Master polygons

Lattices

The square and triangular lattices in the plane have the nice quality that all of their faces remain similar to each other with any affine transformation of the plane. This makes them a good candidate to use if, in a vague sense, we want to subdivide a face of a tiling or polyhedron into smaller faces that are somewhat similar to the original face.

The hexagonal lattice is dual to the triangular lattice. The square lattice’s dual is a lattice staggered with respect to the original lattice.

Relation to Gaussian and Eisenstein integers

The vertices of square and triangular lattices match with the elements of certain integral domains [1] in the complex plane. The Gaussian integers form a square lattice, and the Eisenstein integers form a triangular lattice. To make the parameterization of the \(\Delta(a,b)\) operator match the traditional parameterization used by [Goldberg] and [Coxeter], the Eisenstein integers can be paramaterized with \(u = e^{\pi i/3}\) instead of the usual \(w = e^{2\pi i/3}\). The relationship between the two is simple: \(a + b w = a - b + bu\).

Type	Gaussian	Eisenstein
Form	\(a + b i\)	\(a + b u\)
Adjoined element	\(i =\sqrt{-1}\)	\(u = \frac{1}{2}(1 + i\sqrt 3) = e^{\pi i/3}\)
Units \((a, b)\)	\((1, 0) = 1\), \((0, 1) = i\), \((-1, 0) = -1\), \((0, -1) = -i\)	\((1, 0) = 1\) \((0, 1) = u\), \((-1, 1) = u^2 = u-1 = w\), \((-1, 0) = -1\), \((0, -1) = -u\), \((1, -1) = -u^2=1-u = -w\)
Algebraic Norm (Euclidean Function) \(|x|^2=x\overline x\)	\(a^2 + b^2\)	\(a^2 + ab + b^2\)
Multiplication \((a+bz) (c+dz)\)	\((ac-bd) + (bc+ad)i\)	\((ac-bd)+(bc+ad+bd)u\)
Division \((a+bz)/(c+dz)\)	\(\left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\)	\(\left({ac+ad+bd \over c^2 + cd+ d^2}\right) + \left({bc-ad \over c^2 + cd+ d^2}\right)u\)
Divisor t for prime test	4	3
Class sine	\(C = \sin^2(2\theta) = \frac{4a^2b^2}{(a^2+b^2)^2} = -\frac{(x^2-x^2)^2}{4|x|^4}\)	\(C = \sin^2(3\theta) = \frac{27a^2b^2(a+b)^2}{4(a^2+ab+b^2)^3} = -\frac{(x^3-x^3)^2}{4|x|^6}\)
Master polygon vertices	0, \(x\), \(x(1+i)\), \(xi\)	0, \(x\), \(xu\)
Master polygon center	\(x\frac{1+i}{2}\)	\(x\frac{1+u}{3}\)
Example master polygon (3,2) construction (center marked with white circle)

Elements \(x\) and \(y\) of a domain such that \(x = zy\), where \(z\) is a unit, are called associates. “Normal elements” are defined so that we have a “normal” form for associates: those elements having \(a > 0\) and \(b \ge 0\). All non-zero elements of the domain can be expressed as a normal element times a unit. All the units have \(|z| = 1\), so the associate elements can be thought of as the elements rotated around the origin (by steps of 90 degrees for the Gaussians, 60 degrees for the Eisenstein.)

An equivalence class can be defined from the associate relationship (excluding the zero element): all associated elements belong to the same equivalence class. The “normal element” is the preferred label \(x\) for that equivalence class.

Both these domains are Euclidean domains. That means that all non-zero elements have a unique factorization into prime elements and units, and what’s more, they can be factored using an extension of the Euclidean division algorithm on the natural integers. A prime element can only be divided by itself, an associate of itself, or a unit. More specifically, non-zero elements of these domains can be factored as:

\[x = zp_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}\]

where \(z\) is a unit, \(n_i\) is a natural integer, and \(p_i\) is a normal prime element of the domain.

An element of these domains is prime if:

• \(x = pz\), where \(z\) is a unit, \(p\) is a natural prime, and \(p = -1\mod t\), or
• \(|x|^2\) is a natural prime.

Relation between operators and master polygons

We define \(\Delta(a,b)\) for the triangular GC operator and \(\Box(a,b)\) for the quadrilateral GC operator. The GC operation can be thought of as taking each face of the polyhedron and replacing it with the corresponding master polygon described above, using the Gaussian integers for \(\Box(a,b)\) and Steineisen integers for \(\Delta(a,b)\). (The section on arbitrary surfaces will expand on this definition.). In fact, it has been shown that \(\Delta(a,b)\) corresponds to the equivalence class of associates in the Eisenstein integers, and \(\Box(a,b)\) corresponds to the equivalence class of associates in the Gaussian integers. There are some very nice consequences of this:

• Composition of operators corresponds to multiplication of complex numbers. If \(\Delta(a,b)\Delta(c,d) = \Delta(e,f)\), then \((a + bu)(c + du) = z(e + fu)\) for some unit z. A similar relation holds for \(\Box\).
• Since elements of these domains can be factored, operators can be “factored” into a sequence of operators. When \(x = a + bi\) is an element of the Gaussian integers, and \(x = z p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}\), then \(\Box(a,b) = \Box^{n_1}(p_1)\Box^{n_2}(p_2)\cdots\Box^{n_k}(p_k)\), and similarly for the Eisenstein integers and \(\Delta(a,b)\). It also makes sense to talk about “prime” operators that can’t be decomposed further.

\(\Delta(1,0)\) and \(\Box(1,0)\) are identity operators: where defined, they cause no change to the subdivision.

In the terminology of geodesic domes,

• Class I operators have \(b=0\).
• Class II operators have \(b=a\).
• Class III operators are all others, and occur in chiral pairs. \(\Delta(a,b)\) is the chiral pair of \(\Delta(b,a)\), and the same for \(\Box(a,b)\) and \(\Box(b,a)\).

Here we also introduce the ‘’class sine’‘, to have a smooth measure of how much like a class I or class II operator a given operator is. If an operator is Class I, \(C=0\); if Class II, \(C=1\), and Class III is in between.

Chiral pairs correspond to conjugation of the complex number \(a+bi\) or \(a+bu\). That is, there exists some unit z such that \(z(a-bi) = b+ai\), or \(z(a+b\bar{u}) = b+au\). Interestingly, the notation used in Conway notation for the chiral pair of an operator is an overbar, also used to denote complex conjugation. A consequence of this relation between chiral pairs is that the composed operator \(\Delta(a,b)\Delta(b,a)\) will always be a Class I operator, since a complex number times its conjugate is a real number. For every Class III \((a,b)\), there will exist three operators \(\Delta(a,b)\Delta(b,a)\), \(\Delta(a,b)\Delta(a,b)\), and \(\Delta(b,a)\Delta(b,a)\) with the same algebraic norm. The same holds for \(\Box(a,b)\).

Footnotes

[1]	A type of ring.