## Honeycombs as 2-sided infinite lace towers of tilings

Higher-dimensional geometry (previously "Polyshapes").

### Honeycombs as 2-sided infinite lace towers of tilings

Hi,

most of us know about Wendy's invention of lace prisms and lace towers.
For a short intro refer to

Yesterday it occured to me, that we not only could consider 2 parallel layers of polytopes, both having a common symmetry description (lace prism), nor (finite) multistratic stacks of such (lace towers), but 2-sided likewise unbounded such stacks (kind of 2-sided infinite towers). And applying that not only to polytopes, but also to tilings. I.e. describing an honeycomb just by an infinite series of sectioning planes.

--- rk
Last edited by Klitzing on Fri Nov 01, 2013 5:40 pm, edited 1 time in total.
Klitzing
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

Lace prism:
xx4oo&#x is a cube: x.4o. (square) atop of .x4.o (square), laced by (&#) x-edges, that is unity again.

Lace tower:
xxo3oxx&#xt is a cuboctahedron: x..3o.. (triangle) atop pseudo .x.3.x. (equatorial hexagon) atop ..o3..x (opposite triangle in dual orientation), laced by (&#) x-edges again. Here being a t (tower).
Alternate description of cuboctahedron: xox4oqo&#xt, i.e. x..4o.. (unit square) atop pseudo .o.4.q. (sqrt2 square in dual orientation) atop ..x4..o (square again).

For an infinite stack we would have accordingly an infinte amount of layers, that is an infinite number of node symbols at every node position. Sure impossible. But we would like to consider periodic stackings only. So we could use some period sign in addition - I thought here of a ":". And, additionally, it might serve useful in some occasions, that we not only consider subelements, which run across the boundaries of that period, but which are infinite themselves (e.g. for an apeirogon, being part of an apeirogonal prism, considered as infinite stack of squares). So we might add an i (infinite) onto the suffix (&#xti).

Providing a first easy explicite example:
Code: Select all
aze = x-infin-o = :o:&#xti o       | N | 2---------+---+--:x:&#x   | 2 | N

Now a second one, again the same apeirogon:
Code: Select all
aze = x-infin-x =:oo:&#xti o.       | N * | 1 1 .o       | * N | 1 1----------+-----+---- oo &#x   | 1 1 | N * inner:oo:&#x   | 1 1 | * N outer

Next the mentioned apeirogonal prism:
Code: Select all
azip = x x-infin-o =:x:&#xti o       | 2N | 1  2 | 2 1---------+----+------+---- x       |  2 | N  * | 2 0:o:&#x   |  2 | * 2N | 1 1---------+----+------+----:x:&#x   |  4 | 2  2 | N *:o:&#xti |  N | 0  N | * 2

Or likewise with inner and outer edges (wrt to the chosen period):
Code: Select all
x x-infin-x =:xx:&#xti o.       | 2N  * | 1  1  1 0 | 1 1 1 .o       |  * 2N | 0  1  1 1 | 1 1 1----------+-------+-----------+------ x.       |  2  0 | N  *  * * | 1 1 0 oo &#x   |  1  1 | * 2N  * * | 1 0 1 inner:oo:&#x   |  1  1 | *  * 2N * | 0 1 1 outer .x       |  0  2 | *  *  * N | 1 1 0----------+-------+-----------+------ xx &#x   |  2  2 | 1  2  0 1 | N * * inner:xx:&#x   |  2  2 | 1  0  2 1 | * N * outer:oo:&#xti |  N  N | 0  N  N 0 | * * 2

(In all these cases N would have to run to infinity, for sure. But the prefixed numbers then still show the relative frequencies.)

--- rk
Klitzing
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

Okay, you might say, this works for quite simple examples. But beyond?

Well, next we like to consider some 2D tilings, given as infinite stacks of accordingly spaced apeirogons:

Example 1:
Code: Select all
squat = x4o4o =:x:-infin-:o:&#xti(N -> infin:  within each stacked section,  M -> infin:  for the count of sections) o -infin- o       | NM |  2  2 |  4-------------------+----+-------+--- x         .       |  2 | NM  * |  2:o:-infin-:o:&#x   |  2 |  * NM |  2-------------------+----+-------+---:x:        . &#x   |  4 |  2  2 | NM

Example 2:
Code: Select all
tosquat = x4x4o =:wxw:-infin-:oqo:&#xti o.. -infin- o..       | NM   *  * |   2  1  0   0 |  2  1 .o. -infin- .o.       |  * 2NM  * |   1  0  1   1 |  2  1 ..o -infin- ..o       |  *   * NM |   0  1  0   2 |  2  1-----------------------+-----------+---------------+------ oo. -infin- oo. &#x   |  1   1  0 | 2NM  *  *   * |  1  1:o.o:-infin-:o.o:&#x   |  1   0  1 |   * NM  *   * |  2  0 .x.         ...       |  0   2  0 |   *  * NM   * |  2  0 .oo -infin- .oo &#x   |  0   1  1 |   *  *  * 2NM |  1  1-----------------------+-----------+---------------+------:wxw:        ... &#xt  |  2   4  2 |   2  2  2   2 | NM  * ...         oqo &#xt  |  1   2  1 |   2  0  0   2 |  * NM

(Here w is a (pseudo) edge of length 1+sqrt2, while q is of length sqrt2, and x is unity.)

--- rk
Klitzing
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

Still not convinced?
Okay, then let's turn to the cases mentioned in my introductory mail: i.e. honeycombs, being considered as infinite stacks of tilings!

Example 1:
Code: Select all
chon = x4o3o4o =:x:4:o:4:o:&#xti(N -> infin:  within each stacked section,  M -> infin:  for the count of sections) o 4 o 4 o       | NM |   4  2 |  4   8 |  8-----------------+----+--------+--------+--- x   .   .       |  2 | 2NM  * |  2   2 |  4:o:4:o:4:o:&#x   |  2 |   * NM |  0   4 |  4-----------------+----+--------+--------+--- x 4 o   .       |  4 |   4  0 | NM   * |  2:x:  .   . &#x   |  4 |   2  2 |  * 2NM |  2-----------------+----+--------+--------+---:x:4:o:  . &#x   |  8 |   8  4 |  2   4 | NM  cube

Example 2:
Code: Select all
batch = o4x3x4o =:qooo:4:xuxu:4:ooqo:&#xti o... 4 o... 4 o...       | 4NM   *   *   * |   2   1   0   0   0   1 |  1   2   1   2   0  0 |  2  2 .o.. 4 .o.. 4 .o..       |   * 2NM   *   * |   0   2   2   0   0   0 |  0   4   1   0   1  0 |  2  2 ..o. 4 ..o. 4 ..o.       |   *   * 4NM   * |   0   0   1   2   1   0 |  0   2   0   2   1  1 |  2  2 ...o 4 ...o 4 ...o       |   *   *   * 2NM |   0   0   0   0   2   2 |  0   0   1   4   1  0 |  2  2--------------------------+-----------------+-------------------------+-----------------------+------ ....   x...   ....       |   2   0   0   0 | 4NM   *   *   *   *   * |  1   1   0   1   0  0 |  1  2 oo.. 4 oo.. 4 oo.. &#x   |   1   1   0   0 |   * 4NM   *   *   *   * |  0   2   1   0   0  0 |  2  1 .oo. 4 .oo. 4 .oo. &#x   |   0   1   1   0 |   *   * 4NM   *   *   * |  0   2   0   0   1  0 |  1  2 ....   ..x.   ....       |   0   0   2   0 |   *   *   * 4NM   *   * |  0   1   0   1   0  1 |  2  1 ..oo 4 ..oo 4 ..oo &#x   |   0   0   1   1 |   *   *   *   * 4NM   * |  0   0   0   2   1  0 |  1  2:o..o:4:o..o:4:o..o:&#x   |   1   0   0   1 |   *   *   *   *   * 4NM |  0   0   1   2   0  0 |  2  1--------------------------+-----------------+-------------------------+-----------------------+------ ....   x... 4 o...       |   4   0   0   0 |   4   0   0   0   0   0 | NM   *   *   *   *  * |  0  2  {4} ....   xux.   .... &#xt  |   2   2   2   0 |   1   2   2   1   0   0 |  * 4NM   *   *   *  * |  1  1  {6}:qo.o:  ....   .... &#xt  |   2   1   0   1 |   0   2   0   0   0   2 |  *   * 2NM   *   *  * |  2  0  {4} ....  :x.xu:  .... &#xt  |   2   0   2   2 |   1   0   0   1   2   2 |  *   *   * 4NM   *  * |  1  1  {6} ....   ....   .oqo &#xt  |   0   1   2   1 |   0   0   2   0   2   0 |  *   *   *   * 2NM  * |  0  2  {4} ..o. 4 ..x.   ....       |   0   0   4   0 |   0   0   0   4   0   0 |  *   *   *   *   * NM |  2  0  {4}--------------------------+-----------------+-------------------------+-----------------------+------:qooo:4:xuxu:  .... &#xt  |   8   4   8   4 |   4   8   4   8   4   8 |  0   4   4   4   0  2 | NM  *  toe ....  :xuxu:4:ooqo:&#xt  |   8   4   8   4 |   8   4   8   4   8   4 |  2   4   0   4   4  0 |  * NM  toe

(Here u is a pseudo edge of length 2, q of length sqrt2, and x of unity.)

--- rk
Klitzing
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

It is certianly known about making lace towers etc out of honeycombs. The most useful of these is the A_n lattice, which consists of a ring of nodes. But other structures are used too. Some simple lace towers on hyperbolic lattices have been done too. Laminate theory involves tiling-layers as lace-towers.

In any case, a simple lace tower is xo4xx4ox&#x, which occurs as a slice of x3o4xAo. The latter is a laminate of thae first and xx4xx4oo&#x, the square cupolae becooming a slice of rCO, and the tetrahedra and cubes coming through from different layers.

One can get the vertices of simplex-axis first figures from the t-basic, semicubic and gosset symmetries, by realising that they are in fact, derived from stepping subsequent layers of the t-basic by 1, 2, and 3 steps. 0 and 4 also work, the latter shows us that the 3_31 is in fact a 'body-centred A_7.

When you figure out where the centre and radius is on these figures, one simply draws a sphere in this figure, and where it crosses the rings, one gets the sections. So, eg for x3o3o3x3o3o3o, the vertices lie in t-basic, and we see that this figure has a diameter of 7+4+4+35 = 60. Since the first of the four examples represents all of the verticies of A_7, we first need to calculate where the centre is (1+0+0+4+0+0 mod 7 = 5), we look for pointers where 5 is the centre, and 33 is the radius.

Richard's confusion with my notation in the other thread, where i give 1s2s0s0, is partly derived from this lace-tower thing. It is the ultimate expression of pure numerics in the nodes (which are written as numbers), and the branches are reduced to commas, so s=3. So this figure might be written as x3u3o3o in a notation that Richard is more familiar with. But the C-D diagram can be presented as a free coordinate system, with position-polytopes, and this is what the 1s2s etc notation is saying.
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the dream we dream together is reality.

wendy
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

wendy wrote:It is certianly known about making lace towers etc out of honeycombs. [...]

In any case, a simple lace tower is xo4xx4ox&#x, which occurs as a slice of x3o4xAo. The latter is a laminate of thae first and xx4xx4oo&#x, the square cupolae becooming a slice of rCO, and the tetrahedra and cubes coming through from different layers. [...]

Okay, the mere possibility then was known. In fact, not too surprising.
At least the described full display (instead of mere single segments), and the according notational elemnents, introduced for period selection, seem new.

Here then is the according incidence matrix of ratoh, in the way as you intended (i.e. orthogonal to squat symmetry):
Code: Select all
ratoh = x3o3o *b4x =:xoox:4:xxxx:4:oxxo:&#xti   (N → ∞) o... 4 o... 4 o...       | 4N  *  *  * |  1  2  2  0  0  0  0  0  0  0  0  1 | 1  2  2  1 0  0  0 0  0  0  0 0  1  2 |  1 1 0  0 2 1 .o.. 4 .o.. 4 .o..       |  * 4N  *  * |  0  0  2  2  1  1  0  0  0  0  0  0 | 0  1  2  2 1  2  1 0  0  0  0 0  0  0 |  1 2 1  0 1 0 ..o. 4 ..o. 4 ..o.       |  *  * 4N  * |  0  0  0  0  0  1  2  1  2  0  0  0 | 0  0  0  0 0  2  1 1  1  2  2 0  0  0 |  0 2 1  1 1 0 ...o 4 ...o 4 ...o       |  *  *  * 4N |  0  0  0  0  0  0  0  0  2  1  2  1 | 0  0  0  0 0  0  0 0  2  2  1 1  1  2 |  0 1 0  1 2 1--------------------------+-------------+-------------------------------------+---------------------------------------+-------------- x...   ....   ....       |  2  0  0  0 | 2N  *  *  *  *  *  *  *  *  *  *  * | 0  2  0  0 0  0  0 0  0  0  0 0  1  0 |  1 0 0  0 2 0 ....   x...   ....       |  2  0  0  0 |  * 4N  *  *  *  *  *  *  *  *  *  * | 1  0  1  0 0  0  0 0  0  0  0 0  0  1 |  0 1 0  0 1 1 oo.. 4 oo.. 4 oo.. &#x   |  1  1  0  0 |  *  * 8N  *  *  *  *  *  *  *  *  * | 0  1  1  1 0  0  0 0  0  0  0 0  0  0 |  1 1 0  0 1 0 ....   .x..   ....       |  0  2  0  0 |  *  *  * 4N  *  *  *  *  *  *  *  * | 0  0  1  0 1  1  0 0  0  0  0 0  0  0 |  0 1 1  0 1 0 ....   ....   .x..       |  0  2  0  0 |  *  *  *  * 2N  *  *  *  *  *  *  * | 0  0  0  2 0  0  1 0  0  0  0 0  0  0 |  1 2 0  0 0 0 .oo. 4 .oo. 4 .oo. &#x   |  0  1  1  0 |  *  *  *  *  * 4N  *  *  *  *  *  * | 0  0  0  0 0  2  1 0  0  0  0 0  0  0 |  0 2 1  0 0 0 ....   ..x.   ....       |  0  0  2  0 |  *  *  *  *  *  * 4N  *  *  *  *  * | 0  0  0  0 0  1  0 1  0  1  0 0  0  0 |  0 1 1  0 1 0 ....   ....   ..x.       |  0  0  2  0 |  *  *  *  *  *  *  * 2N  *  *  *  * | 0  0  0  0 0  0  1 0  0  0  2 0  0  0 |  0 2 0  1 0 0 ..oo 4 ..oo 4 ..oo &#x   |  0  0  1  1 |  *  *  *  *  *  *  *  * 8N  *  *  * | 0  0  0  0 0  0  0 0  1  1  1 0  0  0 |  0 1 0  1 1 0 ...x   ....   ....       |  0  0  0  2 |  *  *  *  *  *  *  *  *  * 2N  *  * | 0  0  0  0 0  0  0 0  2  0  0 0  1  0 |  0 0 0  1 2 0 ....   ...x   ....       |  0  0  0  2 |  *  *  *  *  *  *  *  *  *  * 4N  * | 0  0  0  0 0  0  0 0  0  1  0 1  0  1 |  0 1 0  0 1 1:o..o:4:o..o:4:o..o:&#x   |  1  0  0  1 |  *  *  *  *  *  *  *  *  *  *  * 4N | 0  0  0  0 0  0  0 0  0  0  0 0  1  2 |  0 0 0  0 2 1--------------------------+-------------+-------------------------------------+---------------------------------------+-------------- ....   x... 4 o...       |  4  0  0  0 |  0  4  0  0  0  0  0  0  0  0  0  0 | N  *  *  * *  *  * *  *  *  * *  *  * |  0 1 0  0 0 1 xo..   ....   .... &#x   |  2  1  0  0 |  1  0  2  0  0  0  0  0  0  0  0  0 | * 4N  *  * *  *  * *  *  *  * *  *  * |  1 0 0  0 1 0 ....   xx..   .... &#x   |  2  2  0  0 |  0  1  2  1  0  0  0  0  0  0  0  0 | *  * 4N  * *  *  * *  *  *  * *  *  * |  0 1 0  0 1 0 ....   ....   ox.. &#x   |  1  2  0  0 |  0  0  2  0  1  0  0  0  0  0  0  0 | *  *  * 4N *  *  * *  *  *  * *  *  * |  1 1 0  0 0 0 .o.. 4 .x..   ....       |  0  4  0  0 |  0  0  0  4  0  0  0  0  0  0  0  0 | *  *  *  * N  *  * *  *  *  * *  *  * |  0 0 1  0 1 0 ....   .xx.   .... &#x   |  0  2  2  0 |  0  0  0  1  0  2  1  0  0  0  0  0 | *  *  *  * * 4N  * *  *  *  * *  *  * |  0 1 1  0 0 0 ....   ....   .xx. &#x   |  0  2  2  0 |  0  0  0  0  1  2  0  1  0  0  0  0 | *  *  *  * *  * 2N *  *  *  * *  *  * |  0 2 0  0 0 0 ..o. 4 ..x.   ....       |  0  0  4  0 |  0  0  0  0  0  0  4  0  0  0  0  0 | *  *  *  * *  *  * N  *  *  * *  *  * |  0 0 1  0 1 0 ..ox   ....   .... &#x   |  0  0  1  2 |  0  0  0  0  0  0  0  0  2  1  0  0 | *  *  *  * *  *  * * 4N  *  * *  *  * |  0 0 0  1 1 0 ....   ..xx   .... &#x   |  0  0  2  2 |  0  0  0  0  0  0  1  0  2  0  1  0 | *  *  *  * *  *  * *  * 4N  * *  *  * |  0 1 0  0 1 0 ....   ....   ..xo &#x   |  0  0  2  1 |  0  0  0  0  0  0  0  1  2  0  0  0 | *  *  *  * *  *  * *  *  * 4N *  *  * |  0 1 0  1 0 0 ....   ...x 4 ...o       |  0  0  0  4 |  0  0  0  0  0  0  0  0  0  0  4  0 | *  *  *  * *  *  * *  *  *  * N  *  * |  0 1 0  0 0 1:x..x:  ....   .... &#x   |  2  0  0  2 |  1  0  0  0  0  0  0  0  0  1  0  2 | *  *  *  * *  *  * *  *  *  * * 2N  * |  0 0 0  0 2 0 ....  :x..x:  .... &#x   |  2  0  0  2 |  0  1  0  0  0  0  0  0  0  0  1  2 | *  *  *  * *  *  * *  *  *  * *  * 4N |  0 0 0  0 1 1--------------------------+-------------+-------------------------------------+---------------------------------------+-------------- xo..   ....   ox.. &#x   |  2  2  0  0 |  1  0  4  0  1  0  0  0  0  0  0  0 | 0  2  0  2 0  0  0 0  0  0  0 0  0  0 | 2N * *  * * *  tet ....   xxxx 4 oxxo &#xt  |  4  8  8  4 |  0  4  8  4  4  8  4  4  8  0  4  0 | 1  0  4  4 0  4  4 0  0  4  4 1  0  0 |  * N *  * * *  sirco .oo. 4 .xx.   .... &#x   |  0  4  4  0 |  0  0  0  4  0  4  4  0  0  0  0  0 | 0  0  0  0 1  4  0 1  0  0  0 0  0  0 |  * * N  * * *  cube ..ox   ....   ..xo &#x   |  0  0  2  2 |  0  0  0  0  0  0  0  1  4  1  0  0 | 0  0  0  0 0  0  0 0  2  0  2 0  0  0 |  * * * 2N * *  tet:xoox:4:xxxx:  .... &#xt  |  8  4  4  8 |  4  4  8  4  0  0  4  0  8  4  4  8 | 0  4  4  0 1  0  0 1  4  4  0 0  4  4 |  * * *  * N *  sirco ....  :x..x:4:o..o:&#x   |  4  0  0  4 |  0  4  0  0  0  0  0  0  0  0  4  4 | 1  0  0  0 0  0  0 0  0  0  0 1  0  4 |  * * *  * * N  cube

--- rk
Klitzing
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### Re: Honeycombs as 2-sided infinite lace towers of tilings

Many of the sections of polytopes come from lace-towers of tilings, or varously, 'stations' and layered tilings. According to conway and slone, layered tilings are the most efficient up to 24 dimensions, except for the coxeter-todd lattice in 12 dimensions.

Suppose you take a layer representing A_n, made of spheres. In 3d, this would be a set of spheres arranged as spheres. You can stack these in different ways to get different 3d lattices. We shall look at the six-dimensional case for this exercise.

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      12  5  8  9  8  5       12  5  8  9  8  5        12  5  8  9  8  5   0   x  o  o  o  o  o    0   x  o  o  a  o  o     0   x  o  a  o  e  o   7   o  x  o  o  o  o    4   o  o  x  o  o  a     3   o  e  o  x  o  a  28   o  o  x  o  o  o   16   o  a  o  o  x  o    12   x  o  a  o  e  o  63   o  o  o  x  o  o   36   x  o  o  a  o  o    27   o  e  o  x  o  a      T-basic A_6               Semicubic B_6          Gosset 2_22      12  5  8  9  8  5  0    x  o  o  o  o  o  7    o  x  o  o  o  o 28    x  o  o  o  o  o 63    o  x  o  o  o  o      Laminate LB5

The above tables show sections of the three trigonal groups against the simplex group A_x. You can read the across row as an unrolled loop group 6:, and the vertical as a linear column. The numbers indicate the nearest point of the same kind, to the origion (0, 12). The lattice edge is sqrt(12), the distances are squares.

In the semi-cubic, the 'a' nodes represent the semuicubic at the 'other end', ie if x represents the vertices of /E4A, then a is that at E4/A, for example. The combination of the two represents a squashed body-centred cubic.

The third shows the 2_22, where the ends of the three tails show as x, a, and e. The cell centres are at a and e, so we can find the cell by drawing a circle of size 8 around it. We can get this by 2 steps on the same row (8, 0), or one step across, and one step up (5, 3). Both of these add to 8. The presentation of the 2_21 down a simplex-first, is xox3ooo3ooo3oxo3ooo&#xt, which shows that it contains the vertices and edges of a simplex-line prism. Note the simplexes point the same way.

The vertex figure 1_22 gives in two rows up (0, 12) = point, one row up = (9, 3) = o3o3x3o3o, same row = x3o3o3o3o3x, and symmetrically below. So it comes to a lace prism ooxoo3ooooo3oxoxo3ooooo3ooxoo&#xt.

The spheres of radius 12 contain the 'eutactuc stars' of the various trigonal groups. Since these represent the normals to the mirror-planes, every polytope that can be written in 5, 4A and 4B, with integer coordinates, have their vertices in these vertices, and no others. So if you can determine the radius and the cell centres, it is possible to use height and spheres (as we did for 2_21 and 1_22), to find the simplex rings of the various figures.

For the simplex group, the base here is 6:, and the stack in the first frame represents 7:. The cell centres of 7: lie at intervals of 6/7 of the row heights in the diagram above (where the rows are evenly spaced). One can determine at what point the centre of any given figure constructed from this symmetry lies, by noting that the stott-vectors correspond to sums of 5,-1,-1,-1,-1,-1, in strict cyclic permutations. So one gets

These are the plane, in 6D, represented by \sum x_i = 0. This gives a perefectly euclidean tiling, giving a lattice of edge sqrt(2).

x3o3o3o3o3o = 5, -1, -1, -1, -1, -1
o3x3o3o3o3o = 4, 4, -2, -2, -2, -2
o3o3x3o3o3o = 3, 3, 3, -3, -3, -3
o3o3o3o3x3o = 2, 2, 2, 2, -4, -4
o3o3o3o3o3x = 1, 1, 1, 1, 1, -5.
x3o3o3o3o3x = 6, 0, 0, 0, 0, 0, -6

Note that the stott vectors are 1, 2, 3, 4, 5, 6 modulo 6, and so if you add these together, the modulo of 6 is still preserved across all coordinates. One is then for any polytope constructed by integer multiples of these numbers, a central cell which is one of these polytopes: that is, the centre lies in one of these cells.
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