Hyperbolic Celtic Knot Patterns University of （双曲凯尔特结大学模式）.pdf

Hyperbolic Celtic Knot Patterns Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@ Abstract 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. Introduction In about the 6th century Irish monks started using what we now call Celtic knot patterns as ornamentation for religious texts. The monks also created spiral patterns, key patterns, zoomorphic patterns, and decorated lettering, but we will only consider knot patterns. Figure 1 shows a simple example of a knot pattern. The Figure 1: A simple Celtic knot pattern use of this kind of decoration went out of style in about the 10th century, and the methods for creating such patterns were lost as well. Subsequently, people who wanted to make Celtic