Why settle for anything less! The orbifold theory is a modern and complete approach of planar symmetry and Conway’s orbifold notation describes symmetrical patterns in a natural way.
Each symmetrical pattern has various kinds of “features” which we can learn to recognize, and the orbifold symbol is just a list of these. Each feature has a cost, and the Magic Theorem asserts that the orbifold symbols that describe planar symmetry patterns are those that cost exactly $2.
In other words, all we must do to find all of the possible planar symmetry types is just to work out how these costs come out to the right sum. This is powerful! Moreover, this smoothly generalizes across the two-dimensional geometries: Spherical symmetry types cost less than $2, and hyperbolic ones cost more.
The orbifold notation names are well-defined. They are fully systematic and smoothly generalize. The symbols themselves can be calculated with and manipulated in meaningful ways. And the orbifold theory is approachable, with many hands-on activities to bring the Magic Theorem to life.
Every symmetrical pattern like the ones that you will find throughout this book and in the world around you is associated with a particular surface, its orbifold, which we imagine by considering points of the same kind to actually be the same point. For example, folding over and fusing the two halves of the shape at left gives us its orbifold, the half-heart shape at right. The new boundary of this half-heart shape corresponds to the mirror-line across the original pattern. We denote this boundary with a *.
In a similar way, in any pattern, we may bring together points of the same kind and form the pattern’s orbifold. On its orbifold features such as a boundary correspond to the geometry of the pattern. These are what the orbifold notation records.
The power of this approach comes from the well-understood classification and geometrization of surfaces — the topology and geometry of a surface are tightly bound, each constraining the other. The Magic Theorem leverages simple yet powerful tools for understanding the topology of surfaces into classifying their geometries, and thence the patterns that have those as their orbifold surfaces.
Overall, the topology of the orbifold captures symmetries in the pattern.
In fact, the complete flow-chart for analyzing a symmetry type may be summed up as:
Find the orbifold!
Because this approach is fundamentally topological, it smoothly generalizes across the non-Euclidean geometries. Here’s a picture we like, of three woven patterns, in the geometries of the sphere, plane and hyperbolic plane. Although these patterns are in different kinds of geometry, up close they look the essentially same, triangles arranged around holes. As we take up in the book these patterns’ orbifolds are basically the same too, changing just one parameter 5, 6, then 7.
But this is just the beginning! We invite you to come along and learn more!