A Worked Example: Colourings from Subgroups
A colouring is not something you invent arbitrarily: it is something you choose a subgroup for. Copies of an asymmetric motif are labelled by group elements, the colours are the left cosets of a chosen subgroup H, and all properties — the palette size, the stabilisers, and whether the colouring can be run as a clock — follow directly from group theory.
1. Copies are labelled by group elements
Draw one asymmetric motif inside the fundamental domain. The dihedral group Dn produces 2n copies — exactly one per group element. Half read as right-handed and half as left-handed (the reflections).
2. Cosets partition the group
Fix a subgroup H ≤ G. The left coset of an element g is:
gH = { g · h : h ∈ H }
Every element lies in exactly one coset, and there are |G| / |H| cosets in total. Painting copy g with the colour of its coset gH guarantees that the group action consistently permutes the colour palette.
• Stabiliser of colour gH: The conjugate subgroup g H g⁻¹.
• Kernel core(H): The intersection ⋂g g H g⁻¹ of all stabilisers.
• Colour group: The quotient G / core(H). When H is normal,
stabiliser = kernel and the action is regular.
3. Connection to Spacetime Clockwork Crystals
When the colour group G / core(H) is cyclic, the colours can be arranged as phases j/N on a clock. Advancing the phase continuously in time produces a periodic animation — and lifts the 2D pattern into a 3D polar space group.