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Abstract: Mathematics has been called the science of patterns. Some of the oldest, most symmetric, and most beautiful patterns are the tilings of the sphere and of the plane. As examples, we consider the facets of a jewel or the cells of the bees' honeycomb. Mathematicians have generalized these tilings to the hyperbolic plane; the most famous illustrations of these are MC Escher's "Circle Limit" prints. Tilings of the sphere are always finite. Tilings of the plane and of the hyperbolic plane are necessarily infinite - they require an unbounded number of tiles. Since infinity is a difficult idea to understand or to use, we search for ways to "wrap-up" the tilings into a finite, bounded object. For frieze patterns this idea dates back to antiquity. Wrapping up the plane is more subtle; wrapping up the hyperbolic plane is still an area of active research! We'll illustrate all of these ideas with several examples; the last of these will be wrapping the (2,3,7) triangle tiling of the hyperbolic plane around Felix Klein's famous quartic curve.