17-19.12 Magic Theorems

New Math this week:  the main results this week involved the so-called Magic Theorems, that establish important results about the orbifold notation we have been learning to use to name symmetry groups.  To understand the theorems we need to introduce the basic symbols for the notation.

  • $\large *$ Represents a mirror line
  • X Represents a glide reflection
  • $n\in \mathbb{N}$ represents a natural number $n > 1$, a rotation center of order $n$.
  • $O$ represents a pattern without any of the above features.

To each of these symbols there is a cost associated.  By considering all possible valid strings consisting of these symbols, one can then associate to each string a cost by summing over the costs associated to the constituent symbols.  One obtains some beautiful theorems.  But first we need to introduce some further concepts.

The Euler characteristic of a surface.   Suppose I have a set of polygons with the property that every edge is shared by exactly two polygons.  This set of polygons is called a 2-complex and determines a two-dimensional surface without boundary. Consider 2-complexes which yield the 2-sphere as surface.  Let V, E, and F denote the number of vertices, edges, and faces, resp. in the 2-complex. Then we showed by considering the platonic solids that $V-E+F=2$ for these surfaces, and gave reasons to believe that this relation is always true.  We define the associated function for a 2-complex $c$: the Euler characteristic $\lambda(c) := V-E+F$.  For a 2-sphere, $\lambda(c) = 2$.  Considering the 1-holed torus we showed that $\lambda(c)=0$ and, in general, for n-holed tori, $\lambda(c) = 2-2n$.

Euler characteristic of orbifolds. By considering the orbifolds we have been studying, we were able to extend the notion of Euler characteristic to these spaces, too.  The key idea is to introduce fractional points and edges at points and edges which are fixed by some group element.  One determines the fractional weights so that the resulting weights add up to 1 wherever the stabilizer of the group is non-trivial.  That is, make sure that the statement: “The copies of the fundamental domain cover the plane (or sphere) without overlap” remains true.

The cost function.  The cost function is defined as follows:

  • $k(*) = k(X) = 1$,
  • $k(n) = \dfrac{n-1}{n}$ if it comes before any $*$ in the string, otherwise $k(n) = \dfrac{n-1}{2n}$.
  • $k(O) = 0.

Example: Let $s = 235$.  Then $k(x) = \dfrac{1}{2}+ \dfrac{2}{3} + \dfrac{4}{5} = \dfrac{118}{60}$.

Magical theorem for spherical groups:  Let $s$ be a valid string consisting of the above symbols.  Then the cost function $k(s) < 2$ exactly when $s$ describes a spherical orbifold $\mathcal{O}$.  In this case, the Euler characteristic of $\mathcal{O}$ is $\lambda({\mathcal{O}}) = 2-k(s)$, and the symmetry group of  $\mathcal{O}$ has order $\dfrac{1}{\lambda({\mathcal{O}})}$.

Example:  Applied to $235$, one obtains that this orbifold has Euler characteristic $\dfrac{1}{60}$ and the symmetry group has order 60.

There is a similar, simpler theorem for wallpaper groups.

Magical theorem for spherical groups:  Let $s$ be a valid string consisting of the above symbols.  Then the cost function $k(s) = 2 $ exactly when $s$ describes a spherical orbifold $\mathcal{O}$.

Further research: Note that the theorem makes no claims regarding the order of the group, since they are all infinite groups.One important invariant of a wallpaper group is the order $O_T$ of the translation group within the full group.  Can you find a way to derive this invariant from the Conway-Thurston orbifold notation for the group?

Further reading:  See the book Symmetry of Things by John Conway, et. al., Chs. 3, 4, and 6 for details and proofs of all the above themes and claims.  The book can be found  in the math library “Apparat”.

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