Acronym | squat | ||||||||||||||||||||||||||||
Name |
square tiling, 2D hypercubical honeycomb (δ2), Voronoi complex of square lattice, Delone complex of square lattice | ||||||||||||||||||||||||||||
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Vertex figure | [44] = q4o | ||||||||||||||||||||||||||||
Vertex layers
(first ones only) |
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General of army | (is itself convex) | ||||||||||||||||||||||||||||
Colonel of regiment | (is itself locally convex – other uniform tiling member: sha ) | ||||||||||||||||||||||||||||
Dual | (selfdual) | ||||||||||||||||||||||||||||
Confer |
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External links |
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|xRo, 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
qo4xx4oq&#zx → height = 0 q = sqrt(2) = 1.414214 (pseudo) (tegum sum of 2 antialigned (q,x)-tosquats) o.4o.4o. | 4N * | 2 2 0 | 1 1 2 0 .o4.o4.o | * 4N | 0 2 2 | 0 1 2 1 -------------+-------+----------+---------- .. x. .. | 2 0 | 4N * * | 1 1 0 0 oo4oo4oo&#x | 1 1 | * 8N * | 0 1 1 0 .. .x .. | 0 2 | * * 4N | 0 0 1 1 -------------+-------+----------+---------- .. x.4o. | 4 0 | 4 0 0 | N * * * qo .. oq&#zx | 2 2 | 0 4 0 | * 2N * * .. xx ..&#x | 2 2 | 1 2 1 | * * 4N * .o4.x .. | 0 4 | 0 0 4 | * * * N
or o.4o.4o. & | 4N | 2 2 | 1 1 2 ---------------+----+-------+------- .. x. .. & | 2 | 4N * | 1 0 1 oo4oo4oo&#x | 2 | * 4N | 0 1 1 ---------------+----+-------+------- .. x.4o. & | 4 | 4 0 | N * * qo .. oq&#zx | 4 | 0 4 | * N * .. xx ..&#x | 4 | 2 2 | * * 2N
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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