Mathematical Masterpieces: Making Art From Equations

Mathematicians enthralled with unending fractals and flux patterns have been known to call math beautiful — but, increasingly, they aren't the only ones. For many artists, calculations and numerical analyses provide a rich source of ideas and methods for their creations.

The annual Bridges conference showcases the connections between art and mathematics. The conference features, among other things, a juried art exhibition full of a staggering range of mathematically-inspired artworks, where you can see sunsets, lampshades, and more examples of the golden ratio than you can shake a shell at.

Here are some of our favorites from the 2013 entrants.

This visualization is based on the foraging behavior of the seed harvesting ant P. barbatus, a species which does not use pheromone trails for finding food.

Here, 1000 ants stream out of the nest along six patroller trails and then intermittently break off to initiate random searches for seeds. When seeds are encountered ants collect them and return directly to the nest. The simulation lasts for only 500 time steps.

In nature, typically 1800 ants are foraging at any one time, the patroller trails are not uniform in length or uniformly separated, and foraging continues for several thousand time steps — thus obscuring the foraging patterns and structure we are able to observe and recover using our more modest setup. Color gradations are used to disambiguate the search phase from the return phase and reinforce the dynamic aspects of the process. 

Bunny-Squared

This is a self-referential bunny — a sculpture of a bunny, the surface of which is tiled by 72 copies of the word "Bunny."

This piece is part of a larger series of "autologlyphs," following on from HS's "Sphere Autologlyph" from the 2010 Bridges art exhibition. An autologlyph is a word written or represented in a way which is described by the word itself. This style of autologlyph combines Escher-style tessellation with typographical ideas related to ambigrams.

The bunny was created using a technique published by CSK for transferring a symmetric design to a suitably parameterized mesh surface. We modified the technique to require a quarter as many copies of the fundamental domain as compared with the original version. This allowed us to send a smaller (and more affordable) model to the 3D printer. The design of the word "Bunny" was produced using Adobe Illustrator, then thickened in 2D, triangulated, mapped to the 3D surface, and extruded into a thin shell for manufacture.

Seeing Sounds

Musical Flocks is a project in the field of music visualization. It produces animations by simulating the behavior of agents that react to the sound of music. Swarm-like behavior is attained by following rules of separation, alignment and cohesion. This process produces reactive animations and static artifacts that constitute representations of the pieces.

Slow music makes the flock react gently and move slowly, while a high tempo results in fast movement and abrupt changes. Sounds with high volume and rich frequency spectrum affect the majority of the boids, while low volume level and less quantity of active frequencies produces subtle visual variations and a slower graphic evolution.

This series is composed of three visualizations of the piece "Five Armies" by Kevin MacLeod. This musical piece with an epic undertone features a wide variety of instruments (cellos, French horns, trombones, violins, marimba, trumpets, percussion, oboes, clarinets, flutes, basses). The escalating ending provokes large perturbations of the flock leaving a distinct signature in the static and animated pieces.

Star Chart

Stars of the Mind's Sky is the title of a series of works exploring the space of regular star polygons.

Here we see 300 stars "in orbit" along concentric circles. The number of points on a star increases with the radius, and stars of a given number of points are spaced evenly along their circle according to "density," or the "jump number" used in generating them.

Algebraically, these represent the subgroups and cosets generated by elements of a cyclic group. They have been colored on a gradient to indicate the number of cosets; a red star signifies a generating element. As a consequence of these structural choices, we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.

Braiding Equations

This scarf depicts the Yang-Baxter equation of statistical mechanics, a variation of which is the braid equation in algebra, or the 3rd Reidemeister (equivalence) move in knot theory. Assigning the numbers 1, 2, 3 to the colors blue, green, gold, respectively, the Yang-Baxter equation reads: R12R13 R23 = R23 R13 R12, where Rij denotes strand i crossing strand j — the two sides of the equation being depicted on the two ends of the scarf, with equality represented by the middle portion.

For the equation to read correctly, the fabric needed to be double-sided. To create the two-sided fabric and traveling cables simultaneously, the artist introduced a hybrid of double-knitting and tubular knitting. As a result of this technique, the colored strands are produced as embedded I-cords in the brown body of the scarf.

Fractal Fringe

This digital artwork features four views of the same three-dimensional object, a fractal tiling in which every tile is a similar dart shape.

The starting point is a pair of tiles matched along one long edge. A pair of smaller tiles is fit into each of the V-shaped openings in these starting tiles.

The sum of the angles at each vertex where two of the smaller tiles meet a larger tile is greater than 360 degrees, so the three tiles cannot lie in the same plane. The pairs of smaller tiles alternately buckle up and down to accommodate the fit.

This same simple rule is applied repeatedly through ten generations, resulting in the object shown. This object was created in Mathematica, and PhotoShop was used to create the montage. This is a demonstration of the way in which a complex organic structure can result from a simple set of rules being applied over and over again to a simple starting structure. The resulting object is an intriguing blend of the organic and the geometric.

Exploring Symmetry

This work is part of a series of visual meditations on the structure of the alternating group on 5 elements, also known as the icosahedral group.

This group is the smallest non-abelian simple group, and it characterizes the orientation-preserving symmetries of the regular icosahedron and dodecahedron. It also has interesting historical significance as one of the first groups to be studied abstractly, through its connection with the theory of quintic equations.

This image explores the structure of the icosahedral group through a particular presentation by two generators. The group's elements, which appear as yellow disks in this image, are arranged at the vertices of a truncated icosahedron, shown here in stereographic projection, while the group's generators, of orders 2 and 5, correspond to the regions between the disks, colored red and blue, respectively. The image is composed of multiple hand-drawn images which are digitally composited and output as an archival digital print.

Weaving Webs

Artist Annie Verhoeven has developed a very unusual technique to make her artwork, which she has termed “thread art.”

The foundations of this thread-art technique are found in the traditional craft of lace-making. But the finished products are far from traditional: by completely reinventing the old methods and letting go of some of the traditions, lace no longer functions as just a piece of two-dimensional fabric, but becomes art in an enormous variation of sizes, shapes and dimensions.

Mathematics are an integral part of these artworks. It is visible in the finished work; from the way different sections relate to one another to in the proportions of the thread and open space in the work.

“Touch-graph” visualizes a computer program used to show connections between people. The differently colored lines symbolize the different types of connections between people, such as contacts used to consult or, symbolized by another color, the contacts used for discussion. All these different connections are used in our life on every level, be it scientific, religious, political or otherwise, and can help us determine where we stand in the world.