L | * * NMK snubbed forms: s4o4s *b4o
x4o4x4o4*a (N,M,K,L,P → ∞) . . . . | 2NMKLP ♦ 4 4 | 2 4 2 2 2 | 2 1 2 1 -----------+--------+---------------+--------------------------------+-------------------- x . . . | 2 | 4NMKLP * | 1 1 1 0 0 | 1 1 1 0 . . x . | 2 | * 4NMKLP | 0 1 0 1 1 | 1 0 1 1 -----------+--------+---------------+--------------------------------+-------------------- x4o . . | 4 | 4 0 | NMKLP * * * * | 1 1 0 0 x . x . | 4 | 2 2 | * 2NMKLP * * * | 1 0 1 0 x . . o4*a | 4 | 4 0 | * * NMKLP * * | 0 1 1 0 . o4x . | 4 | 0 4 | * * * NMKLP * | 1 0 0 1 . . x4o | 4 | 0 4 | * * * * NMKLP | 0 0 1 1 -----------+--------+---------------+--------------------------------+-------------------- x4o4x . ♦ 4M | 4M 4M | M 2M 0 M 0 | NKLP * * * x4o . o4*a ♦ 2K | 4K 0 | K 0 K 0 0 | * NMLP * * x . x4o4*a ♦ 4L | 4L 4L | 0 2L L 0 L | * * NMKP * . o4x4o ♦ 2P | 0 4P | 0 0 0 P P | * * * NMKL snubbed forms: s4o4s4o4*a
o4xØx4*a4xØx4*a . . . . . | 4NMKLPQ ♦ 2 2 2 2 | 1 1 1 1 2 2 2 2 | 1 1 1 1 2 ----------------+---------+---------------------------------+-------------------------------------------------------------+------------------------------- . x . . . | 2 | 4NMKLPQ * * * | 1 0 0 0 1 1 0 0 | 1 1 0 0 1 . . x . . | 2 | * 4NMKLPQ * * | 0 1 0 0 0 0 1 1 | 0 0 1 1 1 . . . x . | 2 | * * 4NMKLPQ * | 0 0 1 0 1 0 1 0 | 1 0 1 0 1 . . . . x | 2 | * * * 4NMKLPQ | 0 0 0 1 0 1 0 1 | 0 1 0 1 1 ----------------+---------+---------------------------------+-------------------------------------------------------------+------------------------------- o4x . . . | 4 | 4 0 0 0 | NMKLPQ * * * * * * * | 1 1 0 0 0 o . x4*a . . | 4 | 0 4 0 0 | * NMKLPQ * * * * * * | 0 0 1 1 0 o . . *a4x . | 4 | 0 0 4 0 | * * NMKLPQ * * * * * | 1 0 1 0 0 o . . . x4*a | 4 | 0 0 0 4 | * * * NMKLPQ * * * * | 0 1 0 1 0 . x . x . | 4 | 2 0 2 0 | * * * * 2NMKLPQ * * * | 1 0 0 0 1 . x . . x | 4 | 2 0 0 2 | * * * * * 2NMKLPQ * * | 0 1 0 0 1 . . x x . | 4 | 0 2 2 0 | * * * * * * 2NMKLPQ * | 0 0 1 0 1 . . x . x | 4 | 0 2 0 2 | * * * * * * * 2NMKLPQ | 0 0 0 1 1 ----------------+---------+---------------------------------+-------------------------------------------------------------+------------------------------- o4x . *a4x . ♦ 4M | 4M 0 4M 0 | M 0 M 0 2M 0 0 0 | NKLPQ * * * * o4x . . x4*a ♦ 4K | 4K 0 0 4K | K 0 0 K 0 2K 0 0 | * NMLPQ * * * o . x4*a4x . ♦ 4L | 0 4L 4L 0 | 0 L L 0 0 0 2L 0 | * * NMKPQ * * o . x4*a . x4*a ♦ 4P | 0 4P 0 4P | 0 P 0 P 0 0 0 2P | * * * NMKLQ * . xØx xØx ♦ 4Q | 2Q 2Q 2Q 2Q | 0 0 0 0 Q Q Q Q | * * * * 2NMKLP
octahedral Coxeter domain with boundary pattern:
b e
g a
h c
d f
xØxØxØxØxØxØxØxØ*aØ*cØ*gØ*eØ*hØ*fØ*bØ*dØ*a (N,M,K,L,P,Q,R → ∞)
a b c d e f g h
. . . . . . . . | 4NMKLPQR ♦ 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
x . . . . . . . | 2 | 2NMKLPQR * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
. x . . . . . . | 2 | * 2NMKLPQR * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 | 1 0 0 1 1 0
. . x . . . . . | 2 | * * 2NMKLPQR * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 | 0 1 0 1 0 1
. . . x . . . . | 2 | * * * 2NMKLPQR * * * * | 0 0 0 0 0 0 0 0 0 1 1 1 | 0 0 1 0 1 1
. . . . x . . . | 2 | * * * * 2NMKLPQR * * * | 1 0 0 1 0 0 1 0 0 0 0 0 | 1 1 0 1 0 0
. . . . . x . . | 2 | * * * * * 2NMKLPQR * * | 0 1 0 0 0 0 0 1 0 1 0 0 | 0 1 1 0 0 1
. . . . . . x . | 2 | * * * * * * 2NMKLPQR * | 0 0 1 0 1 0 0 0 0 0 1 0 | 1 0 1 0 1 0
. . . . . . . x | 2 | * * * * * * * 2NMKLPQR | 0 0 0 0 0 1 0 0 1 0 0 1 | 0 0 0 1 1 1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
x . . . x . . . | 4 | 2 0 0 0 2 0 0 0 | NMKLPQR * * * * * * * * * * * | 1 1 0 0 0 0
x . . . . x . . | 4 | 2 0 0 0 0 2 0 0 | * NMKLPQR * * * * * * * * * * | 0 1 1 0 0 0
x . . . . . x . | 4 | 2 0 0 0 0 0 2 0 | * * NMKLPQR * * * * * * * * * | 1 0 1 0 0 0
. x . . x . . . | 4 | 0 2 0 0 2 0 0 0 | * * * NMKLPQR * * * * * * * * | 1 0 0 1 0 0
. x . . . . x . | 4 | 0 2 0 0 0 0 2 0 | * * * * NMKLPQR * * * * * * * | 1 0 0 0 1 0
. x . . . . . x | 4 | 0 2 0 0 0 0 0 2 | * * * * * NMKLPQR * * * * * * | 0 0 0 1 1 0
. . x . x . . . | 4 | 0 0 2 0 2 0 0 0 | * * * * * * NMKLPQR * * * * * | 0 1 0 0 0 0
. . x . . x . . | 4 | 0 0 2 0 0 2 0 0 | * * * * * * * NMKLPQR * * * * | 0 1 0 1 0 1
. . x . . . . x | 4 | 0 0 2 0 0 0 0 2 | * * * * * * * * NMKLPQR * * * | 0 0 0 1 0 1
. . . x . x . . | 4 | 0 0 0 2 0 2 0 0 | * * * * * * * * * NMKLPQR * * | 0 0 1 0 0 1
. . . x . . x . | 4 | 0 0 0 2 0 0 2 0 | * * * * * * * * * * NMKLPQR * | 0 0 1 0 1 0
. . . x . . . x | 4 | 0 0 0 2 0 0 0 2 | * * * * * * * * * * * NMKLPQR | 0 0 0 0 1 1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
xØx . . x . x . *gØ*e ♦ 4M | 2M 2M 0 0 2M 0 2M 0 | M 0 M M M 0 0 0 0 0 0 0 | NKLPQR * * * * *
x . x . xØx . . *aØ*c ♦ 4K | 2K 0 2K 0 2K 2K 0 0 | K K 0 0 0 0 K K 0 0 0 0 | * NMLPQR * * * *
x . . x . xØx . *dØ*a ♦ 4L | 2L 0 0 2L 0 2L 2L 0 | 0 L L 0 0 0 0 0 0 L L 0 | * * NMKPQR * * *
. xØx . x . . x *eØ*h ♦ 4P | 0 2P 2P 0 2P 0 0 2P | 0 0 0 P 0 P 0 P P 0 0 0 | * * * NMKLQR * *
. x . x . . xØx *bØ*d ♦ 4Q | 0 2Q 0 2Q 0 0 2Q 2Q | 0 0 0 0 Q Q 0 0 0 0 Q Q | * * * * NMKLPR *
. . xØx . x . x *hØ*f ♦ 4R | 0 0 2R 2R 0 2R 0 2R | 0 0 0 0 0 0 0 R R R 0 R | * * * * * NMKLPQ
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