Experiencing Hyperbolic Geometry
Summary: How can we help our students begin to grasp the idea(s) of hyperbolic geometry, specifically a geometry with constant negative curvature—everywhere—that strongly shows the independence of Euclid’s Fifth Postulate for Euclidean geometry. There have been several models of hyperbolic geometry that mathematicians use—Poincare disk, Poincare half plan, Beltrami pseudosphere, and others, but students just beginning to learn about hyperbolic geometry from these models struggle with several concepts—not all lines (geodesics) appear straight, distances are not the “same” everywhere (except in the pseudosphere model), and the spaces have “edges” whether these edges are part of the space or not. In the 1970s William Thurston created a model using annular strips of paper taped together that, in a limiting sense had constant negative curvature and on which distances did not change as you approached a edge, lines were straight using symmetry notions from the Euclidean plane, and the space could be extended forever in any direction. But these paper models were fragile. Enter Daina Taimina, wife of David Henderson. Why not crochet these? Annular strips (all with the same radius and “width”) were easy crochet ideas. And we have the birth of sturdy crocheted hyperbolic planes suitable for student explorations. In this seminar we will look at how to crochet these hyperbolic plane models and pursue at some of the concepts that are easily discovered by students—symmetries of lines, different types of “parallel” lines, triangles where the sum of the interior angles is always less than 180 degrees and can even approach 0 degrees, and whatever else comes up in our discussions.
Duration: 60 minutes
Format: Online seminar via Zoom web meeting software with questions and discussion. Detailed instructions for joining the seminar will be emailed to registered participants.
Mary Shepherd, PhD, Arizona State University (on sabbatical) and Northwest Missouri State University