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 are [1]:

I. To draw a straight line from any point to any point.

II. To produce a finite straight line continuously in a straight line.

III. To describe a circle with any center and distance.

IV. That all right angles are equal to one another.

V. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

So where is the vital not that creates a bifurcation between Euclidean and hyperbolic geometry? The first four axioms, often called the neutral axioms, are common to both hyperbolic and Euclidean geometry. The difference lurks in the fifth axiom, also called the parallel postulate. This axiom can be stated alternatively as: “for every line there is a…