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An Introduction to Orbifolds

These pages are loosely based on a talk titled "Symmetries, Lattices, and Tilings" given by John Conway at the Regional Geometry Institute at Smith College on July 12, 1993. (1)

After this brief introduction, in which we give the long awaited definition of an orbifold, we go on to compute the orbifolds of several plane patterns. Following that, we look at orbifolds of spherical symmetry groups and describe when handles and cross caps might appear in an orbifold. We then give rules for deciding which orbifold symbols correspond to plane and spherical patterns. These rules can be explained in terms of the Euler characteristic of the surface the orbifold is drawn on. After a brief conclusion, there's yet another homework assignment! There is also a supplementary page of examples of orbifolds of plane patterns if you need more practice.

When looking at an object (like the girl Alice in Through the Looking Glass) and its mirror image, you may be unable to distinguish between the original and its reflection. Similarly, a fly crawling on a brick wall knows where it is on its brick, but has no way of telling one brick from the next. The two sides of the mirror and the different bricks in the wall are indistinguishable, or "symmetric".

Changes we can make that don't affect our observations of an object are called symmetries of the object, and they constitute its symmetry group. For instance, swapping Alice with her reflection is an element of the symmetry group of Alice's universe. By moving from one brick to the next, the fly sees a symmetry of his brick wall.

We are studying the symmetry groups of surfaces. These surfaces may include Kali printouts, the objects displayed by KaliedoTile, floors, walls, and furniture.

In the study of orbifolds, identify ourselves with the image we see behind the looking glass -- we stop distinguishing between the two "different" copies of ourselves. The mirror halves the world (if we extend it indefinitely in every direction), equating each point on one side with a point on the other. We only need to understand one half.

The fly can learn everything it needs to know about the brick wall by studying a small region around itself. Two points on the wall that look the same to the fly may be regarded as identical. The smallest region that, repeated, makes up the fly's entire wall can be described by an orbifold. The pattern on that region is the motif that generates the pattern of the brick wall.

The orbifold of a symmetric surface is obtained by regarding as identical all the points of a surface that look alike. We will visualize orbifolds by folding and twisting our symmetric patterns until all identical points have been brought into contact with each other.

Notes

(1) The Regional Geometry Institute at Smith College was funded by NSF grant #DMS-90 13220.

Many thanks to the suppliers of the software used to illustrate and enhance these pages.

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