Acronym | hexat | ||||||||||||||||||||||||
Name |
hexagonal tiling, Voronoi complex of hexagonal lattice A2 | ||||||||||||||||||||||||
© o3o6x x3x6o x3x3x3*a uBxx3uxBx3uxxB3*a&#zx . o6x : red x3x . : yellow, x3x . : blue, .... .... uxx.&#xt : green, .... .x..3.x.. : yellow . x6o : red x . x3*a : red, .... ux.x ....&#xt : blue, ..x. .... ..x.3*a : teal . x3x : yellow u.xx .... ....&#xt : red, ...x3...x .... : rose | |||||||||||||||||||||||||
Vertex figure | [63] | ||||||||||||||||||||||||
Snub derivation |
+---+---+---+ . + . + | | | | | | +---+---+---+ . + . + | | | | => / \ / +---+---+---+ + . + . | | | | | | +---+---+---+ + . + . | ||||||||||||||||||||||||
Vertex layers
(first ones only) |
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Dual | trat | ||||||||||||||||||||||||
Confer |
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External links |
Both, trat and the tri-hexagonal rhombic tiling (derived from the former by combining 2 triangles each) could be considered as a vertex overlay of hexat plus 3 shifted copies of trats with edge size 3 (see there). Then this decomposition allows for a 3-step partial Stott expansion of the latter running through rhombic uBxx3xoAo3xooA3*a&#zx and rhombic uBxx3uxBx3xooA3*a&#zx towards a pure (6-colored) hexat (A = 3x, B = 4x).
Incidence matrix according to Dynkin symbol
o3o6x (N → ∞) . . . | 2N | 3 | 3 ------+----+----+-- . . x | 2 | 3N | 2 ------+----+----+-- . o6x | 6 | 6 | N snubbed forms: o3o6s
x3x6o (N → ∞) . . . | 6N | 1 2 | 2 1 ------+----+-------+----- x . . | 2 | 3N * | 2 0 . x . | 2 | * 6N | 1 1 ------+----+-------+----- x3x . | 6 | 3 3 | 2N * . x6o | 6 | 0 6 | * N snubbed forms: β3x6o, s3s6o
x3x3x3*a (N → ∞) . . . | 6N | 1 1 1 | 1 1 1 ---------+----+----------+------ x . . | 2 | 3N * * | 1 1 0 . x . | 2 | * 3N * | 1 0 1 . . x | 2 | * * 3N | 0 1 1 ---------+----+----------+------ x3x . | 6 | 3 3 0 | N * * x . x3*a | 6 | 3 0 3 | * N * . x3x | 6 | 0 3 3 | * * N snubbed forms: s3s3s3*a
o3o6/5x (N → ∞) . . . | 2N | 3 | 3 --------+----+----+-- . . x | 2 | 3N | 2 --------+----+----+-- . o6/5x | 6 | 6 | N
o3/2o6x (N → ∞) . . . | 2N | 3 | 3 --------+----+----+-- . . x | 2 | 3N | 2 --------+----+----+-- . o6x | 6 | 6 | N
o3/2o6/5x (N → ∞) . . . | 2N | 3 | 3 ----------+----+----+-- . . x | 2 | 3N | 2 ----------+----+----+-- . o6/5x | 6 | 6 | N
x3x6/5o (N → ∞) . . . | 6N | 1 2 | 2 1 --------+----+-------+----- x . . | 2 | 3N * | 2 0 . x . | 2 | * 6N | 1 1 --------+----+-------+----- x3x . | 6 | 3 3 | 2N * . x6/5o | 6 | 0 6 | * N
s6x3x (N → ∞) demi( . . . ) | 6N | 1 1 1 | 1 1 1 --------------+----+----------+------ demi( . x . ) | 2 | 3N * * | 1 1 0 demi( . . x ) | 2 | * 3N * | 0 1 1 sefa( s6x . ) | 2 | * * 3N | 1 0 1 --------------+----+----------+------ s6x . ♦ 6 | 3 0 3 | N * * demi( . x3x ) | 6 | 3 3 0 | * N * sefa( s6x3x ) | 6 | 0 3 3 | * * N starting figure: x6x3x
x∞s2s∞o (N → ∞) demi( . . . . ) | 2N | 1 2 | 3 ----------------+----+------+-- demi( x . . . ) | 2 | N * | 2 . s2s . | 2 | * 2N | 2 ----------------+----+------+-- sefa( x∞s2s∞o ) | 6 | 2 4 | N starting figure: x∞x2x∞o
ho3oo3oh3*a&#zx (N → ∞) o.3o.3o.3*a | N * | 3 | 3 .o3.o3.o3*a | * N | 3 | 3 ----------------+-----+----+-- oo3oo3oo3*a&#x | 1 1 | 3N | 2 ----------------+-----+----+-- ho .. oh3*a&#zx | 3 3 | 6 | N
or o.3o.3o.3*a & | 2N | 3 | 3 ------------------+----+----+-- oo3oo3oo3*a&#x | 2 | 3N | 2 ------------------+----+----+-- ho .. oh3*a&#zx | 6 | 6 | N
uBxx3uxBx3uxxB3*a&#zx (N → ∞, B = 4x) o...3o...3o...3*a | 6N * * * | 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0 0 .o..3.o..3.o..3*a | * 6N * * | 1 0 0 1 1 0 0 0 0 | 1 1 0 1 0 0 ..o.3..o.3..o.3*a | * * 6N * | 0 1 0 0 0 1 1 0 0 | 1 0 1 0 1 0 ...o3...o3...o3*a | * * * 6N | 0 0 1 0 0 0 0 1 1 | 0 1 1 0 0 1 ----------------------+-------------+----------------------------+--------------- oo..3oo..3oo..3*a&#x | 1 1 0 0 | 6N * * * * * * * * | 1 1 0 0 0 0 o.o.3o.o.3o.o.3*a&#x | 1 0 1 0 | * 6N * * * * * * * | 1 0 1 0 0 0 o..o3o..o3o..o3*a&#x | 1 0 0 1 | * * 6N * * * * * * | 0 1 1 0 0 0 .... .x.. .... | 0 2 0 0 | * * * 3N * * * * * | 0 1 0 1 0 0 .... .... .x.. | 0 2 0 0 | * * * * 3N * * * * | 1 0 0 1 0 0 ..x. .... .... | 0 0 2 0 | * * * * * 3N * * * | 0 0 1 0 1 0 .... .... ..x. | 0 0 2 0 | * * * * * * 3N * * | 1 0 0 0 1 0 ...x .... .... | 0 0 0 2 | * * * * * * * 3N * | 0 0 1 0 0 1 .... ...x .... | 0 0 0 2 | * * * * * * * * 3N | 0 1 0 0 0 1 ----------------------+-------------+----------------------------+--------------- .... .... uxx. &#xt | 2 2 2 0 | 2 2 0 0 1 0 1 0 0 | 3N * * * * * .... ux.x .... &#xt | 2 2 0 2 | 2 0 2 1 0 0 0 0 1 | * 3N * * * * u.xx .... .... &#xt | 2 0 2 2 | 0 2 2 0 0 1 0 1 0 | * * 3N * * * .... .x..3.x.. | 0 6 0 0 | 0 0 0 3 3 0 0 0 0 | * * * N * * ..x. .... ..x.3*a | 0 0 6 0 | 0 0 0 0 0 3 3 0 0 | * * * * N * ...x3...x .... | 0 0 0 6 | 0 0 0 0 0 0 0 3 3 | * * * * * N
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