Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of a hyperbolic plane. Escher’s wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: “Escher got it absolutely right to the millimeter.”
Hyperbolic planes are difficult to explain. In fact, hyperbolic geometry is an extremely huge topic. Many visualizations of hyperbolic planes have been discovered (including the circle limits afore mentioned). Taking Circle Limit III for example (the one with the fishes), here is the gist of what these artworks have to do with hyperbolic geometry:
The number of fishes within a distance of n from the center rises exponentially.
The fishes have equal hyperbolic area. Yes, the tiny fishes on the very edge of the circle are the same size as the fishes in the center (on an actual hyperbolic plane, anyway).
So, the area of a ball of radius n must rise exponentially in n.
Experiments in Processing: curious eels in hyperbolic sea.
I love hyperbolic geometry, heptagons, and Escher.
The eels move on the sides of a triheptagonal tiling of the hyperbolic plane in the Poincaré disk model. You better appreciate this because getting the coordinates is a real pain in the *** (done with a python script, <a href=“https://github.com/rantonels/poincare“>source</a>).
These pictures are pieces of art that have been made based off of the Klein and Poincare disk models for hyperbolic geometry (where through any given point there are an infinite number of parallel lines)
This is a tiling of regular right angled dodecahedron in three dimensional hyperbolic space shown through 4 iterations. 12 generators of the tiling are reflections in each of the 12 dodecahedron’s faces.
Escher’s Circle Limit I - IV and Hyperbolic Geometry
All four of M.C. Escher’s Circle Limit woodcuts are great works of art and superb examples of hyperbolic geometry. Hyperbolic geometry is a form of geometry where the parallel postulate of Euclidean geometry does not hold. This is reflected in the world represented in each of the woodcuts. In each one, the figures are not getting smaller as the reach the edge of the circle as they would in Euclidean geometry. From the figures’ perspective, they are all the same size, regardless of where they are in the circle. Their entire world lies completely within the flat plain of the circle, the edge of which to them remains infinitely far away.