Until now, most methods for making a hyperbolic plane from crochet or
similar fabrics have fallen into one of two categories. In one type, the
work starts from a point or line and expands in a sequence of increasingly
long rows, creating a constant negative curvature. In the other, polygonal
tiles are created out of a more or less Euclidean fabric and then attached
in such a way that the final product approximates a hyperbolic plane on the
large scale with an average negative curvature. On the small scale,
however, the curvature of the fabric will be closer to zero near the center
of the tiles and more negative near the vertices and edges, depending on
the amount of stretch in the fabric. The goal of this project is to show
how crochet can be used to create polygonal tiles which themselves have
constant negative curvature and can therefore be joined into a large region
of a hyperbolic plane without significant stretching. Formulas from
hyperbolic trigonometry are used to show how, in theory, any regular tiling
of the hyperbolic plane can be produced in this way.