Notation
Colour notation and four uncovered polar distinctions
The correspondence atlas uses Chaim Goodman–Strauss’s coloured-orbifold notation—the Conway–Burgiel–Goodman-Strauss convention in The Symmetries of Things—for cyclic colourings of the plane.
Convention
A full colour type is GN/K: G is the uncoloured plane group, N is the number of colours, and K fixes every colour. In a short signature, raised numbers are orders of induced colour permutations—not time shifts. Fix A at phase 0, B at phase 1/N, and so on, and write CN = (ABC…) for one forward tick. A generator with offset +k/N acts by CNk, with k taken modulo N. The identity and powers that split into several disjoint cycles remain powers of this one clock; none reverses time.
Four uncovered oriented cases
Traditional plane-colour equivalence allows a global relabelling of the palette, so it identifies two fixed-time phase characters χ and −χ. This is an external classification equivalence, not a time-reversing operation in either group. Thus four colour types each cover two polar groups with complementary forward skips:
| Chaim colour type | Two polar members | Standard fibrifold entry |
|---|---|---|
| 4424/◦ | g96 · P41/g97 · P43 | (414121) ‡ |
| 3333/◦ | g226 · P31/g225 · P32 | (313131) ‡ |
| 6326/◦ | g248 · P61/g247 · P65 | (613121) ‡ |
| 6323/2222 | g244 · P62/g245 · P64 | (623220) ‡ |
These are four uncovered distinctions, not four missing colourings: the 68 polar groups map onto 64 cyclic plane-colouring classes.
Fibrifold comparison
Conway–Delgado Friedrichs–Huson–Thurston fibrifold notation is the closest three-dimensional analogue of orbifold notation. A decoration nk couples an order-n rotation to a fibre translation by k/n. Its standard equivalence also permits fibre reversal, sending k to n−k; it therefore records each pair above as one entry marked ‡. This too is a naming equivalence, not an operation inside either forward group. The atlas keeps both polar members and uses the classical space-group name and catalog ID to select one.
Sources
The Symmetries of Things Conway et al., On Three-Dimensional Space Groups Correspondence atlas