Nonlinear Tessellations

Background

To construct tilings of the Poincare disc we shall use five concepts from hyperbolic geometry:

Poles and polars, useful for constructing geodesics.

Parallel lines, why the Poincare disc is non-Euclidean

Hyperbolic angles are just like Euclidean angles

Sums of interior angles of polygons. Regular hyperbolic n-gons can have any angle sum between 0 and that of the regular Euclidean n-gon.

Hyperbolic distance and triangle congruency. Despite the greater flexibility of angles of regular hyperbolic n-gons, we shall see that hyperbolic similarity is more rigid than Euclidean similarity.

Return to background.