Xah Lee, 2004-01, 2006-08, 2008
With the inception of computers, there began a movement of computer generated mathematical art. In the early 1990s, they were merely visualization aids in the study of mathematics. Gradually, the complexity and artistry of the images becomes a end itself.
In this exposition, i try to examine the various methods of algorithmic mathematical art, and indicate the various states of the art and possibilities. At the end, i give a definition of Algorithmic Mathematical Art.
above: “Borg cube”, a plot of the equation: Sin[x*y]+Sin[y*z]+Sin[z*x]==0.
above: A section of Riemann's Surface. A minimal surface from Richard Palais's 3D-XplorMath ↗ software.
Plots of arbitrary 3D surfaces for visual arts purposes has not been explored much. The exception is visualization of some special mathematical surfaces, especially in differential geometry. (for example: Klein bottle, minimum surfaces.)
As of today, it is scarcely known how surfaces look like of arbitrary equations of 3 variables. Equations with more than 3 variables can be projected or sliced to 3 dimensional space, and this is completely unexplored.
above: A dodecahedron formed by pentagonal rings, by Michael Trott. (http://www.graphica.com/gallery/).
In 2 dimensions we have regular polygons, such as square and pentagon and hexagon. The 3-dimensional version is called regular polyhedron. For example: cube, octahedron, icosahedron. The general name for such objects in any dimension is called polytope.
Polyhedrons↗ as artistic themes have been explored greatly, way back in the Greek times and in different cultures, often exhibited as toys, sculptures, paper models (origami), or hanging decoration. In modern times, polyhedrons have been exhibited as architecture element as well. (Examples: the Disney's geodesic dome, Eden Project hemi-spheres, Atomium, Biosphere2↗ )
above: left: Paper models by Robert Webb↗. Above right: A Islamic lamp in the shape of a stellated polyhedron. (at the entrance of Aladdin hotel resort in Las Vegas).
Computer models as visual arts have been very popular. However, most of them are direct rendering of the mathematical object with almost no artistic value. Commonly seen are polyhedrons rendered with latest ray-tracing computer graphics advances, with some stellation or truncation. Highly artist algorithmic exploration of polytopes is almost non-existent. The image by Michael Trott above, of a dodecahedron rendered as linked pentagonal rings, is a basic artistic exploration.
For some examples of artistic algorithmic exploration, one can systematically examine ways to render a space by linked rings. For the centrally symmetric dodacahedron, for example, by some cleverness in linking, the links can extend to infinity as to fill the whole space with these links, and a cut-away view can be made to result in a artistic image. Or, polygonal tubes can be made to have larger radius, so that each polygonal “side” of the tube is a large flat area that a Maze↗ can be incribed upon. The number of sides of the ring can increase or decrease according to certain parameters, such as symmetry location or distance from origin. Or, one can use spheres instead of rings. The coloring of balls around the dodecahedron can be exploited systematically, so to result in different symmetry groups of different groups of balls. Imagine a 3-D space with a cluster of differently colored balls that exhibits subtly different symmetry groups as in an elaborate intricate tiling work. Or, imagine 3-D space tiled uniformly by regular polyhedrons, with some walls open, as to form a 3-dimensional maze. If well done, a cut-away view or translucent view of this would be fantastic. A walk-thru as implemented in video games also has great potential. Solid stacks of cubes is called Polycube↗ (a 3-dimensional version of polymino↗). Solid block sculptures with other regular polyhedron are basically unexplored. Block of connected tetrahedron are particularly interesting.
Below are some examples of polytopes:
above: “fiery”: plots of the equation Sin[x*Sin[y]]-Cos[y*Cos[x]]==0.
above: A density plot version.
Two-dimensional plots for visual art purposes have not been explored much. Algebraic curves of more than 4 degrees are almost unexplored. One could create a program that systematically generates and plots all possible equations by degree or type, including non-algebraic ones.
Almost all ornamental elements in architecture or interior decorations are based in geometry, and most of them based on curves. Examples of curve based traditional art include: dome, arch, vault↗ (based on arcs of circle), volutes↗ (as scrolls on columns or violins) (based on various spirals), curlicue (e.g. iron wrought filigree found on gates) (based on Cornu spiral↗, lituus), toy Spirograph↗ (based on roses, cycloidal curves).
Methods of artwork based on curves can include: tangents, envelope, caustics (String art); osculating circles and inversion, evolute/involute, pedal curves, parallel curves, pursuit curves.
above: Nested inversion of circles. Nested circles can result in many esthetic images.
above: Stereographic projection of a circles on a sphere to the plane (produced by software KriviznaPlus↗ by Viktor Massalogin).
above: left: Nested inversion of circles (by William Gilbert↗). Above right: Circles generated by Mobius transform, by Ed Pegg Jr in his Math Games column↗ (March 15, 2004).
Traditional artworks based on geometric process include that of circles of tangency ( Appollonius circles↗ ), various plane transformations applied to a grid or regular tilings (linear, affine, projective, geometric inversion, mobiüs transformation, stereographic projection...). A particular example in recreation is anamorphosis↗, which is a drawing that is not recognizable until its reflection is viewed on a shiny cylinder or metal ball. (See artwork “Trash Reflection”)
above: A wallpaper pattern with star motif, cut into a diamond outline, then a fish-eye lens transformation is applied.
For some images of traditional geometric processes, see:
above: “mushroom triangle”, generated by recursive line-replacement
above: A snowflake generated by recursive line-replacement. (from An eye for Fractals, Michael McGuire. 1991. p.16.)
above: A “Pinwheel tiling”: recursive dissection of a right triangle with sides 1, 2*Sqrt[5]. (Two photos of the Federation Square↗ , Melbourne, Australia, where the entire building's flat surfaces are rendered as pinwheel tiling: photo 1;photo 2)
L-System↗ is a recursive string replacement system originally devised to model plant growth. It is often used to generate self-similar images by interpreting the string as drawing commands or geometric objects. Turtle Graphics is from the programing language Logo↗, which algorithmically controls the movement of a pen, by specifying directions and pen down or up.
L-System and Turtle Graphics have been somewhat popular among recreational programers. However, there have not been serious studies of visual art possibilities with these methods. Commonly found are illustration of famous plane-filling curves↗ in mathematics, plant-growth modeling, or simplistic symmetric drawings for children as a demo of Logo. Few of them are ingenious.
Many functions in mathematics can be visualized as a plot. For curves and surfaces, the plotting scheme is simple. Often just on a 2D grid or 3D grid, with marked axes as coordinate. For other functions, such as vector valued functions, complex valued functions, mathematicians have developed other schemes to visualize these functions. Here are some examples.
above: “The real part of the discriminant as a function of the nome q on the unit disk”, by Linas Vepstas, 2005. (gallery↗)
above: A plot of the complex-valued function (z-2)^2(z+1-2*i)(z+2+2*i)/z^3. By Hans Lundmark, 2004. (gallery↗)
