Acronym squat
Name square tiling,
2D hypercubical honeycomb2),
Voronoi complex of square lattice,
Delone complex of square lattice
 
 
x4o4o o4x4o x4o4x
©
Vertex figure [44] = q4o
Vertex layers
(first ones only)
LayerSymmetrySubsymmetries
 o4o4oo4o .. o4o
1x4o4ox4o .
{4} first
. o4o
vertex first
2x4q .. q4o
vertex figure
3ax4Q .. o4u
3bd4o .
4ax4D .. q4u
4bd4x .
.........
(d=u+x=3x, Q=2q, D=3q)
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform tiling member: sha )
Dual (selfdual)
Confer
more general:
xPoPo   x4oPo   x4o2Po  
general duoprisms:
n,m-dip   n,n-dip   n-azedip  
related tesselations:
sha  
general polytopal classes:
hypercubical honeycomb   partial Stott expansions   regular   noble polytopes  
External
links
wikipedia

Squat also can be seen as an extension of the general (n-gonal, apeirogonal) duoprism (n → ∞), thereby becoming a flat tiling.

The body-centered square tiling would be a (smaller) square tiling in turn.

There exist regular modwraps of this tiling, x4o4o|r, obtained by identifying every r-th vertex on each hole. These then allow representations as finite regular skew polyhedra, which happen to be the torical facial subsets, built by the lacing squares of the r-prismatic cells, of the (r,r)-duoprism.


Incidence matrix according to Dynkin symbol

x4o4o   (N → ∞)

. . . | N |  4 | 4
------+---+----+--
x . . | 2 | 2N | 2
------+---+----+--
x4o . | 4 |  4 | N

snubbed forms: s4o4o

o4x4o   (N → ∞)

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

snubbed forms: o4s4o

x4o4x   (N → ∞)

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

snubbed forms: s4o4x, s4o4s

x4o4/3o   (N → ∞)

. .   . | N |  4 | 4
--------+---+----+--
x .   . | 2 | 2N | 2
--------+---+----+--
x4o   . | 4 |  4 | N

x4/3o4o   (N → ∞)

.   . . | N |  4 | 4
--------+---+----+--
x   . . | 2 | 2N | 2
--------+---+----+--
x4/3o . | 4 |  4 | N

x4/3o4/3o   (N → ∞)

.   .   . | N |  4 | 4
----------+---+----+--
x   .   . | 2 | 2N | 2
----------+---+----+--
x4/3o   . | 4 |  4 | N

o4x4/3o   (N → ∞)

. .   . | 2N |  4 | 2 2
--------+----+----+----
. x   . |  2 | 4N | 1 1
--------+----+----+----
o4x   . |  4 |  4 | N *
. x4/3o |  4 |  4 | * N

o4/3x4/3o   (N → ∞)

.   .   . | 2N |  4 | 2 2
----------+----+----+----
.   x   . |  2 | 4N | 1 1
----------+----+----+----
o4/3x   . |  4 |  4 | N *
.   x4/3o |  4 |  4 | * N

x4/3o4/3x   (N → ∞)

.   .   . | 4N |  2  2 | 1  2 1
----------+----+-------+-------
x   .   . |  2 | 4N  * | 1  1 0
.   .   x |  2 |  * 4N | 0  1 1
----------+----+-------+-------
x4/3o   . |  4 |  4  0 | N  * *
x   .   x |  4 |  2  2 | * 2N *
.   o4/3x |  4 |  0  4 | *  * N

x∞o x∞o   (N → ∞)

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

x∞x x∞o   (N → ∞)

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

snubbed forms: x∞s2s∞o

x∞x x∞x   (N → ∞)

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

s4o4o   (N → ∞)

demi( . . . ) | N |  4 | 4
--------------+---+----+--
      s4o .    2 | 2N | 2
--------------+---+----+--
sefa( s4o4o ) | 4 |  4 | N

snubbed forms: x4o4o

s4x4o   (N → ∞)

demi( . . . ) | 4N |  2  2 |  2 1 1
--------------+----+-------+-------
demi( . x . ) |  2 | 4N  * |  1 1 0
sefa( s4x . ) |  2 |  * 4N |  1 0 1
--------------+----+-------+-------
      s4x .     4 |  2  2 | 2N * *
demi( . x4o ) |  4 |  4  0 |  * N *
sefa( s4x4o ) |  4 |  0  4 |  * * N

starting figure: x4x4o

o4s4o   (N → ∞)

demi( . . . ) | N | 2 2 | 4
--------------+---+-----+--
      o4s .    2 | N * | 2
      . s4o    2 | * N | 2
--------------+---+-----+--
sefa( o4s4o ) | 4 | 2 2 | N

snubbed forms: o4x4o

s4o4s   (N → ∞)

demi( . . . ) | 2N | 1  2 1 |  4
--------------+----+--------+---
      s4o .     2 | N  * * |  2
      s 2 s     2 | * 2N * |  2
      . o4s     2 | *  * N |  2
--------------+----+--------+---
sefa( s4o4s ) |  4 | 1  2 1 | 2N

snubbed forms: x4o4x

s4x4s   (N → ∞)

demi( . . . ) | 4N |  1  1  1  1 | 1 1  2
--------------+----+-------------+-------
      s 2 s     2 | 2N  *  *  * | 0 0  2
demi( . x . ) |  2 |  * 2N  *  * | 1 1  0
sefa( s4x . ) |  2 |  *  * 2N  * | 1 0  1
sefa( . x4s ) |  2 |  *  *  * 2N | 0 1  1
--------------+----+-------------+-------
      s4x .     4 |  0  2  2  0 | N *  *
      . x4s     4 |  0  2  0  2 | * N  *
sefa( s4x4s ) |  4 |  2  0  1  1 | * * 2N

starting figure: x4x4x

x4s4o   (N → ∞)

demi( . . . ) | 2N | 1 1  2 | 2 2
--------------+----+--------+----
      . s4o     2 | N *  * | 0 2
demi( x . . ) |  2 | * N  * | 2 0
sefa( x4s . ) |  2 | * * 2N | 1 1
--------------+----+--------+----
      x4s .     4 | 0 2  2 | N *
sefa( x4s4o ) |  4 | 2 0  2 | * N

starting figure: x4x4o

x4s4x   (N → ∞)

demi( . . . ) | 4N |  1  1  1  1 | 1 1 1 1
--------------+----+-------------+--------
demi( x . . ) |  2 | 2N  *  *  * | 1 0 1 0
demi( . . x ) |  2 |  * 2N  *  * | 0 1 1 0
sefa( x4s . ) |  2 |  *  * 2N  * | 1 0 0 1
sefa( . s4x ) |  2 |  *  *  * 2N | 0 1 0 1
--------------+----+-------------+--------
      x4s .     4 |  2  0  2  0 | N * * *
      . s4x     4 |  0  2  0  2 | * N * *
demi( x . x ) |  4 |  2  2  0  0 | * * N *
sefa( x4s4x ) |  4 |  0  0  2  2 | * * * N

starting figure: x4x4x

s4s4x   (N → ∞)

demi( . . . ) | 4N |  1  1  2 | 1 1  2
--------------+----+----------+-------
demi( . . x ) |  2 | 2N  *  * | 0 1  1
sefa( . s4x ) |  2 |  * 2N  * | 0 1  1
sefa( s4s . ) |  2 |  *  * 4N | 1 0  1
--------------+----+----------+-------
      s4s .     4 |  0  0  4 | N *  *
      . s4x     4 |  2  2  0 | * N  *
sefa( s4s4x ) |  4 |  1  1  2 | * * 2N

starting figure: x4x4x

s4x4s'   (N → ∞)
(x4x4x -s--> s4x4x  : alternating . x4x)
(s4x4x -s'-> s4x4s' : alternating . x .)

demi( demi( . . . ) ) | 4N |  1  2  1 | 1 1  2
----------------------+----+----------+-------
demi( demi( . x . ) ) |  2 | 2N  *  * | 1 0  1
sefa( sefa( s4x4x ) ) |  2 |  * 4N  * | 0 1  1
sefa( demi( . x4x ) ) |  2 |  *  * 2N | 1 0  1
----------------------+----+----------+-------
snub( demi( . x4x ) ) |  4 |  2  0  2 | N *  *
snub( sefa( s4x4x ) ) |  4 |  0  4  0 | * N  *
sefa(       s4x4x   ) |  4 |  1  2  1 | * * 2N

starting figure: s4x4x

ss'4o4x   (N → ∞)
(x4o4x -s-->  s4o4x  : alternating . o4x)
(s4o4x -s'-> ss'4o4x : alternating . o4x)

demi( demi( . . . ) ) | 4N |  2  2 | 1  2 1
----------------------+----+-------+-------
demi( demi( . . x ) ) |  2 | 4N  * | 1  1 0
sefa( sefa( s4o4x ) ) |  2 |  * 4N | 0  1 1
----------------------+----+-------+-------
demi( demi( . o4x ) ) |  4 |  4  0 | N  * *
snub( sefa( s4o4x ) ) |  4 |  2  2 | * 2N *
sefa(      ss'4o4x  ) |  4 |  0  4 | *  * N

starting figure: s4o4x

:x:∞:o:&##x   (N → ∞)   → heights = 1

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

:qo:∞:oq:&##x   (N → ∞)   → heights = 1/sqrt(2) = 0.707107

 o. ∞ o.      | N * |  2  2 | 2 2
 .o ∞ .o      | * N |  2  2 | 2 2
--------------+-----+-------+----
 oo ∞ oo &#x  | 1 1 | 2N  * | 1 1
:oo:∞:oo:&#x  | 1 1 |  * 2N | 1 1
--------------+-----+-------+----
:qo: :..:&#xt | 2 2 |  2  2 | N *
:..: :oq:&#xt | 2 2 |  2  2 | * N

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