MORE TO DO!!
Each pattern is associated with a special surface, the pattern’s orbifold. Points that appear the same in the pattern are to be fused together and henceforth are the same point on the pattern’s orbifold. For example, the two sides of the heart-shaped pattern below at left are identical (at least ideally). By folding the heart over and fusing identical points together, we have the half-hearted figure at right, the orbifold of the heart-shaped pattern. This orbifold has a newly formed boundary, corresponding to the mirror line down the center of the heart. It is this boundary, a feature of the topology of the orbifold surface, that we record with an * in his orbifold notation.
The power of this approach comes from the well-understood classification and geometrization of surfaces — the topology of a surface strictly constrains the possible geometries it may have, and this is what the Magic Theorem is counting up.
Here are some more examples of orbifolds:
In fact, there is a family of Magic Theorems: The spherical symmetrical patterns are those with a cost of less than $2. More expensive totals give patterns in the hyperbolic plane. Even frieze patterns fit into the scheme.