Master polygons

Lattices

n-Regular	3	4
Domain	Eisenstein ${\displaystyle a+bu}$	Gaussian ${\displaystyle a+bi}$
Adjoined unit	${\displaystyle {\begin{aligned}u&={\frac {1}{2}}\left(1+i{\sqrt {3}}\right)\\&=e^{{\frac {1}{3}}\pi i}=\omega +1\end{aligned}}}$	${\displaystyle i={\sqrt {-1}}}$
Norm ${\displaystyle t(a,b)}$	${\displaystyle a^{2}+ab+b^{2}}$	${\displaystyle a^{2}+b^{2}}$.
Master polygon	${\displaystyle (0,x,xu)}$	${\displaystyle (0,x,x(1+i),xi)}$

Regular lattices over the complex plane can be used to create "master polygons". In geodesic dome terminology, this is the "breakdown structure" or "principal polyhedral triangle" (PPT). The 4-regular case uses the square lattice over the Gaussian integers, and the 3-regular case uses triangular lattice over the Eisenstein integers. For convenience, an alternate parameterization of the Eisenstein integers is used, based on the sixth root of unity instead of the third.^{[lower-alpha 1]} The usual definition of Eisenstein integers uses the element ${\textstyle \omega ={\frac {1}{2}}\left(-1+i{\sqrt {3}}\right)=e^{{\frac {2}{3}}\pi i}=u-1}$. A norm, ${\displaystyle t(k,\ell )}$, is defined as the square of the absolute value of the complex number. For 3-regular graphs this norm is the T-number or triangulation number used in virology.

The master polygon is an equilateral triangle or square laid over the lattice. The table to the right gives formulas for the vertices of the master polygons in the complex plane, and the gallery below shows the (3,2) master triangle and square. So that the polygon can be described by a single complex number, one vertex is fixed at 0. There are multiple numbers that can describe the same polygon: these are associates of each other: if ${\displaystyle x}$ and ${\displaystyle y}$ are associates, then ${\displaystyle x=u^{n}y}$ in the Eisensteins or ${\displaystyle x=i^{n}y}$ in the Gaussians for some integer ${\displaystyle n}$. The set of elements that are associates of each other is an equivalence class, and the element of each equivalence class that has ${\displaystyle a>0}$ and ${\displaystyle b\geq 0}$ is the normal form.

• 
  (3,2) master triangle over triangular grid
• 
  (3,2) master square over square grid

Master polygons, and the operator ${\displaystyle GC_{k,\ell }\left(G_{0}\right)}$, can be classified as follows:

• Class I: ${\displaystyle \ell =0.}$
• Class II: ${\displaystyle k=\ell .}$
• Class III: all other. Class III operators exist in chiral pairs: ${\displaystyle GC_{k,\ell }\left(G_{0}\right)}$ is the chiral pair of ${\displaystyle GC_{\ell ,k}(G_{0})}$.

Below are tables of master triangles and squares. Class I corresponds to the first column, and Class II corresponds to the diagonal with a slightly darker background.

Master polygons for triangles

Master triangles through (8,8)
	0	1	2	3	4	5	6	7	8
1
2
3
4
5
6
7
8

Master polygons for squares

Master squares through (8,8)
	0	1	2	3	4	5	6	7	8
1
2
3
4
5
6
7
8

Composition of Goldberg–Coxeter operations corresponds to multiplication of complex numbers. If and only if ${\displaystyle GC_{a,b}\left(GC_{c,d}\left(G_{0}\right)\right)=GC_{e,f}\left(G_{0}\right)}$ (i.e. the series of operations on the left produces a graph isomorphic to the one on the right), then for a 3-regular graph ${\displaystyle (a+bu)(c+du)}$ is in the equivalence class of ${\displaystyle (e+fu)}$, and for a 4-regular graph ${\displaystyle (a+bi)(c+di)}$ is in the equivalence class of ${\displaystyle (e+fi)}$. There are some useful consequences of this:

• The application of repeated Goldberg–Coxeter operations is commutative and associative.
• Complex conjugation of the element ${\displaystyle a+bi}$ or ${\displaystyle a+bu}$ corresponds to reflection of the constructed graph.
• Since the Gaussian integers and Euclidean integers are both Euclidean domains, elements of those domains can be uniquely factored into prime elements. Therefore, it also makes sense to decompose a Goldberg–Coxeter operator into a sequence of "prime" Goldberg–Coxeter operators, and this sequence is unique (up to rearrangement).

Performing the GC construction

The steps of performing the GC construction ${\displaystyle GC_{k,\ell }(G_{0})}$are as follows:

1. Determine the master polygon, based on ${\displaystyle k}$, ${\displaystyle \ell }$, and ${\displaystyle G_{0}}$
2. If operating on a 3- or 4-regular graph (instead of a graph with triangular/quadrilateral faces), take its dual graph. This dual graph will have triangular or quadrilateral faces.
3. Replace the faces of the triangulated/quadrangulated graph with the master polygon. Be aware that planar graphs have an "external" face that must be replaced as well. In the below example, this is done by attaching it to one side of the graph and connecting other sides as necessary. This temporarily introduces overlapping edges into the graph, but the resulting graph is planar. The vertices can be rearranged so that there are no overlapping edges.
4. If the original graph was a 3- or 4-regular graph, take the dual of the result of step 3. Otherwise, the result of step 3 is the GC construction.

Below is an example, where ${\displaystyle GC_{1,1}(G_{0})}$ is constructed on the skeleton of a cube. In the last two graphs, blue lines are edges of ${\displaystyle G_{0}}$, while black lines are edges of ${\displaystyle GC_{1,1}(G_{0})}$. (Dotted lines are normal graph edges, just drawn differently to make overlapping graph edges more visible.) Red vertices are in ${\displaystyle G_{0}}$ and remain in ${\displaystyle GC_{1,1}(G_{0})}$, while blue vertices are newly created by the construction and are only in ${\displaystyle GC_{1,1}(G_{0})}$.

• 
  Master square (1,1)
• 
  Starting polyhedron (Cube)
• 
  ${\displaystyle G_{0}}$, the skeleton of the cube
• 
  Intermediate step of constructing ${\displaystyle GC_{1,1}(G_{0})}$.
• 
  The result ${\displaystyle GC_{1,1}(G_{0})}$, after rearrangement
• 
  Embedding of the result (rhombic dodecahedron)