Worked example

Color group notation on a rosette

Chapter 12 of The Symmetries of Things, illustrated on a \(D_6\)-rosette.

1. The uncoloured rosette

Let \(P\) be an asymmetric motif in generic position. Let \(r_6\) denote counterclockwise rotation through \(60^\circ\), and let \(R_x\) denote reflection in the \(x\)-axis. The orbit of \(P\) has twelve elements and realizes

\[ G=\left\langle r_6,R_x\,\middle|\, r_6^6=e,\ R_x^2=e,\ R_xr_6R_x=r_6^{-1}\right\rangle \cong D_6,\qquad |G|=12. \]

Set \(r_3=r_6^2\), \(r_2=r_6^3\), and \(R_y=R_xr_2\). Thus \(R_y\) is reflection in the \(y\)-axis and \(R_xR_y=r_2\). Moreover, \(G=\{r_6^k,R_xr_6^k:0\leq k<6\}\). Operations act on the right of \(P\); consequently \(P\mathbin{\cdot}(R_xr_6)\) means “reflect in the \(x\)-axis, then rotate through \(60^\circ\).”

An earlier description says “reflection in the \(y\)-axis” twice; the subscripts and diagram show that the intended pair is \(R_x\) and \(R_y\).

As a finite rosette, the uncoloured pattern has orbifold signature \(\ast6\mathord{\bullet}\). The terminal bullet records the common fixed centre.

The \(G\)-orbit of an asymmetric motif \(P\).

2. A three-colouring by right cosets

Choose the blue subgroup \(H=\langle R_x,R_y\rangle=\{e,R_x,R_y,r_2\}\). Its three right cosets partition \(G\). Assigning one colour to each coset gives the displayed colouring.

Colour	Right coset	Elements
Blue	\(H\)	\(e,\ R_x,\ R_y,\ r_2\)
Red	\(Hr_6\)	\(er_6,\ R_xr_6,\ R_yr_6,\ r_2r_6\)
Green	\(Hr_3\)	\(er_3,\ R_xr_3,\ R_yr_3,\ r_2r_3\)

Since \([G:H]=3\), there are three colours. Only the reference coset \(H\) is itself a subgroup. In particular, \(Hr_6\) is the red colour class, not the stabilizer of red.

The right-coset partition \(H\backslash G\).

3. The single-colour stabilizer \(H\) and the colour-fixing subgroup \(K\)

The subgroup \(H\) stabilizes the blue class setwise. Its two reflections preserve blue while interchanging the other two colour classes. Hence \(H\) is larger than the subgroup that preserves all three colours separately.

One chosen colour

\(H=\operatorname{Stab}(\mathrm{blue}) =\{e,R_x,R_y,r_2\},\qquad H\cong\) .

Every colour

\(K=\{e,r_2\}=\langle r_2\rangle,\qquad K\cong\) .

\(H\): the blue set is preserved by two perpendicular reflections.

\(K\): the half-turn exchanges antipodal motifs of the same colour.

The red and green stabilizers are the conjugates \(H_{\mathrm{red}}=r_6^{-1}Hr_6\) and \(H_{\mathrm{green}}=r_3^{-1}Hr_3\). Their common intersection is \(K=H\cap r_6^{-1}Hr_6\cap r_3^{-1}Hr_3\). In orbifold notation, \(H\) has signature \(\ast2\mathord{\bullet}\) and \(K\) has signature \(2\mathord{\bullet}\).

4. Chaim’s \(G^3/H/K\) notation

Group	Order	Role	Standard description / rosette signature
\(G\)	12	All uncoloured symmetries	dihedral(6); \(\ast6\mathord{\bullet}\)
\(H\)	4	Stabilizer of one colour	dihedral(2); \(\ast2\mathord{\bullet}\)
\(K\)	2	Fixes every colour class	cyclic(2); \(2\mathord{\bullet}\)

The full description is

\[(\ast6\mathord{\bullet})^3/(\ast2\mathord{\bullet})/(2\mathord{\bullet})\]

Since \(H\neq K\), Chaim suppresses \(H\) and uses a double slash:

\[(\ast6\mathord{\bullet})^3//(2\mathord{\bullet})\]

The superscript \(3\) is the number of colours, equivalently \([G:H]\). It is not \([G:K]\), which equals \(6\) in this example. The induced behaviour of the generators on the colour names is recorded in the representations below.

5. Colourings with at most three colours

Because the motif orbit is regular, a transitive \(p\)-colouring is determined by a subgroup \(H\) of index \(p\). Its colour classes are the right cosets of \(H\), and \(K=\displaystyle\bigcap_{g\in G}g^{-1}Hg\) is the subgroup fixing every colour. Up to three colours, the complete list is as follows.

Colours	Representative \(H\)	Kernel \(K\)	Full Chaim type \(G^p/H/K\)	Coset partitions
1	\(G\)	\(G\)	\((\ast6\mathord{\bullet})^1/(\ast6\mathord{\bullet})/(\ast6\mathord{\bullet})\)	1
2	\(\langle r_6\rangle\cong C_6\)	\(\langle r_6\rangle\)	\((\ast6\mathord{\bullet})^2/(6\mathord{\bullet})/(6\mathord{\bullet})\)	1
2	\(\langle r_3,R_x\rangle\cong D_3\)	\(\langle r_3,R_x\rangle\)	\((\ast6\mathord{\bullet})^2/(\ast3\mathord{\bullet})/(\ast3\mathord{\bullet})\)	2
3	\(\langle r_2,R_x\rangle\cong D_2\)	\(\langle r_2\rangle\cong C_2\)	\((\ast6\mathord{\bullet})^3/(\ast2\mathord{\bullet})/(2\mathord{\bullet})\)	3

“Coset partitions” counts distinct partitions of the twelve group-labelled motif positions. Ambient equivalence and simultaneous relabelling of the colours identify the partitions in each row as one colour type.

One colour

The monochromatic type

One class: all twelve operations lie in \(G\).

\((\ast6\mathord{\bullet})^1/(\ast6\mathord{\bullet})/(\ast6\mathord{\bullet})\)

Representation

\(r_6\)60° rotation	\(\mathrm{id}\)
\(R_x\)Reflection in the \(x\)-axis	\(\mathrm{id}\)

Relations\(G=\langle r_6,R_x\mid r_6^6=R_x^2=e,\ R_xr_6R_x=r_6^{-1}\rangle\)

\(H\): \(G\)
\(K\): \(G\)
Classes: \(G\)

Two colours

Two inequivalent kernels

Rotations form one colour class; reflections form the other.

Rotation kernel

\((\ast6\mathord{\bullet})^2/(6\mathord{\bullet})/(6\mathord{\bullet})\)

Representation

\(r_6\)60° rotation	\(\mathrm{id}\)
\(R_x\)Reflection in the \(x\)-axis	\((A\ B)\)

Relations\(G=\langle r_6,R_x\mid r_6^6=R_x^2=e,\ R_xr_6R_x=r_6^{-1}\rangle\)

\(H\): \(\langle r_6\rangle\)
\(K\): \(\langle r_6\rangle\)
Classes: \(\langle r_6\rangle,\ \langle r_6\rangle R_x\)

The parity of the exponent of \(r_6\) determines the colour.

Dihedral kernel

\((\ast6\mathord{\bullet})^2/(\ast3\mathord{\bullet})/(\ast3\mathord{\bullet})\)

Representation

\(r_6\)60° rotation	\((A\ B)\)
\(R_x\)Reflection in the \(x\)-axis	\(\mathrm{id}\)

Relations\(G=\langle r_6,R_x\mid r_6^6=R_x^2=e,\ R_xr_6R_x=r_6^{-1}\rangle\)

\(H\): \(\langle r_3,R_x\rangle\)
\(K\): \(\langle r_3,R_x\rangle\)
Classes: \(H,\ Hr_6\)

The second concrete partition is obtained from \(\langle r_3,R_xr_6\rangle\) and belongs to the same ambient-equivalence class.

Three colours

The unique three-colour type

The classes are \(H\), \(Hr_6\), and \(Hr_3\).

\((\ast6\mathord{\bullet})^3/(\ast2\mathord{\bullet})/(2\mathord{\bullet})\)

Representation

ABC

\(r_6\)60° rotation	\((A\ B\ C)\)
\(R_x\)Reflection in the \(x\)-axis	\((B\ C)\)

Relations\(G=\langle r_6,R_x\mid r_6^6=R_x^2=e,\ R_xr_6R_x=r_6^{-1}\rangle\)

\(H\): \(\langle r_2,R_x\rangle=\langle R_x,R_y\rangle\)
\(K\): \(\langle r_2\rangle\)
Classes: \(H,\ Hr_6,\ Hr_3\)

The three concrete partitions arise from the three conjugate choices of the point stabilizer \(H\).