Tesselations of the Hyperbolic Plane and the Art of M.C.Escher

Short of hallucinogens, viewing the art of M .C. Escher is the fastest trip to infinity available in this dimension.

Escher produced four patterns using hyperbolic geometry: Circle Limit I, Circle Limit II, Circle Limit III, Circle Limit IV.

We will proceed to dive into the hyperbolic plane and the math of hyperbolic tessellations. After this, we’ll return to these works with the ability to understand the mathematical scaffolding Escher used to create the Circle Limits.

But what is hyperbolic geometry? Revealed to the world by János Bolyai and Nikolai Lobachevsky in the 19th century, hyperbolic geometry is built from axioms closely related to the axioms of Euclidean geometry [1]. Between the axioms of the Euclidean plane and the axioms of the hyperbolic plane, there is a difference of only one word. Harold Coxeter called this “the vital word not” [1]. It follows that a brief explanation of Euclidean geometry is helpful to understanding hyperbolic geometry.

Euclidean geometry was introduced by Euclid in his seminal textbook Elements, published circa 300 B.C. [1]. Euclidean geometry is built from five postulates or axioms. That is, every theorem of Euclidean geometry can be derived from these axioms. From Sir Thomas Heath’s translation of Elements, these axioms…