Mini-tutorial · finite ambient group

The 3×3 torus: three ways to name a group

The smallest setting where colour symmetry is completely computable. The ambient group has nine elements, it has exactly six subgroups, and every subgroup can be named three independent ways — as the stabilizer of a pattern, by a presentation, and in the colour notation of The Symmetries of Things. All three agree.

The ambient group

Take a 3×3 grid that wraps in both directions and let it act on itself by translation. Writing X for “shift one row” and Y for “shift one column”:

A = ⟨ X, Y | X³ = Y³ = 1, XY = YX ⟩ = (ℤ/3)², of order 9.

This is the finite version of the handle symbol ◦ from Chapter 10 of The Symmetries of Things, whose reduced annotation is ◦_X,Y with the single relation XY = YX. The torus adds X³ = Y³ = 1. Because A is abelian, every subgroup is normal, so every quotient is a group and every colour group is regular: the number of colours is exactly the index.

Three ways to name the same group

1 · As a stabilizer Choose a coloured pattern and keep the translations that leave it unchanged. Equivalently, an equivalence relation on cells that is invariant under translation. The two are the same data: the subgroup is the class of the identity, and the classes are its cosets.
2 · By generators Give generators and relations. Imposing the extra relations that kill K turns the ambient presentation into a presentation of the colour group A/K — relations describe the quotient, never the subgroup.
3 · Chaim notation Replace each generator in the signature by the colour permutation it induces. The short form keeps only the order of that permutation; the type G/K records the full group and the kernel.

The six subgroups

Cells are filled by their coset of K, which is exactly the colour class. Ringed cells are K itself. The number of distinct letters is the index, which is the number of colours.

K = A

the whole torus

|K| = 9 index 1 1 colour

tiled 2×2

1 · As a stabilizer: The translations fixing the constant pattern: one colour everywhere. As an equivalence relation on cells: same class when none — every cell is equivalent to every other agrees.
2 · By generators: K = ⟨X, Y⟩. Imposing X = 1, Y = 1 on the ambient presentation gives the colour group A/K = trivial group.
3 · Chaim notation: Short signature ◦^1,1, long form ◦^1,1, colour type ◦.

K = ⟨Y⟩

rows

|K| = 3 index 3 3 colours

tiled 2×2

1 · As a stabilizer: The translations fixing three horizontal stripes. As an equivalence relation on cells: same class when a mod 3 — the row index agrees.
2 · By generators: K = ⟨Y⟩. Imposing Y = 1 on the ambient presentation gives the colour group A/K = C₃, generated by the image of X.
3 · Chaim notation: Short signature ◦^3,1, long form ◦^(ABC), 1, colour type ◦³/◦.

K = ⟨X⟩

columns

|K| = 3 index 3 3 colours

tiled 2×2

1 · As a stabilizer: The translations fixing three vertical stripes. As an equivalence relation on cells: same class when b mod 3 — the column index agrees.
2 · By generators: K = ⟨X⟩. Imposing X = 1 on the ambient presentation gives the colour group A/K = C₃, generated by the image of Y.
3 · Chaim notation: Short signature ◦^1,3, long form ◦^1, (ABC), colour type ◦³/◦.

K = ⟨XY⟩

the diagonal

|K| = 3 index 3 3 colours

tiled 2×2

1 · As a stabilizer: The translations fixing three diagonal stripes, leaning one way. As an equivalence relation on cells: same class when (b − a) mod 3 agrees.
2 · By generators: K = ⟨XY⟩. Imposing XY = 1, so Y = X⁻¹ on the ambient presentation gives the colour group A/K = C₃, generated by the image of X.
3 · Chaim notation: Short signature ◦^3,3, long form ◦^{(ACB), (ABC)}, colour type ◦³/◦.

K = ⟨XY²⟩

the anti-diagonal

|K| = 3 index 3 3 colours

tiled 2×2

1 · As a stabilizer: The translations fixing three diagonal stripes, leaning the other way. As an equivalence relation on cells: same class when (a + b) mod 3 agrees.
2 · By generators: K = ⟨XY²⟩. Imposing XY² = 1, so X = Y on the ambient presentation gives the colour group A/K = C₃, generated by the common image of X and Y.
3 · Chaim notation: Short signature ◦^3,3, long form ◦^{(ABC), (ABC)}, colour type ◦³/◦.

K = {0}

a single cell

|K| = 1 index 9 9 colours

tiled 2×2

1 · As a stabilizer: The translations fixing nine distinct colours, no repetition. As an equivalence relation on cells: same class when the cell itself — nothing is identified agrees.
2 · By generators: K = ⟨— (empty)⟩. Imposing nothing on the ambient presentation gives the colour group A/K = C₃ × C₃, the whole torus.
3 · Chaim notation: Short signature ◦^3,3, long form ◦^{(ABC…), (ADG…)}, colour type ◦⁹/◦.

Summary

subgroup	shape	index	colours	short signature	colour type
`K = A`	the whole torus	1	1	◦^1,1	◦
`K = ⟨Y⟩`	rows	3	3	◦^3,1	◦³/◦
`K = ⟨X⟩`	columns	3	3	◦^1,3	◦³/◦
`K = ⟨XY⟩`	the diagonal	3	3	◦^3,3	◦³/◦
`K = ⟨XY²⟩`	the anti-diagonal	3	3	◦^3,3	◦³/◦
`K = {0}`	a single cell	9	9	◦^3,3	◦⁹/◦

Three subgroups share the short signature ◦^3,3. The short form records only permutation orders, so it cannot separate them — the long form and the colour group can. This is the same lossiness that forces Conway to print ∗^(AB)3^(BC)3^(CA)3 in full for ∗333³//◦.

Six subgroups, three types

Aut(A) = GL(2,𝔽₃) has order 48 and acts on the six subgroups with three orbits: the trivial subgroup, the four lines, and the whole torus. The four lines are one orbit — one line for each slope 0, 1, 2, ∞ — which is Conway’s count of p + 1 subgroups of index p. You can see the group acting: transposing swaps rows and columns, and shearing carries columns to each diagonal in turn.

There is no two-colour entry anywhere on this page, and the reason is arithmetic: a colour group must have order dividing 9, and 2 ∤ 9. Equivalently, no pattern on this torus has a colour-swapping symmetry, since the two classes would have to be the same size.

Where this sits

Lifting to the plane, every one of these six kernels is a lattice, so every colour type here is ◦ with some number of colours — the family at the top of the periodic colour-pattern catalog. The torus classification is finer than the wallpaper one: fixing a finite ambient removes the ability to rescale, so index matters where affine equivalence would forget it.

Generated by scripts/generate_torus_tutorial.py. Notation follows Conway, Burgiel and Goodman-Strauss, The Symmetries of Things, Chapters 10–13.