Doug Dunham
I was originally inspired to make hyperbolic patterns by M.C. Escher's four "Circle Limit" prints. After succeeding in duplicating Escher's hyperbolic prints using computer methods, I noticed that such patterns came in "families" that were indexed by the multiplicities p, and q of the rotation centers. Such patterns are based on the regular tessellations {p,q} of regular p-gons meeting q at each vertex. This observation led to the creation of new hyperbolic Escher-inspired 2-dimensional patterns based on his Euclidean and even spherical repeating patterns. Some of Escher's patterns contain sublteties in their crystallographic classification, such as his Regular Division Drawing number 42 which appear to have symmetry group p4.
This hyperbolic pattern is inspired by Escher's Regular Division Drawing number 42. There are obvious 5-fold rotation centers at the meeting points of the conchs. However the apparent 4-fold meeting points of the cockles are actually only 2-fold rotation centers since the openings of the adjacent snails alternately point toward and away from those centers. This preserves Escher's symmetry in Drawing 42 in which there are two kinds of 4-fold rotation centers at the meeting points of the conchs. Similarly, in this pattern there are two kinds of 5-fold rotation centers at those meeting points, those at which the tips of the conchs touch the openings of the snails and those at which the tips point away from the snail openings.