Beauty - Mathematics

Richa Gupta, The International School Bangalore

Introduction

Regularly repeating patterns extending to infinity can be some of the most aesthetically appealing motifs to the human eye. Such regular divisions of the plane are known as tessellations; tessellations are arrangements of closed shapes that immaculately cover a plane, without overlapping or gaps 1. Such motifs can be seen in numerous areas—such as in honeycombs, ancient Roman mosaics, and the contemporary woodcuts of M. C. Escher.  

Due to their infinite and repetitive nature, tessellations are some of the most beautiful facets of art. However, most of us are only familiar with Euclidean tessellations, which are patterns on the three-dimensional space postulated by Euclid; Non-Euclidean tessellations, and in particular, hyperbolic tessellations, are not as widely known. So, in this essay, I will be exploring the mathematics behind hyperbolic tessellations, with a special emphasis on the art of Escher; through this essay, I will attempt to answer the question: “to what extent do hyperbolic tessellations play a role in developing notions of beauty?”

Euclidean Tessellations

On the Euclidean plane, tessellations are generally formed by transforming a convex polygon by isometries, consequently creating a repeating pattern over the plane 1; an isometry is a linear transformation which includes rotation, reflection, translation, and glides 2. Using congruent copies of a regular polygon (with p sides) to form the tessellation creates a regular tiling 3. In the Euclidean plane, a regular tiling is denoted by {p,q}, where p represents the number of sides of the polygon, and q represents the number of polygons meeting at a single vertex 4. This denotation is also known as the Schläfli Symbol 5.

The sum of internal angles in a regular polygon can be found by the following formula: n = pi (p - 2)16, where is the number of sides. Hence, the value of each angle = pi (p - 2) over p. So, the following equality can be derived where pi is in radians:

pi (p - 2) over p = 2 pi over q

On further solving this equality, the following expressions are obtained:

pq - 2q - 2p = 0

pq - 2q - 2p +4 = 4

(q - 2)(p - 2) = 4

There are only 3 solutions to this equation: {3,6}, {4,4} and {6,3}. This implies that 6 equilateral triangles can form a regular, Euclidean tesselation, as can 4 squares and 3 regular hexagons meeting at a single point.

Non-Euclidean Geometry

In three-dimensional space, there are three categories of curvature geometries: Euclidean, spherical, and hyperbolic 7. Spherical and hyperbolic geometries fall under Non-Euclidean geometry as they each use their own version of Euclid’s 5th postulate—the parallel line postulate.  

Spherical geometry, also known as Riemannian Geometry, is the study of curved spaces; so, while Euclidean space can be studied on a flat piece of paper, spherical geometry is studied on the surface of a sphere 8. In spherical geometry, the sum of the angles of a triangle is greater than 180°.

Hyperbolic geometry, on the other hand, is the study of saddle-shaped space 8; so, working on a hyperbolic plane is similar to working on the surface of a saddle. However, unlike the surface of a sphere, hyperbolic space is infinite. In hyperbolic geometry, the sum of the three angles of a triangle is less than 180°; this plane is the only one with a constant, negative curvature 9. The expression for regular tessellations is provided by the formula (q - 2)(p - 2) > 4, for the Schläfli Symbol {p,q} 10; hence, unlike the cases of Euclidean and spherical geometry, an infinite number of regular tessellations can be constructed.

Hyperbolic geometry is a predominant concept in the contemporary art of M. C. Escher, for it is through his woodcuts that the effects of hyperbolic space on the formation of tessellations can be seen.

M. C. Escher’s Circle Limit III

Maurits Cornelis Escher was a Dutch artist who made mathematically influenced lithographs and woodcuts; one of his most aesthetically pleasing works of art is the third of a series of four woodcuts, known as Circle Limit III.  This woodcut presents a brilliant example of the incorporation of hyperbolic tessellations in art. 

Escher based his woodcut Circle Limit III on the Poincare disk model of hyperbolic geometry 10; in this model, objects in hyperbolic geometry are represented on the Euclidean plane 12. As said by Escher regarding this woodcut, “all these strings of fish shoot up like rockets from infinitely far away, […] and fall back again whence they came, not one single component ever reaches the edge.” 13 As the fish approach the boundary of the disk, it is seen that equal hyperbolic distances correspond with shorter Euclidean distances—for with respect to the hyperbolic plane, all fish are exactly the same size.

Circle Limit III represents a regular tessellation on the hyperbolic plane 12. At first sight, it may be believed that this woodcut is a regular tessellation of triangles and squares; this is demonstrated by examining the white backbone lines of the fish. It looks like three triangles and three squares join at a vertex; however, given that this woodcut is a regular tessellation, this would imply that each corner angle is 60°—and hence that the sum of the angles of a triangle is 180° 5. However, this cannot be possible, given that Escher’s Circle Limit III has the visual indications of a hyperbolic tessellation—and that the sum of angles of a triangle on a hyperbolic plane is less than 180°.

As said by Escher regarding Circle Limit III, “all these strings of fish shoot up like rockets from infinitely far away, perpendicularly from the boundary […]”; Escher’s comment implies that the white backbone lines are hyperbolic lines—which are circular arcs orthogonal to the bounding circle of the Poincare disk 9. However, on using hyperbolic trigonometry, Harold Coxeter, one of the greatest geometers of the 20th century, revealed that the white backbone lines make angles of approximately 80° with the disk; hence, these lines are not hyperbolic lines or geodesics, but rather equidistant curves in the hyperbolic plane 9. In the Poincare disk model, equidistant lines make acute or obtuse angles with the bounding circle, and each point is at an equal hyperbolic distance from a hyperbolic line with the same endpoints 9. In differential geometry, a geodesic refers to a “straight line” in curved spaces 14; so, since the white backbone lines are not geodesics, the squares and triangles cannot be deemed polygons in the hyperbolic plane.

So, another regular tessellation for Circle Limit III has been provided 5, denoted by the Schläfli Symbol {8,3} — which indicates that three regular octagons meet at a single point. 

This Schläfli Symbol fits the condition of hyperbolic tessellations, since (3 - 2)(8 - 2) > 4.

The hyperbolic tessellations in Circle Limit III can also be generalised in the form of (p,q,r), where p denotes the number of fish meeting at right fins, q at the left fins, and r at the noses. So, by this generalisation, Escher’s woodcut can be denoted in the form (4,3,3). As written in a paper by Douglas Dunham regarding patterns in Circle Limit III 12, another hyperbolic tessellation can be obtained if the octagons in the regular tessellation are divided into 4 kites; in a kite, two sets of adjacent sides are congruent 15. The diagram can be found in his paper: More “Circle Limit III” Patterns 12. Based on the (p,q,r) generalisation, the vertex angles of the kite come up to be 2 pi over p, 2 pi over q, pi over r and pi over r; for example, when p right fins meet at a point, the angle of that vertex of the kite = 2 pi over p. Two angles of the kite are pi over r, rather than 2 pi over r, because r noses and r tails meet at a single point; hence, the noses and tails only occupy an angle of pi each.

In the hyperbolic plane, the sum of angles of a triangle is less than p; therefore, the sum of angles of a quadrilateral is less than pi radians. Hence, for Escher’s Circle Limit III to be a hyperbolic tessellation, the following inequality must be fulfilled: 

2 pi over p + 2 pi over q + pi over r + pi over r < 2 pi

1 over p + 1 over q + 1 over r < 1

Here, 1/4 + 1/3 + 1/3 = 11/12 < 1

Hence, analysis of this tessellation further proves that Escher’s woodcut is a prime example of hyperbolic art. Thus, mathematics is an integral part of analysing tessellations in the three classical planes of geometry—Euclidean, spherical, and hyperbolic.

Conclusion

Therefore, it may be concluded that mathematics plays a great role in developing notions of beauty, especially with respect to tessellations in hyperbolic planes. Circle Limit III was Escher’s way of bounding the concept of infinity within a single disk, and simultaneously creating beautiful patterns. Mathematics and beauty are inextricably intertwined—we just have to find the link that connects them, and appreciate the fact that beauty is a result of every mathematical observation and calculation.