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Hyperbolic Celtic Knot Patterns

Centuries ago, Celtic knot patterns were used to decorate religious texts. Celtic knots are formed by weaving bands in an alternating over-and-under pattern. Originally, these were finite patterns on the Euclidean plane. Recently such patterns have also been drawn on spheres, thus utilizing a second of the three “classical geometries”. We complete the process by exhibiting Celtic knot patterns in hyperbolic geometry, the third classical geometry. Our methods lead to a unified framework for discussing knot patterns in each of the classical geometries. Because of the precision and many calculations required to construct hyperbolic patterns, it is natural to generate such patterns by computer. Thus, the patterns we show are created by using computers, mathematics, and aesthetic considerations.