Hyperbolic Geometry and Exploration of Mathematical Topology Through Crochet

The goal of exploring mathematical forms through crochet is to produce and explain mathematical objects and concepts to a general audience by allowing them to interact with a variety of crocheted items. Crochet was chosen as the method for representing the objects due to the sturdiness of crocheted surfaces and the ability to manipulate the crocheted objects in a way computer graphics cannot be manipulated. By crocheting different variations of a hyperbolic plane, with varying stitch ratios, one can observe how a hyperbolic plane is affected by exponential growth rate. One can observe that the less stitches a hyperbolic crochet pattern has between stitch increases, the more pronounced a hyperbolic plane’s “curls”. By having two crocheted mobius bands that are mirror-imaged, one can also illustrate the concept of how a Klein bottle is formed. Where one might have difficulty imagining a surface that has no distinction between inside and outside, by providing a craft-based visual representation, one can highlight the mathematical properties that go into composing the object