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Tessellations of the Hyperbolic Plane and M.C. Escher

(this post is by Allyson Redhunt)

“For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” -M. C. Escher

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Figure 1. M.C. Escher [7].

Escher’s beginnings
M. C. Escher (1898-1972) (Figure 1) is known for his mind-boggling artwork that challenges our sense of space. Although many of his works are artistic renditions of deep mathematical ideas, he had no formal training in mathematics. In fact, he was a poor student in high school–struggling to earn a diploma. In his post-secondary education, he was trained at the School of Architecture and Decorative Arts in Haarlem. This is where he developed his fascination with structure.
Although he began as an architecture student, he was soon switched to a decorative arts path. During his schooling, he did some traveling and was exposed to Moorish motifs. These works of design are created from mathematically careful patterns. From their influence, Escher’s art branched into tessellations of the plane. He was inspired by the idea of “approaches to infinity” and began by playing with flat surfaces and spheres.

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Figure 3. Coxeter’s tessellation [2].


Escher’s ideas about structure, pattern, and infinity were suddenly enhanced when he came across the work of geometer H. S. M. Coxeter (1907-2003). Coxeter and Escher struck up a correspondence when Coxeter hoped to use Escher’s unique depictions of symmetry in a presentation for the Royal Society of Canada. Coxeter sent Escher a copy of the talk, which included an illustration depicting a tessellation of the hyperbolic plane (Figure 3) [2]. This image sparked a new area of Escher’s exploration of infinity [6]. To fully understand the beauty of his works, it is helpful to have a basic understanding of hyperbolic geometry.

A crash course in hyperbolic geometry
So what is hyperbolic space? Grade school mathematics is taught using Euclidean geometry. This assumes Euclid’s axioms, which he intended to be the basis of all geometry. However, one of them was a great source of debate between mathematicians. The “Parallel Postulate,” which states that if one straight line crosses two other straight lines to make both angles on one side less than 90˚, then the two lines meet. Proving that triangles have 180˚ angle sums is an application of this postulate [4].
However, the Parallel Postulate need not hold true in all cases, such as on the surface of a sphere. Proving that the postulate need not hold led to the discovery of an important “non-Euclidean” geometry called hyperbolic geometry. Although it at first seems unnatural to think about parallel lines performing in “new” ways, hyperbolic surfaces can be found in nature. Two common examples are sea slugs (Figure 4) and lettuce (Figure 5). The wavy structure is the tip-off that their surfaces exhibit hyperbolic geometry. This is because, while a Euclidean surface has curvature equal to zero everywhere, a hyperbolic surface has constant negative curvature (for comparison, a sphere has constant positive curvature).

Close up of a colorful nudibranch.

Figure 4. Close up of a colorful nudibranch [8].lettuce2-700x525 Figure 5. Leafy lettuce [9].


Hyperbolic geometry has many interesting properties that counter our ingrained Euclidean intuition. To understand them, we will explore an important model of hyperbolic space: the Poincaré disc model.
The Poincaré disc model
We can imagine hyperbolic space as an open disk in the complex plane $\C$. We think of the space in the disk getting infinitely more “dense” as we approach the boundary of the disc, so the distance of a straight line between two points get longer as we approach the boundary. Thus, the shortest path between two points may be curved to take advantage of the “less dense” area towards the center of the disc. It turns out that the shortest distance between two points lies along the arc of a circle that is perpendicular to the boundary. In Figure 6, the shortest distance, called a geodesic, between A and B is the arc length of the given circle. Thus, the idea of “lines” in Euclidean space is generalized when in hyperbolic space to include circles.
This disc model is precisely what Escher saw in Coxeter’s book, and is what he used to create his art.

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Figure 6. A geodesic between points A and B (created with GeoGebra).

Polygons in hyperbolic space
Since lines in hyperbolic space differ from our intuition about lines in Euclidean space, we must adjust our understanding of polygons as well. A hyperbolic polygon is a region of the hyperbolic plane whose boundary is decomposed into finitely many generalized line segments (recall that this includes circle segments), called edges, meeting only at their endpoints, called vertices. At most two edges can meet at any one point [1].
Earlier we mentioned that Euclid’s axiom that fails in hyperbolic space is used to prove that triangles have an interior angle sum of 180˚. Since this axiom does not hold, we need a new framework for thinking about polygons in hyperbolic space. In particular, Euclidean rectangles–quadrilaterals having four 90˚ angles–do not exist in hyperbolic space.
The basis for the polygon framework in hyperbolic space is the Gauss-Bonnet formula, which tells us that the area of a convex geodesic n-gon is (n-2)\pi minus the sum of the interior angles [10]. Let’s take a moment for this to sink in–the area of polygons in hyperbolic space is dependent completely on angles! This goes against our Euclidean intuition, which tells us that a triangle with longer side lengths has a larger area than a triangle with shorter side lengths, despite the fact that both have angle sums of 180˚.
Armed with this knowledge that hyperbolic space gets more “dense” along with these new ideas about polygons, we are ready to dig in to the math of Escher’s works: hyperbolic tessellations.

Tessellations of hyperbolic space
Anyone who has looked at a tiled bathroom floor is familiar with the idea of tessellations: If we repeat a pattern of polygons, we can create a pattern over a large space. Someone who has attempted to tile their own bathroom floor may have noticed that not all tile shapes fit together nicely in a pattern. Note that a bathroom floor is an example of a Euclidean space, a geometry in which it turns out to be relatively difficult to happen upon a true tessellation. In contrast, hyperbolic space is relatively easy to tessellate.
Formally, a tessellation is a polygonal tiling of a plane that covers the entire plane. The Tessellation Theorem states that any polygon tiling in a complete space with the angles around any vertex adding to \frac{2\pi}{n} for some integer n > 0 will be a tessellation of the plane. This holds true for Euclidean, spherical, and hyperbolic geometries.

When thinking about the increased “density” as we approach the edge, it should not be surprising that polygons appear smaller as we approach the boundary. Despite this, the areas stay constant (recall that it depends only on the angles!).

Escher’s tessellations
Escher created five works inspired by hyperbolic plane tessellations: Circle Limits I-IV and Snakes. While Circle Limit II and Snakes are beautiful (I highly recommend looking them up), in them Escher took more artistic license in tessellating the hyperbolic plane than the others. Thus, we will investigate Circle Limits I, III, and IV, all of which are wood cuts. His goal with creating these was to depict infinity in a finite space.

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Figure 7. Polygons on Circle Limit I [5].

Circle Limit I
This is Escher’s first attempt at his exploration of hyperbolic tessellations, and my least favorite. I find it so be harsh and too sharp to be aesthetically pleasing.

As you can see from Figure 7, we can find both hexagonal and quadrilateral tessellations [5].

Notice that the Euclidean side lengths and areas of the polygons diminish as we approach the edge. Just how much do they scale? The answer comes from the distance formula for hyperbolic geometry–analogous to d=\sqrt{x^2+y^2} in Euclidean geometry. It turns out that lengths are inversely proportional to the distance to the boundary. That is to say, a segment half way between the center and the boundary has twice the Euclidean length of one that is one quarter of the way from the boundary to the center. Since Euclidean area is related to edge lengths, the areas share this same relationship.

Escher was displeased with the result of this piece because the fish aren’t all facing the same direction, the coloring doesn’t alternate well, and the fish do not look realistic [3]. His Circle Limit III will resolve these self-criticisms.

 

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Figure 8. Circle Limit III with the central tile of the octagonal tessellation [7], (polygon made using GeoGebra).

Circle Limit III
Circle Limit III is my favorite of the tessellations. In this work, the fish are nicely organized, their coloring properly alternates, and the images look more like real animals. This is achieved by alternating between triangles and quadrilaterals (as you can see from the defined white spines).

Although this image is more aesthetically pleasing, it did not pass the mathematical scrutiny of our very own Coxeter. As we discussed earlier, geodesic polygons must be made of perpendicular circle segments, but the arcs in Circle Limit III meet the boundary at about 80˚. An alternative view on the polygons, though, gives a true tessellation with octagons [5]. The center polygon is indicated in Figure 8, and the rest of the tiles can be found in the same way, by connecting the nose vertices and the fin vertices. Notice that the edges of the polygon are not straight lines, and are instead segments of circles perpendicular to the boundary of the disc.

 

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Figure 9. Circle Limit IV [7].

Circle Limit IV
Circle Limit IV (Figure 9) is the last piece in the series, and plays with negative space to alternate angels and devils. It is commonly referred to as “Heaven and Hell,” and creates a hexagonal tiling–can you find it? (Hint: look at toes and wing tips to find vertices). Check this link to see if you found it. Escher also made a similar Euclidean tiling: Notebook Pattern or Symmetry Work 45 (1941) [6].

Today, most geometers use computers to study and create tessellations and other geometrical phenomena. The fact the Escher created these using only a compass and ruler is nothing short of miraculous. (This  column explores the nuts and bolts of how he could have achieved that–an especially hard task considering he was working in wood cuts). And recall that Escher was working not only without the aide of technology, but also without the aide of a formal math education. Keeping that in mind, although the technicalities of non-Euclidean spaces can be daunting, the mathematical beauty that arises from hyperbolic geometry are within your grasp, too.

 

References

[1] Bonahon, Francis. Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots, American Mathematical Society, 2009.
[2] Casselman, Bill. How Did Escher Do It?, America Mathematical Society Feature Column.
[3] Dunham, Douglas. Transformation of Hyperbolic Escher Patterns, University of Minnesota-Duluth.
[4] Mackenzie, Dana. The Universe in Zero Words, Elwin Street Press, 2012.
[5] Math & the Art of MC Escher. Hyperbolic Geometry, EscherMath, 2016.
[6] Math Explorer Club. M. C. Escher and Hyperbolic Geometry, Cornell University, funded by the National Science Foundation.
[7] MC Escher, website.
[8] National Geographic. Nudibranchs Photo Gallery.
[9] Organic Facts. Health Benefits of Lettuce.
[10] Schwartz, Richard Evan. Mostly Surfaces, American Mathematical Society, 2011.

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