Acronym rich
Name rectified cubic honeycomb
 
 © ©    ©
Vertex figure
 ©
Coordinates (i/sqrt(2), j/sqrt(2), k/sqrt(2))           for integers i,j,k with i+j+k even, but i,j,k not all even themselves
Confer
related CRF honeycombs:
gyrich   pexrich   6Q3-2S3-gyro   6Q3-2S3-ortho   3Q3-S3-2P6-2P3-gyro   3Q3-S3-2P6-2P3-ortho  
general polytopal classes:
partial Stott expansions  
External
links
wikipedia

Parallel dissections of this uniform honeycomb at its thats, i.e. cutting each into a pair of tricues, allows for several (non-elementary) scaliform honeycombs.

Whereas parallel elongations at its squats, cutting all octs into pairs of squippies, leads (if recombining those with that introduced cube into esquidpies) to pexrich.


Incidence matrix according to Dynkin symbol

o4x3o4o   (N → ∞)

. . . . | 3N    8 |  4  8 | 4 2
--------+----+-----+-------+----
. x . . |  2 | 12N |  1  2 | 2 1
--------+----+-----+-------+----
o4x . . |  4 |   4 | 3N  * | 2 0
. x3o . |  3 |   3 |  * 8N | 1 1
--------+----+-----+-------+----
o4x3o .  12 |  24 |  6  8 | N *
. x3o4o   6 |  12 |  0  8 | * N

o4x3o4/3o   (N → ∞)

. . .   . | 3N    8 |  4  8 | 4 2
----------+----+-----+-------+----
. x .   . |  2 | 12N |  1  2 | 2 1
----------+----+-----+-------+----
o4x .   . |  4 |   4 | 3N  * | 2 0
. x3o   . |  3 |   3 |  * 8N | 1 1
----------+----+-----+-------+----
o4x3o   .  12 |  24 |  6  8 | N *
. x3o4/3o   6 |  12 |  0  8 | * N

o4o3x4/3o   (N → ∞)

. . .   . | 3N    8 |  8  4 | 2 4
----------+----+-----+-------+----
. . x   . |  2 | 12N |  2  1 | 1 2
----------+----+-----+-------+----
. o3x   . |  3 |   3 | 8N  * | 1 1
. . x4/3o |  4 |   4 |  * 3N | 0 2
----------+----+-----+-------+----
o4o3x   .   6 |  12 |  8  0 | N *
. o3x4/3o  12 |  24 |  8  6 | * N

o4/3x3o4/3o   (N → ∞)

.   . .   . | 3N    8 |  4  8 | 4 2
------------+----+-----+-------+----
.   x .   . |  2 | 12N |  1  2 | 2 1
------------+----+-----+-------+----
o4/3x .   . |  4 |   4 | 3N  * | 2 0
.   x3o   . |  3 |   3 |  * 8N | 1 1
------------+----+-----+-------+----
o4/3x3o   .  12 |  24 |  6  8 | N *
.   x3o4/3o   6 |  12 |  0  8 | * N

o3x3o *b4o   (N → ∞)

. . .    . | 6N    8 |  4  4  4 |  2 2 2
-----------+----+-----+----------+-------
. x .    . |  2 | 24N |  1  1  1 |  1 1 1
-----------+----+-----+----------+-------
o3x .    . |  3 |   3 | 8N  *  * |  1 1 0
. x3o    . |  3 |   3 |  * 8N  * |  1 0 1
. x . *b4o |  4 |   4 |  *  * 6N |  0 1 1
-----------+----+-----+----------+-------
o3x3o    .   6 |  12 |  4  4  0 | 2N * *
o3x . *b4o  12 |  24 |  8  0  6 |  * N *
. x3o *b4o  12 |  24 |  0  8  6 |  * * N

x3o3x *b4o   (N → ∞)

. . .    . | 6N    4   4 |  4  4  4 |  4 1 1
-----------+----+---------+----------+-------
x . .    . |  2 | 12N   * |  1  2  0 |  2 1 0
. . x    . |  2 |   * 12N |  0  2  1 |  2 0 1
-----------+----+---------+----------+-------
x3o .    . |  3 |   3   0 | 8N  *  * |  1 1 0
x . x    . |  4 |   2   2 |  * 6N  * |  2 0 0
. o3x    . |  3 |   0   3 |  *  * 8N |  1 0 1
-----------+----+---------+----------+-------
x3o3x    .  12 |  12  12 |  4  6  4 | 2N * *
x3o . *b4o   6 |  12   0 |  8  0  0 |  * N *
. o3x *b4o   6 |   0  12 |  0  0  8 |  * * N

o3x3o *b4/3o   (N → ∞)

. . .      . | 6N    8 |  4  4  4 |  2 2 2
-------------+----+-----+----------+-------
. x .      . |  2 | 24N |  1  1  1 |  1 1 1
-------------+----+-----+----------+-------
o3x .      . |  3 |   3 | 8N  *  * |  1 1 0
. x3o      . |  3 |   3 |  * 8N  * |  1 0 1
. x . *b4/3o |  4 |   4 |  *  * 6N |  0 1 1
-------------+----+-----+----------+-------
o3x3o      .   6 |  12 |  4  4  0 | 2N * *
o3x . *b4/3o  12 |  24 |  8  0  6 |  * N *
. x3o *b4/3o  12 |  24 |  0  8  6 |  * * N

x3o3x *b4/3o   (N → ∞)

. . .      . | 6N    4   4 |  4  4  4 |  4 1 1
-------------+----+---------+----------+-------
x . .      . |  2 | 12N   * |  1  2  0 |  2 1 0
. . x      . |  2 |   * 12N |  0  2  1 |  2 0 1
-------------+----+---------+----------+-------
x3o .      . |  3 |   3   0 | 8N  *  * |  1 1 0
x . x      . |  4 |   2   2 |  * 6N  * |  2 0 0
. o3x      . |  3 |   0   3 |  *  * 8N |  1 0 1
-------------+----+---------+----------+-------
x3o3x      .  12 |  12  12 |  4  6  4 | 2N * *
x3o . *b4/3o   6 |  12   0 |  8  0  0 |  * N *
. o3x *b4/3o   6 |   0  12 |  0  0  8 |  * * N

x3o3x3o3*a   (N → ∞)

. . . .    | 6N    4   4 |  2  4  2  2  2 | 2 1 2 1
-----------+----+---------+----------------+--------
x . . .    |  2 | 12N   * |  1  1  1  0  0 | 1 1 1 0
. . x .    |  2 |   * 12N |  0  1  0  1  1 | 1 0 1 1
-----------+----+---------+----------------+--------
x3o . .    |  3 |   3   0 | 4N  *  *  *  * | 1 1 0 0
x . x .    |  4 |   2   2 |  * 6N  *  *  * | 1 0 1 0
x . . o3*a |  3 |   3   0 |  *  * 4N  *  * | 0 1 1 0
. o3x .    |  3 |   0   3 |  *  *  * 4N  * | 1 0 0 1
. . x3o    |  3 |   0   3 |  *  *  *  * 4N | 0 0 1 1
-----------+----+---------+----------------+--------
x3o3x .     12 |  12  12 |  4  6  0  4  0 | N * * *
x3o . o3*a   6 |  12   0 |  4  0  4  0  0 | * N * *
x . x3o3*a  12 |  12  12 |  0  6  4  0  4 | * * N *
. o3x3o      6 |   0  12 |  0  0  0  4  4 | * * * N

x3o3x3/2o3/2*a   (N → ∞)

. . .   .      | 6N    4   4 |  2  4  2  2  2 | 2 1 2 1
---------------+----+---------+----------------+--------
x . .   .      |  2 | 12N   * |  1  1  1  0  0 | 1 1 1 0
. . x   .      |  2 |   * 12N |  0  1  0  1  1 | 1 0 1 1
---------------+----+---------+----------------+--------
x3o .   .      |  3 |   3   0 | 4N  *  *  *  * | 1 1 0 0
x . x   .      |  4 |   2   2 |  * 6N  *  *  * | 1 0 1 0
x . .   o3/2*a |  3 |   3   0 |  *  * 4N  *  * | 0 1 1 0
. o3x   .      |  3 |   0   3 |  *  *  * 4N  * | 1 0 0 1
. . x3/2o      |  3 |   0   3 |  *  *  *  * 4N | 0 0 1 1
---------------+----+---------+----------------+--------
x3o3x   .       12 |  12  12 |  4  6  0  4  0 | N * * *
x3o .   o3/2*a   6 |  12   0 |  4  0  4  0  0 | * N * *
x . x3/2o3/2*a  12 |  12  12 |  0  6  4  0  4 | * * N *
. o3x3/2o        6 |   0  12 |  0  0  0  4  4 | * * * N

s4x3o4o   (N → ∞)

demi( . . . . ) | 6N    4   4 |  4  4  4 | 1  4 1
----------------+----+---------+----------+-------
demi( . x . . ) |  2 | 12N   * |  1  2  0 | 1  2 0
sefa( s4x . . ) |  2 |   * 12N |  0  2  1 | 0  2 1
----------------+----+---------+----------+-------
demi( . x3o . ) |  3 |   3   0 | 8N  *  * | 1  1 0
      s4x . .   |  4 |   2   2 |  * 6N  * | 0  2 0
sefa( s4x3o . ) |  3 |   0   3 |  *  * 8N | 0  1 1
----------------+----+---------+----------+-------
demi( . x3o4o )   6 |  12   0 |  8  0  0 | N  * *
      s4x3o .    12 |  12  12 |  4  6  4 | * 2N *
sefa( s4x3o4o )   6 |   0  12 |  0  0  8 | *  * N

starting figure: x4x3o4o

o3x3o *b4s

demi( . . .    . ) | 6N    4   4 |  2  2  4  2  2 | 1 2 2 1
-------------------+----+---------+----------------+--------
demi( . x .    . ) |  2 | 12N   * |  1  1  1  0  0 | 1 1 1 0
sefa( . x . *b4s ) |  2 |   * 12N |  0  0  1  1  1 | 0 1 1 1
-------------------+----+---------+----------------+--------
demi( o3x .    . ) |  3 |   3   0 | 4N  *  *  *  * | 1 1 0 0
demi( . x3o    . ) |  3 |   3   0 |  * 4N  *  *  * | 1 0 1 0
      . x . *b4s   |  4 |   2   2 |  *  * 6N  *  * | 0 1 1 0
sefa( o3x . *b4s ) |  3 |   0   3 |  *  *  * 4N  * | 0 1 0 1
sefa( . x3o *b4s ) |  3 |   0   3 |  *  *  *  * 4N | 0 0 1 1
-------------------+----+---------+----------------+--------
demi( o3x3o    . )   6 |  12   0 |  4  4  0  0  0 | N * * *
      o3x . *b4s    12 |  12  12 |  4  0  6  4  0 | * N * *
      . x3o *b4s    12 |  12  12 |  0  4  6  0  4 | * * N *
sefa( o3x3o *b4s )   6 |   0  12 |  0  0  0  4  4 | * * * N

starting figure: o3x3o *b4x

:oq:4:xo:4:oo:&##x   (N → ∞)   → all heights = 1/sqrt(2) = 0.707107

 o. 4 o. 4 o.      | 2N *   4  2  2 | 2  4  2  4 | 4 2
 .o 4 .o 4 .o      |  * N   0  4  4 | 0  4  4  4 | 4 2
-------------------+------+----------+------------+----
 ..   x.   ..      |  2 0 | 4N  *  * | 1  1  0  1 | 2 1
 oo 4 oo 4 oo &#x  |  1 1 |  * 4N  * | 0  2  1  0 | 2 1
:oo:4:oo:4:oo:&#x  |  1 1 |  *  * 4N | 0  0  1  2 | 2 1
-------------------+------+----------+------------+----
 o. 4 x.   ..      |  4 0 |  4  0  0 | N  *  *  * | 2 0
 ..   xo   .. &#x  |  2 1 |  1  2  0 | * 4N  *  * | 1 1
:oq:  ..   .. &#xt |  2 2 |  0  2  2 | *  * 2N  * | 2 0
 ..  :xo:  .. &#x  |  2 1 |  1  0  2 | *  *  * 4N | 1 1
-------------------+------+----------+------------+----
:oq:4:xo:  .. &#xt   8 4 |  8  8  8 | 2  4  4  4 | N *
 ..  :xo:4:oo:&#xt   4 2 |  4  4  4 | 0  4  0  4 | * N

:xxo:3:xox:3:oxx:3*a&##x   (N → ∞)   → all heights = sqrt(2/3) = 0.816497

 o.. 3 o.. 3 o.. 3*a     | 3N  *  *   2  2  2  0  0  0  0  0  2 | 1 1  2  2  1 0 0  0  0  0 0 0  2  2  1 | 1 1 0 2 1 1
 .o. 3 .o. 3 .o. 3*a     |  * 3N  *   0  0  2  2  2  2  0  0  0 | 0 0  2  1  2 1 1  2  1  2 0 0  0  0  0 | 2 1 1 1 0 1
 ..o 3 ..o 3 ..o 3*a     |  *  * 3N   0  0  0  0  0  2  2  2  2 | 0 0  0  0  0 0 0  1  2  2 1 1  1  2  2 | 1 0 1 1 1 2
-------------------------+----------+----------------------------+----------------------------------------+------------
 x..   ...   ...         |  2  0  0 | 3N  *  *  *  *  *  *  *  * | 1 0  1  0  0 0 0  0  0  0 0 0  1  0  0 | 1 0 0 1 1 0
 ...   x..   ...         |  2  0  0 |  * 3N  *  *  *  *  *  *  * | 0 1  0  1  0 0 0  0  0  0 0 0  0  1  0 | 0 1 0 1 0 1
 oo. 3 oo. 3 oo. 3*a&#x  |  1  1  0 |  *  * 6N  *  *  *  *  *  * | 0 0  1  1  1 0 0  0  0  0 0 0  0  0  0 | 1 1 0 1 0 0
 .x.   ...   ...         |  0  2  0 |  *  *  * 3N  *  *  *  *  * | 0 0  1  0  0 1 0  1  0  0 0 0  0  0  0 | 1 0 1 1 0 0
 ...   ...   .x.         |  0  2  0 |  *  *  *  * 3N  *  *  *  * | 0 0  0  0  1 0 1  0  0  1 0 0  0  0  0 | 1 1 0 0 0 1
 .oo 3 .oo 3 .oo 3*a&#x  |  0  1  1 |  *  *  *  *  * 6N  *  *  * | 0 0  0  0  0 0 0  1  1  1 0 0  0  0  0 | 1 0 1 0 0 1
 ...   ..x   ...         |  0  0  2 |  *  *  *  *  *  * 3N  *  * | 0 0  0  0  0 0 0  0  1  0 1 0  0  1  0 | 0 0 1 1 0 1
 ...   ...   ..x         |  0  0  2 |  *  *  *  *  *  *  * 3N  * | 0 0  0  0  0 0 0  0  0  1 0 1  0  0  1 | 1 0 0 0 1 1
:o.o:3:o.o:3:o.o:3*a&#x  |  1  0  1 |  *  *  *  *  *  *  *  * 6N | 0 0  0  0  0 0 0  0  0  0 0 0  1  1  1 | 0 0 0 1 1 1
-------------------------+----------+----------------------------+----------------------------------------+------------
 x..   ...   o.. 3*a     |  3  0  0 |  3  0  0  0  0  0  0  0  0 | N *  *  *  * * *  *  *  * * *  *  *  * | 1 0 0 0 1 0
 ...   x.. 3 o..         |  3  0  0 |  0  3  0  0  0  0  0  0  0 | * N  *  *  * * *  *  *  * * *  *  *  * | 0 1 0 0 0 1
 xx.   ...   ...    &#x  |  2  2  0 |  1  0  2  1  0  0  0  0  0 | * * 3N  *  * * *  *  *  * * *  *  *  * | 1 0 0 1 0 0
 ...   xo.   ...    &#x  |  2  1  0 |  0  1  2  0  0  0  0  0  0 | * *  * 3N  * * *  *  *  * * *  *  *  * | 0 1 0 1 0 0
 ...   ...   ox.    &#x  |  1  2  0 |  0  0  2  0  1  0  0  0  0 | * *  *  * 3N * *  *  *  * * *  *  *  * | 1 1 0 0 0 0
 .x. 3 .o.               |  0  3  0 |  0  0  0  3  0  0  0  0  0 | * *  *  *  * N *  *  *  * * *  *  *  * | 0 0 1 1 0 0
 ...   .o. 3 .x.         |  0  3  0 |  0  0  0  0  3  0  0  0  0 | * *  *  *  * * N  *  *  * * *  *  *  * | 0 1 0 0 0 1
 .xo   ...   ...    &#x  |  0  2  1 |  0  0  0  1  0  2  0  0  0 | * *  *  *  * * * 3N  *  * * *  *  *  * | 1 0 1 0 0 0
 ...   .ox   ...    &#x  |  0  1  2 |  0  0  0  0  0  2  1  0  0 | * *  *  *  * * *  * 3N  * * *  *  *  * | 0 0 1 0 0 1
 ...   ...   .xx    &#x  |  0  2  2 |  0  0  0  0  1  2  0  1  0 | * *  *  *  * * *  *  * 3N * *  *  *  * | 1 0 0 0 0 1
 ..o 3 ..x   ...         |  0  0  3 |  0  0  0  0  0  0  3  0  0 | * *  *  *  * * *  *  *  * N *  *  *  * | 0 0 1 1 0 0
 ..o   ...   ..x 3*a     |  0  0  3 |  0  0  0  0  0  0  0  3  0 | * *  *  *  * * *  *  *  * * N  *  *  * | 1 0 0 0 1 0
:x.o:  ...   ...    &#x  |  2  0  1 |  1  0  0  0  0  0  0  0  2 | * *  *  *  * * *  *  *  * * * 3N  *  * | 0 0 0 1 1 0
 ...  :x.x:  ...    &#x  |  2  0  2 |  0  1  0  0  0  0  1  0  2 | * *  *  *  * * *  *  *  * * *  * 3N  * | 0 0 0 1 0 1
 ...   ...  :o.x:   &#x  |  1  0  2 |  0  0  0  0  0  0  0  1  2 | * *  *  *  * * *  *  *  * * *  *  * 3N | 0 0 0 0 1 1
-------------------------+----------+----------------------------+----------------------------------------+------------
 xxo   ...   oxx 3*a&#xt   3  6  3 |  3  0  6  3  3  6  0  3  0 | 1 0  3  0  3 0 0  3  0  3 0 1  0  0  0 | N * * * * *
 ...   xo. 3 ox.    &#x    3  3  0 |  0  3  6  0  3  0  0  0  0 | 0 1  0  3  3 0 1  0  0  0 0 0  0  0  0 | * N * * * *
 .xo 3 .ox   ...    &#x    0  3  3 |  0  0  0  3  0  6  3  0  0 | 0 0  0  0  0 1 0  3  3  0 1 0  0  0  0 | * * N * * *
:xxo:3:xox:  ...    &#xt   6  3  3 |  3  3  6  3  0  0  3  0  6 | 0 0  3  3  0 1 0  0  0  0 1 0  3  3  0 | * * * N * *
:x.o:  ...  :o.x:3*a&#x    3  0  3 |  3  0  0  0  0  0  0  3  6 | 1 0  0  0  0 0 0  0  0  0 0 1  3  0  3 | * * * * N *
 ...  :xox:3:oxx:   &#xt   3  3  6 |  0  3  0  0  3  6  3  3  6 | 0 1  0  0  0 0 1  0  3  3 0 0  0  3  3 | * * * * * N

© 2004-2019
top of page