User:Helgus/ Intersections of mathematics and arts

Since the first artistic manifestations of humankind, Mathematics has been one of the basic tools used in the creation of many expressions of Art. In Architecture, for example, the volume design has followed rules coming from Euclidean Geometry, while the ornaments of buildings and monuments have been generated using different processes of repetition and symmetries. In Sculpture and Painting, the well known Golden Mean and its rational approximations given by ratios of successive Fibonacci numbers, have been frequently used to get more harmonic and aesthetic pieces. Nowadays, the computations systems used in artistic domains (Drawing and Design assisted by computers, production of images, etc.) bring new ideas for artistic creation. That is why in teaching Mathematics, we have to emphasize the subject of “visualization”, putting in evidence the importance of computer graphics in explaining from the surfaces formed by soap bubbles, fractal structures, knots and chaos up to hyperbolic spaces and more general topological transformations.

The paper by Vera de Spinadel

University of Buenos Aires

Buenos Aires – Argentina

vspinade@fibertel.com.ar

http://www.maydi.org.ar

"Intersections of mathematics and arts" at Vikiznanie (in English)

Introduction
Among the numerous applications of mathematical tools to different sorts of Art expressions, we have selected the following three:
 * 1) Minimal surfaces and sculpture
 * 2) Hyperbolic spaces and tessellations
 * 3) Fractal Art and coloring algorithms

Minimal surfaces and sculpture


Mathematics is, undoubtedly, an essential design language for doing sculpture. Because mathematical equations give birth to new curves, surfaces and bodies. One remarkable example of this symbiosis of Mathematics and Sculpture is Helaman Ferguson, born in 1940 at Salt Lake City, USA. Ferguson is an artist with a PhD. in Mathematics and his sculptures are created in a variety of materials, having a mathematical basis for their designs. In 1999, a snow-sculpting international competition held every January, took place this time in Breckenridge, Colorado, USA, from 19 to 23 January. Each team started with a 10 x 10 x 12-foot block of hard packed artificial snow, weighing approximately 20 tons. Helaman Ferguson and teammates Stan Wagon, Dan Schwalbe and Tamco Nemeth from Macalester College in St. Paul, Minnesota, carved out a 12-foot-tall form of negative Gauss curvature, known as the Costa surface. The Costa surface was discovered in 1983 by the Brazilian mathematician Celso J. Costa [1], who wrote the equations of this surface in his PhD. thesis (see Fig. 1). It belongs to the class of unbounded “minimal” surfaces.

A minimal surface has the property that any small distortion can only increase its surface area. Until the 1980s, mathematicians knew only three examples of unbounded minimal surfaces without self-intersections. One was the trivial plane and the other two were depicted in Fig. 2.
 * 1) cathenoid
 * 2) helicoids



Then Costa proposed equations for a new minimal surface with only one topological hole and three punctures, that is, it was a triply punctured torus. But it was not clear from the equations if Costa’s new surface had self-intersections. This surface enjoyed a non-trivial topology, using for its parameterization Weierstrass elliptic functions: the “pe”, the “zeta” and the “sigma” function. The functions used for the Costa surface pertain to the unit square or the lattice of Gaussian integers or the lemniscate’s case. For more details, see recent versions of Mathematica [2] and [3].



The title of the sculpture of Ferguson’s team (there were 16 teams), was “Invisible Handshake” (Fig. 3), because the empty space around the Costa Surface is topologically similar to the space between two hands that are about to meet.

Helaman Ferguson has already used previous versions of the Costa surface. In 1994, the officials of the Maryland Science Center in Baltimore, USA, invited him to create an exact mathematical ´model of the Costa surface to be exhibited at a fair on Modern Mathematics. Ferguson was very enthusiastic with this idea and he built a 10 feet height model, that children could touch and even slide down it. This was the beginning of sculptures of a family of Costa surfaces, carved in different materials. Costa II, Costa III and Costa IV were carved in aluminum and then cast the result in bronze. Costa V was carved in Carrara marble, Costa VI in Silicon bronze and Costa X was carved in snow for the Snow Sculpture Championship held in 1999.



The key to prove that the Costa surface has no intersections was discovered in 1983 by David Hoffman, Jim Hoffman and Bill Meeks of the University of Massachusetts. It was a success when they were able to visualize the computer generated pictures of the Costa surface (Fig. 4). The big surprise was not only that there were no self-intersections but that the picture possessed a high degree of symmetries. It turned out that the Costa surface was the first of an infinite family of new minimal surfaces, each one with a different number of tunnels, coming into the interior of the form and emerging like big mouths similar to trumpets.

In such a way, the computer visualization opened a door for mathematicians and artists to explore a set of until then unknown geometrical entities. And put in evidence the astonishing structural force of Gauss negative curvature in modeling sculptures. This negative curvature at every point was the key to rendering the Costa surface in snow, Ferguson says in his home-page.

And he added that after the heat wave that hit Breckenridge the second week, when the temperature raised, the only one that retained its structural shape, with walls that got thinner and thinner, was “Invisible Handshake”…

Hyperbolic spaces and tessellations


The hyperbolic plane admits various models and we shall choose from all them “Poincaré model”. In this model the upper half of the complex plane $$H$$ is mapped on the interior of the unitary disk $$D$$
 * $$H=\{z\in C:\mathrm{Im}(z)>0\}\to D=\{z\in C:|z|<1\}.$$

In this hyperbolic geometry, straight lines are half-circumferences whose centers are on the x-axis and as a limit case, they can be normal straight lines to the x-axis. These lines are mapped on $$D$$ as arcs of circumferences normal to the boundary (Fig. 5). The $$x$$-axis is the infinite of the plane. Besides, given a straight line a and an exterior point $$A$$ (Fig. 6), we have two parallels to the line a that connect $$a$$ with the infinite points $$U$$ and $$V$$. It is easy to prove that $$AU = +\infty$$ and $$AV = - \infty$$.



Another interesting property of hyperbolic geometry is that the angle sum of a triangle is always greater than zero and strictly less than 180° (Fig. 7). The consequence of this is that f we consider any hyperbolic tessellation, we will find much more tilings in this space than in the Euclidean plane. Obviously in the Euclidean plane we can tessellate only with triangles, squares or regular hexagons, as well as with topologically equivalent forms. In the hyperbolic universe, instead, we could make a design with so many incident hexagons at a vertex as we want. In example, John F. Rigby has obtained, using a computer program, the tiling  of Fig. 8 and   of Fig. 9 (see Ref. [4]).

Fuchsian groups


A “Fuchsian” group is a discrete group of symmetries described by $$2 \times 2$$ matrices of the form $$\left(\begin{matrix}a&b\\ c&d\end{matrix}\right)$$, where $$a, b, c, d \in R$$. Particularly interesting among Fuchsian groups is the “unimodular group” $$U$$, where $$a, b, c,$$ and $$d$$ are integer numbers and $$ad-bc=1$$.

Let us consider now two samples of hyperbolic tilings, one due to David Epstein [5] depicted in Fig. 10, and another one due to Caroline Series [6], that can be seen in Fig. 11. These tilings on the unitary disk were designed looking for a “fundamental region” in the Fuchsian group $$G$$. A fundamental region is a set $$X$$ such that each transformation in the discrete group $$G$$ carries $$X$$ into a disjoint set $$gX$$ and that each point of the plane is carried into some point in $$X$$ by some element of the group $$G$$. It is easy to prove that polygonal fundamental regions always exist for Fuchsian groups.

Limit sets
Both models of the hyperbolic universe considered, have a natural boundary: the real axis bounds the upper half plane and the unit circle bounds the disk $$D$$. The boundary does not belong to the space: if we look at the tilings of Figs. 9 and 10 we see that the tiles get smaller and smaller as we approach the boundary at infinity. The “limit set” $$L$$ of the Fuchsian group $$G$$ is defined to be the set of points at infinity which are limit points of an orbit. $$L$$ is independent of the particular orbit chosen and there exist two possibilities
 * 1) $$L$$ is the whole circle (see Fig. 10)
 * 2) $$L$$ is a set of zero Lebesgue measure (with holes) like in Fig. 11.

In other words, it is the self-similar fractal obtained by intersecting the smaller and smaller arcs inside $$D$$. These last sorts of tilings have been the base for a deep research about the fractal characteristics of the frontiers of chaos (see [7]).

Hyperbolic tiling


Undoubtedly, the first artist who was dazzled with the infinite possibilities of hyperbolic tilings was the Dutch graphic Maurits C. Escher. G. S. M. Coxeter met Escher in 1954, at the World Congress of Mathematicians, where some of Escher works were exhibited. And this was the beginning of a fruitful interchange that ended when Escher died in 1972. Coxeter sent Escher his paper about Symmetry and this paper inspired Escher the design of four marvelous pictures that are called “Circle Limit I, II, II and IV” and can be seen at Figs. 12, 13, 14 and 15.



These hyperbolic tilings helped in the teaching of Hyperbolic Geometry that was not a familiar subject at that time. For example, Douglas J. Dunham of the Computer Science Department at the University of Minnesota, Duluth, USA, generated a computer program to transform repeating Islamic patterns in hyperbolic geometry [8]. A beautiful example of the use of such a program can be seen at Fig. 16. At the left, there is an Islamic pattern with symmetry group p6m [6,3]. With a slight distortion this pattern is deformed at the right into the hyperbolic hexagon arabesque with symmetry group [6,4].



Last year, Dunham at Granada, Spain (ISAMA-Bridges 2003) presented several hyperbolic spiral patterns. For example, he showed at the left of Fig. 17 a celtic spiral pattern and at the right, a hyperbolic version of this configuration (see [9]). And he even announced that in the near future, he will design hyperbolic versions of Robert Krawczyk’ s well known Spiro laterals [10].



Finally, it is interesting to mention the hyperbolic sculptures of the artist and Art historian Irene Rousseau [11]. Her sculptures use mosaic-tilling patterns and geometry to express the concepts of symmetry, infinity, 3D space and time (see Fig. 18). Yet, as visual art objects, they stand on their own merit.

Fractal Art and coloring algorithms


An object is called a “fractal” when its border, its surface or its internal structure shows a self. similar constitution. The name fractal was introduced by the polish mathematician Benoit B. Mandelbrot, who was born in 1924 at Warsaw, Poland. They appeared when studying nonlinear iterative equations, even in such simple cases as the following one
 * $$z \to z^2+c \ \ \ \ (2)$$

where $$z$$ and $$c$$ are complex numbers. After a first period during the ´80s of creative production of fractals, the mathematical formulas did not produce at the screen of the computer, relevant new sets. Then, a group of mathematicians began to consider the possibilities of interpreting the old formulas through color algorithms.

Each fractal formula produces at the complex plane for each point, an “orbit” $$z_0, z_1, z_2, \ldots, z_n$$ and the question was: which initial values would produce a sequence that tends to infinite and which not? The interface between these two sets is called a “Julia set” (in honor of the French mathematician Gaston Julia, who was already studying the iteration of complex polynomials at the beginning of the XX century). Both sets are either connected or not. Mandelbrot’s set (called Apple-man), obtained with the quadratic transformation (2) is the set of all c for which the Julia set is connected. It is always colored in black and the different colors indicate how fast the iterated grow (see Fig. 19).

The color algorithms interpret this sequence of points and the use of color algorithms allows the introduction of distortions, forms and designs of great originality.

The escape-time algorithm was one of the earliest algorithms, It was based on the number of iterations necessary to determine whether the orbit sequence tends to infinity or not. In 1992, Peitgen et al. [12] proved that when the orbit exceeds a border region R, it diverges to infinity. Stopping as soon as $$z_n$$ is outside $$R$$, the coloring value is the length $$n$$ of the sequence. Normally, for Mandelbrot’s set depicted at Fig. 19, $$R$$ is selected as a circle centered at the origin with radius 2.

One of the main difficulties of the escape-time algorithm is that it is discrete and produces a band effect called “tiger striping”, similar to the contour lines of a topographic map. But this condition can be avoided using, for example, in the case of the “normalized iteration count algorithm”, the following expression
 * $$d=n+1-\frac{\log(\log |z_n|)}{\log 2} \ \ \ \ (3)$$

to evaluate the distance of every point to the fractal set. In such a case, all discrete images can be converted into continuous ones, preserving the original fractal configuration.

Orbit trap


This is one of the largest sets of coloring algorithms because it provides an enormous variety of options. The idea is to choose a region $$T$$ of the complex plane and analyze the relationship between the $$z_n$$ values and $$T$$, stopping the iteration for the $$z_n$$ that fell inside the trap and coloring based on the distance to $$T$$. Mathematicians and programmers began with regions like triangles, ellipses, astroids or rectangles situated at different zones of the complex plane and the figures obtained were astonishing!



By using non-Euclidean distance measures, contours of equal distance were transformed into other shapes. Obviously, these trap shapes could be distorted by rotation, skewing and stretching, even incrementally with each iteration.



But his was again a discrete method. To avoid this circumstance, many variations were created (see for example [13], [14]). These variations consisted in combining all distances below the threshold together or considering the magnitude or angle of the vector  or some combination of $$z_n$$ values related to the trap distances. For example, Javier Barrallo Calonge and Vera W. de Spinadel [15], have obtained the following beautiful designs in black and white (Figs. 20, 21, 22, 23, 24, 25, 26, 27 and 28). All of them correspond to Julia sets of the transformation $$z \to z^2+c$$, where a special technique has been used to get white and black images.







More exotic designs obtained by the same authors can be seen at the colored Figs. 29, 30, 31, 32, 33 and 34. Fig. 29 is one of the typical bulbs of Mandelbrot´s set, converted in a triptych by the use of different color algorithms. At the left, the orbit trap is a circle of unitary diameter with center at the origin and a small tolerance. At the center, an orbit trap array with Gaussian integers is applied. At the right, a beautiful five points star was used as an orbit trap. Fig. 30 shows the filaments of the bulbs of the Mandelbrot´s set using an astroid as an orbit trap. Fig. 31 is another variant of Mandelbrot´s set.





Finally, Figs. 32, 33 and 34 are Julia sets of
 * $$z=z^3+cz^2-z$$

for $$c=0.012-0.44i$$. The three images use the same numerical values but the number and geometrical properties of the orbit traps were different for each image. Several superposed layers have been used, taking as orbit traps astroids, diamonds, hyperbolas, ellipses and the coordinate axis. The selection of the placement of these orbit traps, followed aesthetical criteria.

Undoubtedly, these figures are simultaneously, fascinating artistic images.

Links

 * "Intersections of mathematics and arts" at Vikiznanie (in English)