Algorithmic Mathematical Art, page 1
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.
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.
- Math Surface Gallery. A surface gallery of about 60 surfaces.
Regular Solids of 3 or more dimensions
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 )
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:
- Sandor Kabai has created many fantastic algorithmic mathematical art, many are themed around regular polyhedrons. His gallery is at: http://www.kabai.hu/
- 〈Rob's Polyhedron Models〉 By Robert Webb. @ http://www.software3d.com/MyModels.php
- Beautiful intricate physical models of polyhedrons by Ulrich Mikloweit. @ http://www.polyedergarten.de/e_polyseite1.htm (it would be great if these can be generated on a computer algorithmically.)
- Russell Towle's studies: 〈polytopes〉 @
http://home.inreach.com/rtowle/Polytopes/polytope.html• 〈zonohedra〉 @ http://home.inreach.com/rtowle/Zonohedra/zonohedra.html
- 〈HyperSpace Polytope Slicer Java applet〉 (2000) By Mark Newbold. @ http://dogfeathers.com/java/hyperslice.html
- 〈Hyperspace Star Polytope Slicer〉 (2000) By Mark Newbold. @ http://dogfeathers.com/java/hyperstar.html
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 (based on Cornu spiral, lituus), toy Spirograph (based on roses, cycloidal curves).
Plane Geometry and Processes
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”)
For some images of traditional geometric processes, see:
- Inversion of circles gallery
- Transformation of the Plane • Transformation of the Plane II
- Stereographic Projected Beauties
- 〈Anamorphic Projections〉 (1989). By Paul Bourke. @
L-Systems, Turtle Graphics
L-System is a recursive symbol-sequence replacement system originally devised to model plant growth. It is often used to generate self-similar images by interpreting the symbol-sequence 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.
〔Two-dimensional L-systems By Robert M Dickau. @ http://robertdickau.com/lsys2d.html〕
Plotting of Functions and Processes
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.