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Conclusion

Given a symmetric pattern or object, we can quickly construct an orbifold that completely describes its symmetry group. Conversely, given an orbifold we can reconstruct the action of a symmetry group on a surface. Orbifolds present information about symmetry groups in such a way that we can quickly determine the order and many of the elements of the symmetry group in question. Conway's orbifold notation provides an easy way to describe orbifolds and suggests a comprehensive system for naming the symmetry groups of surfaces.

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