632/333[8]

G

632/333[8].o

632/333[8].1

632/333[8].2

632/333[8].3

632/333[8].4

.o — coloring by orbits gives each H-orbit its own color; it is unique. .1….4 — coloring by cosets depends on where the motif is placed among the 8 copies of G’s domain inside H’s; the 4 placements give 4 distinct colorings of the same uncolored pattern. Any picture enlarges on click.

the color group

group G

632 ⟨ a, b, c | a^2, b^3, (a∗b)^6, b∗c ⟩

subgroup H

333 index 8  ·  generators b (rotation by -0.6667*PI about (0.25,0)), ABAbaBa (rotation by 0.6667*PI about (-0.75,0)), AbAbbaba (rotation by -0.6667*PI about (-0.75,0)), AbAbAbaBaBa (rotation by -0.6667*PI about (-0.25,0.866))

colors

8 one per coset of H, equivalently one per orbit

placements

4 the conjugates of H, [G : NG(H)]

how G permutes the colors

Words act left to right: (a*b)(p) = b(a(p)).

the two families of motions

f₁ … f₇ — where the motif starts

The motions carrying the motif to the other cells of H’s domain (one cell per H-orbit): f1=a, f2=Ba, f3=ba, f4=aBa, f5=aba, f6=BaBa, f7=aBaBa.

g₁ … g₇ — what makes the colors

The coset transversal: gj carries H onto coset j, and the sets H gj are the color classes: g1=a, g2=ab, g3=aB, g4=aba, g5=aBa, g6=aBaBaBa, g7=ababa.

fundamental domains and pairing transforms

	domain	tiling	symmetries
G	G domain the fundamental domain of 632	G tiling the plane tiled by images of G’s domain	G gens rotation centers of G, the hurricane’s arm count = the order
H	H domain the fundamental domain of H: a union of 8 cells of G, internal walls shown	H tiling the plane tiled by H’s domain, over G’s finer tiling	H gens rotation centers of H = 333; inequivalent axis classes in shades of the base color

color partitioning

	f0.1	f1.2	f2.3	f3.4	f4≡ .4	f5≡ .3	f6≡ .2	f7≡ .1	∪all orbits
orbits of H	H	f1 H	f2 H	f3 H	f4 H	f5 H	f6 H	f7 H	all orbits of H
orbits of H g1	H g1	f1 H g1	f2 H g1	f3 H g1	f4 H g1	f5 H g1	f6 H g1	f7 H g1	all orbits of H g1
orbits of H g2	H g2	f1 H g2	f2 H g2	f3 H g2	f4 H g2	f5 H g2	f6 H g2	f7 H g2	all orbits of H g2
orbits of H g3	H g3	f1 H g3	f2 H g3	f3 H g3	f4 H g3	f5 H g3	f6 H g3	f7 H g3	all orbits of H g3
orbits of H g4	H g4	f1 H g4	f2 H g4	f3 H g4	f4 H g4	f5 H g4	f6 H g4	f7 H g4	all orbits of H g4
orbits of H g5	H g5	f1 H g5	f2 H g5	f3 H g5	f4 H g5	f5 H g5	f6 H g5	f7 H g5	all orbits of H g5
orbits of H g6	H g6	f1 H g6	f2 H g6	f3 H g6	f4 H g6	f5 H g6	f6 H g6	f7 H g6	all orbits of H g6
orbits of H g7	H g7	f1 H g7	f2 H g7	f3 H g7	f4 H g7	f5 H g7	f6 H g7	f7 H g7	all orbits of H g7
all colorscoloring by cosets	632/333[8].1	632/333[8].2	632/333[8].3	632/333[8].4	≡ .4	≡ .3	≡ .2	≡ .1

Each panel is fi H gj: columns are the 8 cells of H’s domain, rows the 8 orbit classes — one color per row, and a full row together is the whole pattern: the right margin shows it as the composition of its 8 orbits, each in its own color — the coloring by orbits .o carried by that row’s gj. Only the 4 columns marked by the blue frame carry inequivalent colorings — the placements; the others repeat a marked one: f₄ ≡ .4, f₅ ≡ .3, f₆ ≡ .2, f₇ ≡ .1. Every panel except H also shows, faded, the set it comes from: H gj panels show H, and fi… panels show their row’s H gj. The bottom margin is the coloring by cosets of every column — under a marked column the pictures from the top of the page, under a repeated column the same partition with its colors relabeled (titled ≡ its marked twin). Under and over every panel sit faded context layers — the tilings, fundamental domains and symmetry centers of G and H; the ☰ menu at the top right toggles each decoration.