Acronym chon
Name cubic honeycomb,
3D hypercubical honeycomb3),
Voronoi complex of primitive cubical lattice,
Delone complex of primitive cubical lattice
 
 © ©    ©
Vertex figure
 ©
Vertex layers
(first ones only)
LayerSymmetrySubsymmetries
 o4o3o4oo4o3o .. o3o4o
1x4o3o4ox4o3o .
cell first
. o3o4o
vertex first
2x4o3q .. q3o4o
vertex figure
3x4q3o .. o3q4o
4ax4o3Q .. o3o4u
4bd4o3o .
.........
(Q=2q, d=u+x=3x)
Coordinates (i, j, k)           i.e. all integer touples
Dual (selfdual)
Confer
related tesselations:
Voronoi complex of bcc lattice   Delone complex of fcc lattice  
general polytopal classes:
hypercubical honeycomb   partial Stott expansions   noble polytopes  
External
links
wikipedia   wikipedia
 ©    ©

The Voronoi complex and the Delone complex of the primitive cubic lattice are relatively shifted copies of this cubic honeycomb.

A facial subset, built from the lacing squares of the 4-prismatic cells each, such that just 6 of those squares remain at each vertex, by itself describes the regular infinite skew polyhedron x4o6o|4, a modwrap of the hyperbolical tiling x4o6o.


Incidence matrix according to Dynkin symbol

x4o3o4o   (N → ∞)

. . . . | N   6 | 12 | 8
--------+---+----+----+--
x . . . | 2 | 3N |  4 | 4
--------+---+----+----+--
x4o . . | 4 |  4 | 3N | 2
--------+---+----+----+--
x4o3o .  8 | 12 |  6 | N

snubbed forms: s4o3o4o

x4o3o4x   (N → ∞)

. . . . | 8N    3   3 |  3   6  3 | 1  3  3 1
--------+----+---------+-----------+----------
x . . . |  2 | 12N   * |  2   2  0 | 1  2  1 0
. . . x |  2 |   * 12N |  0   2  2 | 0  1  2 1
--------+----+---------+-----------+----------
x4o . . |  4 |   4   0 | 6N   *  * | 1  1  0 0
x . . x |  4 |   2   2 |  * 12N  * | 0  1  1 0
. . o4x |  4 |   0   4 |  *   * 6N | 0  0  1 1
--------+----+---------+-----------+----------
x4o3o .   8 |  12   0 |  6   0  0 | N  *  * *
x4o . x   8 |   8   4 |  2   4  0 | * 3N  * *
x . o4x   8 |   4   8 |  0   4  2 | *  * 3N *
. o3o4x   8 |   0  12 |  0   0  6 | *  *  * N

snubbed forms: s4o3o4x, s4o3o4s, s4o3o4s'

o3o3o *b4x   (N → ∞)

. . .    . | 2N   6 | 12 | 4 4
-----------+----+----+----+----
. . .    x |  2 | 6N |  4 | 2 2
-----------+----+----+----+----
. o . *b4x |  4 |  4 | 6N | 1 1
-----------+----+----+----+----
o3o . *b4x   8 | 12 |  6 | N *
. o3o *b4x   8 | 12 |  6 | * N

snubbed forms: o3o3o *b4s

x4o3o4/3o   (N → ∞)

. . .   . | N   6 | 12 | 8
----------+---+----+----+--
x . .   . | 2 | 3N |  4 | 4
----------+---+----+----+--
x4o .   . | 4 |  4 | 3N | 2
----------+---+----+----+--
x4o3o   .  8 | 12 |  6 | N

o4o3o4/3x   (N → ∞)

. . .   . | N   6 | 12 | 8
----------+---+----+----+--
. . .   x | 2 | 3N |  4 | 4
----------+---+----+----+--
. . o4/3x | 4 |  4 | 3N | 2
----------+---+----+----+--
. o3o4/3x  8 | 12 |  6 | N

x4o3o4/3x   (N → ∞)

. . .   . | 8N    3   3 |  3   6  3 | 1  3  3 1
----------+----+---------+-----------+----------
x . .   . |  2 | 12N   * |  2   2  0 | 1  2  1 0
. . .   x |  2 |   * 12N |  0   2  2 | 0  1  2 1
----------+----+---------+-----------+----------
x4o .   . |  4 |   4   0 | 6N   *  * | 1  1  0 0
x . .   x |  4 |   2   2 |  * 12N  * | 0  1  1 0
. . o4/3x |  4 |   0   4 |  *   * 6N | 0  0  1 1
----------+----+---------+-----------+----------
x4o3o   .   8 |  12   0 |  6   0  0 | N  *  * *
x4o .   x   8 |   8   4 |  2   4  0 | * 3N  * *
x . o4/3x   8 |   4   8 |  0   4  2 | *  * 3N *
. o3o4/3x   8 |   0  12 |  0   0  6 | *  *  * N

x4/3o3o4/3o   (N → ∞)

.   . .   . | N   6 | 12 | 8
------------+---+----+----+--
x   . .   . | 2 | 3N |  4 | 4
------------+---+----+----+--
x4/3o .   . | 4 |  4 | 3N | 2
------------+---+----+----+--
x4/3o3o   .  8 | 12 |  6 | N

x4/3o3o4/3x   (N → ∞)

.   . .   . | 8N    3   3 |  3   6  3 | 1  3  3 1
------------+----+---------+-----------+----------
x   . .   . |  2 | 12N   * |  2   2  0 | 1  2  1 0
.   . .   x |  2 |   * 12N |  0   2  2 | 0  1  2 1
------------+----+---------+-----------+----------
x4/3o .   . |  4 |   4   0 | 6N   *  * | 1  1  0 0
x   . .   x |  4 |   2   2 |  * 12N  * | 0  1  1 0
.   . o4/3x |  4 |   0   4 |  *   * 6N | 0  0  1 1
------------+----+---------+-----------+----------
x4/3o3o   .   8 |  12   0 |  6   0  0 | N  *  * *
x4/3o .   x   8 |   8   4 |  2   4  0 | * 3N  * *
x   . o4/3x   8 |   4   8 |  0   4  2 | *  * 3N *
.   o3o4/3x   8 |   0  12 |  0   0  6 | *  *  * N

o3o3o *b4/3x   (N → ∞)

. . .      . | 2N   6 | 12 | 4 4
-------------+----+----+----+----
. . .      x |  2 | 6N |  4 | 2 2
-------------+----+----+----+----
. o . *b4/3x |  4 |  4 | 6N | 1 1
-------------+----+----+----+----
o3o . *b4/3x   8 | 12 |  6 | N *
. o3o *b4/3x   8 | 12 |  6 | * N

snubbed forms: o3o3o *b4s

x∞o x4o4o   (N → ∞)

. . . . . | N  2  4 |  8 4 | 8
----------+---+------+------+--
x . . . . | 2 | N  * |  4 0 | 4
. . x . . | 2 | * 2N |  2 2 | 4
----------+---+------+------+--
x . x . . | 4 | 2  2 | 2N * | 2
. . x4o . | 4 | 0  4 |  * N | 2
----------+---+------+------+--
x . x4o .  8 | 4  8 |  4 2 | N

snubbed forms: s∞o2s4o4o

x∞o o4x4o   (N → ∞)

. . . . . | 2N   2  4 |  8 2 2 | 4 4
----------+----+-------+--------+----
x . . . . |  2 | 2N  * |  4 0 0 | 2 2
. . . x . |  2 |  * 4N |  2 1 1 | 2 2
----------+----+-------+--------+----
x . . x . |  4 |  2  2 | 4N * * | 1 1
. . o4x . |  4 |  0  4 |  * N * | 2 0
. . . x4o |  4 |  0  4 |  * * N | 0 2
----------+----+-------+--------+----
x . o4x .   8 |  4  8 |  4 2 0 | N *
x . . x4o   8 |  4  8 |  4 0 2 | * N

snubbed forms: s∞o2o4s4o

x∞o x4o4x   (N → ∞)

. . . . . | 4N   2  2  2 |  4  4 1  2 1 | 2  4 2
----------+----+----------+--------------+-------
x . . . . |  2 | 4N  *  * |  2  2 0  0 0 | 1  2 1
. . x . . |  2 |  * 4N  * |  2  0 1  1 0 | 2  2 0
. . . . x |  2 |  *  * 4N |  0  2 0  1 1 | 0  2 2
----------+----+----------+--------------+-------
x . x . . |  4 |  2  2  0 | 4N  * *  * * | 1  1 0
x . . . x |  4 |  2  0  2 |  * 4N *  * * | 0  1 1
. . x4o . |  4 |  0  4  0 |  *  * N  * * | 2  0 0
. . x . x |  4 |  0  2  2 |  *  * * 2N * | 0  2 0
. . . o4x |  4 |  0  0  4 |  *  * *  * N | 0  0 2
----------+----+----------+--------------+-------
x . x4o .   8 |  4  8  0 |  4  0 2  0 0 | N  * *
x . x . x   8 |  4  4  4 |  2  2 0  2 0 | * 2N *
x . . o4x   8 |  4  0  8 |  0  4 0  0 2 | *  * N

snubbed forms: s∞o2s4o4x

x∞x x4o4o   (N → ∞)

. . . . . | 2N  1 1  4 |  4  4  4 | 4 4
----------+----+--------+----------+----
x . . . . |  2 | N *  * |  4  0  0 | 4 0
. x . . . |  2 | * N  * |  4  0  0 | 0 4
. . x . . |  2 | * * 4N |  1  1  2 | 2 2
----------+----+--------+----------+----
x . x . . |  4 | 2 0  2 | 2N  *  * | 2 0
. x x . . |  4 | 0 2  2 |  * 2N  * | 0 2
. . x4o . |  4 | 0 0  4 |  *  * 2N | 1 1
----------+----+--------+----------+----
x . x4o .   8 | 4 0  8 |  4  0  2 | N
. x x4o .   8 | 0 4  8 |  0  4  2 | N

snubbed forms: s∞x2s4o4o

x∞x o4x4o   (N → ∞)

. . . . . | 4N   1  1  4 |  4  4  2  2 | 2 2 2 2
----------+----+----------+-------------+--------
x . . . . |  2 | 2N  *  * |  4  0  0  0 | 2 2 0 0
. x . . . |  2 |  * 2N  * |  0  4  0  0 | 0 0 2 2
. . . x . |  2 |  *  * 8N |  1  1  1  1 | 1 1 1 1
----------+----+----------+-------------+--------
x . . x . |  4 |  2  0  2 | 4N  *  *  * | 1 1 0 0
. x . x . |  4 |  0  2  2 |  * 4N  *  * | 0 0 1 1
. . o4x . |  4 |  0  0  4 |  *  * 2N  * | 1 0 1 0
. . . x4o |  4 |  0  0  4 |  *  *  * 2N | 0 1 0 1
----------+----+----------+-------------+--------
x . o4x .   8 |  4  0  8 |  4  0  2  0 | N * * *
x . . x4o   8 |  4  0  8 |  4  0  0  2 | * N * *
. x o4x .   8 |  0  4  8 |  0  4  2  0 | * * N *
. x . x4o   8 |  0  4  8 |  0  4  0  2 | * * * N

snubbed forms: s∞x2o4s4o

x∞x x4o4x   (N → ∞)

. . . . . | 8N   1  1  2  2 |  2  2  2  2  1  2  1 | 1  2 1 1  2 1
----------+----+-------------+----------------------+--------------
x . . . . |  2 | 4N  *  *  * |  2  2  0  0  0  0  0 | 1  2 1 0  0 0
. x . . . |  2 |  * 4N  *  * |  0  0  2  2  0  0  0 | 0  0 0 1  2 1
. . x . . |  2 |  *  * 8N  * |  1  0  1  0  1  1  0 | 1  1 0 1  1 0
. . . . x |  2 |  *  *  * 8N |  0  1  0  1  0  1  1 | 0  1 1 0  1 1
----------+----+-------------+----------------------+--------------
x . x . . |  4 |  2  0  2  0 | 4N  *  *  *  *  *  * | 1  1 0 0  0 0
x . . . x |  4 |  2  0  0  2 |  * 4N  *  *  *  *  * | 0  1 1 0  0 0
. x x . . |  4 |  0  2  2  0 |  *  * 4N  *  *  *  * | 0  0 0 1  1 0
. x . . x |  4 |  0  2  0  2 |  *  *  * 4N  *  *  * | 0  0 0 0  1 1
. . x4o . |  4 |  0  0  4  0 |  *  *  *  * 2N  *  * | 1  0 0 1  0 0
. . x . x |  4 |  0  0  2  2 |  *  *  *  *  * 4N  * | 0  1 0 0  1 0
. . . o4x |  4 |  0  0  0  4 |  *  *  *  *  *  * 2N | 0  0 1 0  0 1
----------+----+-------------+----------------------+--------------
x . x4o .   8 |  4  0  8  0 |  4  0  0  0  2  0  0 | N  * * *  * *
x . x . x   8 |  4  0  4  4 |  2  2  0  0  0  2  0 | * 2N * *  * *
x . . o4x   8 |  4  0  0  8 |  0  4  0  0  0  0  2 | *  * N *  * *
. x x4o .   8 |  0  4  8  0 |  0  0  4  0  2  0  0 | *  * * N  * *
. x x . x   8 |  0  4  4  4 |  0  0  2  2  0  2  0 | *  * * * 2N *
. x . o4x   8 |  0  4  0  8 |  0  0  0  4  0  0  2 | *  * * *  * N

x∞o x∞o x∞o   (N → ∞)

. . . . . . | N  2 2 2 | 4 4 4 | 8
------------+---+-------+-------+--
x . . . . . | 2 | N * * | 2 2 0 | 4
. . x . . . | 2 | * N * | 2 0 2 | 4
. . . . x . | 2 | * * N | 0 2 2 | 4
------------+---+-------+-------+--
x . x . . . | 4 | 2 2 0 | N * * | 2
x . . . x . | 4 | 2 0 2 | * N * | 2
. . x . x . | 4 | 0 2 2 | * * N | 2
------------+---+-------+-------+--
x . x . x .  8 | 4 4 4 | 2 2 2 | N

x∞x x∞o x∞o   (N → ∞)

. . . . . . | 2N  1 1  2  2 | 2 2 2 2  4 | 4 4
------------+----+-----------+------------+----
x . . . . . |  2 | N *  *  * | 2 2 0 0  0 | 4 0
. x . . . . |  2 | * N  *  * | 0 0 2 2  0 | 0 4
. . x . . . |  2 | * * 2N  * | 1 0 1 0  2 | 2 2
. . . . x . |  2 | * *  * 2N | 0 1 0 1  2 | 2 2
------------+----+-----------+------------+----
x . x . . . |  4 | 2 0  2  0 | N * * *  * | 2 0
x . . . x . |  4 | 2 0  0  2 | * N * *  * | 2 0
. x x . . . |  4 | 0 2  2  0 | * * N *  * | 0 2
. x . . x . |  4 | 0 2  0  2 | * * * N  * | 0 2
. . x . x . |  4 | 0 0  2  2 | * * * * 2N | 1 1
------------+----+-----------+------------+----
x . x . x .   8 | 4 0  4  4 | 2 2 0 0  2 | N *
. x x . x .   8 | 0 4  4  4 | 0 0 2 2  2 | * N

x∞x x∞x x∞o   (N → ∞)

. . . . . . | 4N   1  1  1  1  2 | 1 1  2 1 1  2  2  2 | 2 2 2 2
------------+----+----------------+---------------------+--------
x . . . . . |  2 | 2N  *  *  *  * | 1 1  2 0 0  0  0  0 | 2 2 0 0
. x . . . . |  2 |  * 2N  *  *  * | 0 0  0 1 1  2  0  0 | 0 0 2 2
. . x . . . |  2 |  *  * 2N  *  * | 1 0  0 1 0  0  2  0 | 2 0 2 0
. . . x . . |  2 |  *  *  * 2N  * | 0 1  0 0 1  0  0  2 | 0 2 0 2
. . . . x . |  2 |  *  *  *  * 4N | 0 0  1 0 0  1  1  1 | 1 1 1 1
------------+----+----------------+---------------------+--------
x . x . . . |  4 |  2  0  2  0  0 | N *  * * *  *  *  * | 2 0 0 0
x . . x . . |  4 |  2  0  0  2  0 | * N  * * *  *  *  * | 0 2 0 0
x . . . x . |  4 |  2  0  0  0  2 | * * 2N * *  *  *  * | 1 1 0 0
. x x . . . |  4 |  0  2  2  0  0 | * *  * N *  *  *  * | 0 0 2 0
. x . x . . |  4 |  0  2  0  2  0 | * *  * * N  *  *  * | 0 0 0 2
. x . . x . |  4 |  0  2  0  0  2 | * *  * * * 2N  *  * | 0 0 1 1
. . x . x . |  4 |  0  0  2  0  2 | * *  * * *  * 2N  * | 1 0 1 0
. . . x x . |  4 |  0  0  0  2  2 | * *  * * *  *  * 2N | 0 1 0 1
------------+----+----------------+---------------------+--------
x . x . x .   8 |  4  0  4  0  4 | 2 0  2 0 0  0  2  0 | N * * *
x . . x x .   8 |  4  0  0  4  4 | 0 2  2 0 0  0  0  2 | * N * *
. x x . x .   8 |  0  4  4  0  4 | 0 0  0 2 0  2  2  0 | * * N *
. x . x x .   8 |  0  4  0  4  4 | 0 0  0 0 2  2  0  2 | * * * N

x∞x x∞x x∞x   (N → ∞)

. . . . . . | 8N   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 | 4N  *  *  *  *  * |  1  1  1  1  0  0  0  0  0  0  0  0 | 1 1 1 1 0 0 0 0
. x . . . . |  2 |  * 4N  *  *  *  * |  0  0  0  0  1  1  1  1  0  0  0  0 | 0 0 0 0 1 1 1 1
. . x . . . |  2 |  *  * 4N  *  *  * |  1  0  0  0  1  0  0  0  1  1  0  0 | 1 1 0 0 1 1 0 0
. . . x . . |  2 |  *  *  * 4N  *  * |  0  1  0  0  0  1  0  0  0  0  1  1 | 0 0 1 1 0 0 1 1
. . . . x . |  2 |  *  *  *  * 4N  * |  0  0  1  0  0  0  1  0  1  0  1  0 | 1 0 1 0 1 0 1 0
. . . . . x |  2 |  *  *  *  *  * 4N |  0  0  0  1  0  0  0  1  0  1  0  1 | 0 1 0 1 0 1 0 1
------------+----+-------------------+-------------------------------------+----------------
x . x . . . |  4 |  2  0  2  0  0  0 | 2N  *  *  *  *  *  *  *  *  *  *  * | 1 1 0 0 0 0 0 0
x . . x . . |  4 |  2  0  0  2  0  0 |  * 2N  *  *  *  *  *  *  *  *  *  * | 0 0 1 1 0 0 0 0
x . . . x . |  4 |  2  0  0  0  2  0 |  *  * 2N  *  *  *  *  *  *  *  *  * | 1 0 1 0 0 0 0 0
x . . . . x |  4 |  2  0  0  0  0  2 |  *  *  * 2N  *  *  *  *  *  *  *  * | 0 1 0 1 0 0 0 0
. x x . . . |  4 |  0  2  2  0  0  0 |  *  *  *  * 2N  *  *  *  *  *  *  * | 0 0 0 0 1 1 0 0
. x . x . . |  4 |  0  2  0  2  0  0 |  *  *  *  *  * 2N  *  *  *  *  *  * | 0 0 0 0 0 0 1 1
. x . . x . |  4 |  0  2  0  0  2  0 |  *  *  *  *  *  * 2N  *  *  *  *  * | 0 0 0 0 1 0 1 0
. x . . . x |  4 |  0  2  0  0  0  2 |  *  *  *  *  *  *  * 2N  *  *  *  * | 0 0 0 0 0 1 0 1
. . x . x . |  4 |  0  0  2  0  2  0 |  *  *  *  *  *  *  *  * 2N  *  *  * | 1 0 0 0 1 0 0 0
. . x . . x |  4 |  0  0  2  0  0  2 |  *  *  *  *  *  *  *  *  * 2N  *  * | 0 1 0 0 0 1 0 0
. . . x x . |  4 |  0  0  0  2  2  0 |  *  *  *  *  *  *  *  *  *  * 2N  * | 0 0 1 0 0 0 1 0
. . . x . x |  4 |  0  0  0  2  0  2 |  *  *  *  *  *  *  *  *  *  *  * 2N | 0 0 0 1 0 0 0 1
------------+----+-------------------+-------------------------------------+----------------
x . x . x .   8 |  4  0  4  0  4  0 |  2  0  2  0  0  0  0  0  2  0  0  0 | N * * * * * * *
x . x . . x   8 |  4  0  4  0  0  4 |  2  0  0  2  0  0  0  0  0  2  0  0 | * N * * * * * *
x . . x x .   8 |  4  0  0  4  4  0 |  0  2  2  0  0  0  0  0  0  0  2  0 | * * N * * * * *
x . . x . x   8 |  4  0  0  4  0  4 |  0  2  0  2  0  0  0  0  0  0  0  2 | * * * N * * * *
. x x . x .   8 |  0  4  4  0  4  0 |  0  0  0  0  2  0  2  0  2  0  0  0 | * * * * N * * *
. x x . . x   8 |  0  4  4  0  0  4 |  0  0  0  0  2  0  0  2  0  2  0  0 | * * * * * N * *
. x . x x .   8 |  0  4  0  4  4  0 |  0  0  0  0  0  2  2  0  0  0  2  0 | * * * * * * N *
. x . x . x   8 |  0  4  0  4  0  4 |  0  0  0  0  0  2  0  2  0  0  0  2 | * * * * * * * N

:x:4:o:4:o:&##x   (N → ∞)   → height = 1

 o 4 o 4 o     | N   4 2 | 4  8 | 8
---------------+---+------+------+--
 x   .   .     | 2 | 2N * | 2  2 | 4
:o:4:o:4:o:&#x | 2 |  * N | 0  4 | 4
---------------+---+------+------+--
 x 4 o   .     | 4 |  4 0 | N  * | 2
:x:  .   . &#x | 4 |  2 2 | * 2N | 2
---------------+---+------+------+--
:x:4:o:  . &#x  8 |  8 4 | 2  4 | N

:qoo:3:oqo:3:ooq:3*a&##x   (N → ∞)   → all heights = 1/sqrt(3) = 0.577350

 o.. 3 o.. 3 o.. 3*a     | N * *   3  0  3 |  3  6  3 | 3 3 2
 .o. 3 .o. 3 .o. 3*a     | * N *   3  3  0 |  6  3  3 | 3 2 3
 ..o 3 ..o 3 ..o 3*a     | * * N   0  3  3 |  3  3  6 | 2 3 3
-------------------------+-------+----------+----------+------
 oo. 3 oo. 3 oo. 3*a&#x  | 1 1 0 | 3N  *  * |  2  2  0 | 2 1 1
 .oo 3 .oo 3 .oo 3*a&#x  | 0 1 1 |  * 3N  * |  2  0  2 | 1 1 2
:o.o:3:o.o:3:o.o:3*a&#x  | 1 0 1 |  *  * 3N |  0  2  2 | 1 2 1
-------------------------+-------+----------+----------+------
 ...   oqo   ...    &#xt | 1 2 1 |  2  2  0 | 3N  *  * | 1 0 1
:qoo:  ...   ...    &#xt | 2 1 1 |  2  0  2 |  * 3N  * | 1 1 0
 ...   ...  :ooq:   &#xt | 1 1 2 |  0  2  2 |  *  * 3N | 0 1 1
-------------------------+-------+----------+----------+------
:qoo:3:oqo:  ...    &#xt  3 3 2 |  6  3  3 |  3  3  0 | N * *
:qoo:  ...  :ooq:3*a&#xt  3 2 3 |  3  3  6 |  0  3  3 | * N *
 ...  :oqo:3:ooq:   &#xt  2 3 3 |  3  6  3 |  3  0  3 | * * N

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