I am really interested in art and aspire to be an artist myself. Therefore, I wanted to benefit from this assignment by the means of exploring the flirtations of art with mathematics. Among all great artists, M.C. Escher is the one who comes to mind when one thinks of mathematics. As I researched deeper into his art, I found out that he is more than just flirting with mathematics but in a marital relationship. So much that he can be referred to as a secret mathematician. Escher’s fascinating work holds inside a great knowledge of Euclidean and Non-Euclidean Geometry and touches many other areas in mathematics and geometry. I’m especially interested in geometry for it explores the mystery of shapes and the material world around us. Since it would be impossible to cover all in this exploration, I chose to take a closer look on the application of hyperbolic geometry in Escher’s artworks. This paper aims to explain and prove the use of hyperbolic geometry laws in Escher’s work.
M.C. Escher was the first artist who created interesting patterns in the hyperbolic plane. He was intrigued by the open question of whether his work belongs to the domain of mathematics or to that of art (“The mathematics behind the art of M.C. Escher”). For his work, he used the Poincare´ disk model and the Poincare´ half plane model of hyperbolic geometry. Interestingly, Escher was able to create all the hyperbolic patterns by hand and hence it was a very long, tedious process. Escher created the Circle Limit I in 1958; an example representation is shown in Fig. 3. After that Escher created three more patterns of Circle Limit known as, Circle Limit II, Circle Limit III (shown Fig. 1), and Circle Limit IV (Bruter, Claude).
Introducing Hyperbolic Geometry
Hyperbolic Space In high school, we have explored Euclidean geometry, which takes place on a flat two-dimensional plane. But hyperbolic geometry describes a curved two-dimensional world called the hyperbolic space. Therefore, different laws apply for this alternative space of geometry. It agrees with Euclid’s 4 postulates but not the parallel postulate. Instead, it suggests,
If l is any line and P is any point not on l , then there exists at least two lines through P that are parallel to l .
Figure 1. Figure 1 shows a piece of hyperbolic space.
Poincaré Disc Model There are many models of hyperbolic space, but this paper will concentrate on one map of the hyperbolic space, which is called the Poincaré disc, also used by Escher to represent the hyperbolic space on flat surface. This disc model shows the inside of a circle with the radius 1, but the circle is not included. Since this is a distorted map, equally sized shapes with equal distances on the hyperbolic space, seem to get closer and smaller near the highly distorted edges. No figure can touch the edge since the distorted plane expands to infinity and there is no actual edge. Therefore distance on a hyperbolic plane is given by (“Hyperbolic Geometry”),
Distance on a hyperbolic plane = 2tanh-1x
tanh-1x is the inverse of the hyperbolic tangent function, which has a graph like;
Figure 2: Graph of a hyperbolic tangent function.
x =1 and x = -1 are the vertical asymptotes for the curve y = tanh-1x, i.e. the curve goes to infinity at those points.
In hyperbolic space, the lines in Euclidean geometry replace geodesics. They are the shortest distances between two points and are like straight lines on the actual hyperbolic space but projected as arcs onto flat in the Poincaré disc model.
For example, the image on the left shows geodesics on the Poincaré disc. The arcs of circles that meet the edge of the disk at 90 degrees are geodesics. Geodesics that pass through the center of the disk are seen as straight lines.
Figure 3: Interpretation of geodesics on the Poincaré disc.
Figure 4: An example of the Poincaré disc model. The left image shows the 6,4 tessellation and the right one is a representation of Circle Limit I.
Euclidean points inside a bounded circle in the Poincare´ disk model (hyperbolic geometry) represent hyperbolic points. Euclidean circular arcs represent hyperbolic lines by orthogonal to the bounding circle, that also includes diameters. Hyperbolic lines are shown on the left image of Fig. 4 as the edges of the triangles and on the right image as the backbone lines of the fish.
Figure 5: Representation of Poincare´ half-plane model. Both images show the half-plane version of the triangle pattern of figure 4. The left image corresponds to the 6,4 tessellation and the right one corresponds to the half-plane version of Circle Limit I in figure 4.
Poincare´ half-plane model Euclidean points (x, y) in the upper half plane (i.e. y> 0) represents hyperbolic points are represented in the Poincare´ half-plane model. Semicircular arcs above the x-axis and with center on it, which also includes vertical half-lines, represent hyperbolic lines. The half-plane versions of images in figure 4 are shown in Figures 5. The edges of the triangles and the backbone lines of the fish in Fig. 4 are all hyperbolic lines in this model.
Escher made use of this model to create two or three patterns, which are called line limit patterns. In this model, Euclidean distance to the x-axis is inversely proportional to the hyperbolic length.
Using congruent copies of a basic sub pattern, a repeating pattern can be made. For example, in the left images in figures 4 and 5, either a black or a white triangle can be considered as a the sub pattern, in the right image of Fig.4, half a white fish with the adjoining half of a black fish is a sub pattern. Repeating patterns are necessary to show the hyperbolic nature of these models. As we can see in fig 4 and 5, if we had shown only one triangle, it was not possible to determine if it was hyperbolic or curvilinear Euclidean triangle.
Polygons and Defect Since the geodesics or the lines of the hyperbolic space are shown as curved on the Euclidean space, the polygons have sides that are curved. This causes the angle sums of the polygons to have a defect. Taking the triangle for example we can see that:
Figure 6: Representation of sum of angle in a hyperbolic triangle.
In a hyperbolic triangle, angles sum up to than 180°. First we get the sum of interior angles of the hyperbolic triangle and then we subtract it from 180 degrees, in order to get the defect of a triangle. It is possible to calculate the defect of any polygon with n sides (hyperbolic space) using,
D = (n-2) x 180 – (sum of angles of the polygon), where n is the number of sides and D represent the defect in degrees.
This defect D is directly proportional to the corresponding area of a triangle in hyperbolic space. Therefore, using a constant of proportionality, we can use, A=c1D, where c1 is a positive number and A is the area of the triangle. If the sums of angles of two different triangles are same, then the triangles have the same area as well.
Ideal Polygons Ideal polygons in the hyperbolic space are those with all vertices going to infinity, appearing to touch at the border of the Poincaré disc, without ever actually touching. They are all perpendicular to the border, so they make 0 degrees of an angle with each other. So their defect is the sum of the interior angles of the same polygon in the Euclidean space.
Figure 7: Ideal polygons.
Above are all ideal polygons, first two triangles and the third a hexagon.
Application of Hyperbolic Geometry in Escher’s Work Escher has used hyperbolic tessellations in many of his pieces. A hyperbolic tessellation is a filling of the hyperbolic space by tiles, leaving no gaps and overlaps. There are infinite regular tessellations on the hyperbolic space. Using these regular tessellations, it is possible to form irregular one’s like Escher’s.
In a regular tessellation, subsequently the number of sides of the polygon (p) and the number of polygons that meet at each vertex (q) is annotated as {p,q}. This is called a Schläfli symbol. Besides applications of symmetry, Escher also used conic in his art masterpieces. It is possible to determine on which plane the tessellation by using the equation 1/p + 1/q, if the sum equals ½, then it becomes Euclidian; if it is less than ½, then it becomes hyperbolic; if it is greater than ½, then it becomes spherical (elliptic). This could be simply explained as there are regular polygons that consist of p vertices. Therefore, one of the exterior angle will be equal to 360°/n. Since the polygon has p equal angles, each being 180° - 360°/n, then sum of interior angles will be equal to (p-2)*180°.
Figure 8: Escher’s Circle Limit 1 above is a tessellation on the Poincare disc model. The spines of fish are geodesics that form quadrilaterals and make a tiling tessellation. The red lines are the 6,4 regular tessellation superimposed on Circle Limit 1.
Figure 9: On the left side is the image of a Circle Limit II and on the right is a Circle Limit IV represented (Bruter, Claude).
Patterns in the Poincare´ Disk Model In fig. 9 we show an example of Escher’s patterns Circle Limit II and Circle Limit IV. ‘Snakes’, which was Escher’s last pattern, shows interlocking hyperbolic rings near the circular boundary and the inner rings form a dilation pattern. A complete pattern of the hyperbolic rings is shown in Fig 10. A pattern like Circle Limit III is shown in fig 10 (Bruter, Claude).
Figure 10: On the left is a interlocking ring pattern inspired by Escher’s ‘Snakes’ and on the right is a pattern like Circle Limit III (Bruter, Claude).
Patterns in the Poincare´ Half-Plane Model Escher created three line limit patterns. Figures 11 shows half-plane versions of Circle Limit IV, and a limit pattern (Bruter, Claude).
Figure 11: Left image shows an Escher one limit pattern and right image shows a half version of Circle Limit IV (Bruter, Claude).
All these interesting patterns can now be created using advanced computational techniques. The book Mathematics and modern art shows a number of these patterns. Escher’s art using mathematics is indeed inspiring; he is sometimes regarded as the father of modern tessellations.
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