632/632[3]

G

632/632[3].o

632/632[3].1

632/632[3].2

632/632[3].3

.o — coloring by orbits gives each H-orbit its own color; it is unique. .1….3 — coloring by cosets depends on where the motif is placed among the 3 copies of G’s domain inside H’s; the 3 placements give 3 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

632 index 3  ·  generators a (rotation by 1*PI about (0,0)), BaB (rotation by 0.3333*PI about (0.75,-0))

colors

3 one per coset of H, equivalently one per orbit

placements

3 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 … f₂ — where the motif starts

The motions carrying the motif to the other cells of H’s domain (one cell per H-orbit): f1=B, f2=b.

g₁ … g₂ — what makes the colors

The coset transversal: gj carries H onto coset j, and the sets H gj are the color classes: g1=b, g2=B.

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 3 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 = 632; inequivalent axis classes in shades of the base color

color partitioning

	f0.1	f1.2	f2.3	∪all orbits
orbits of H	H	f1 H	f2 H	all orbits of H
orbits of H g1	H g1	f1 H g1	f2 H g1	all orbits of H g1
orbits of H g2	H g2	f1 H g2	f2 H g2	all orbits of H g2
all colorscoloring by cosets	632/632[3].1	632/632[3].2	632/632[3].3

Each panel is fi H gj: columns are the 3 cells of H’s domain, rows the 3 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 3 orbits, each in its own color — the coloring by orbits .o carried by that row’s gj. All 3 columns carry inequivalent colorings — the 3 placements. 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.