The main attraction of Celtic knot patterns is their geometric design, but the topological aspect (the “knot”) is also interesting mathematically. One can survey traditional designs to see whether they are really knots, or braids of how many strands, and how they are knotted. In so far as I know, this has not been studied much. There is little systematic study of algorithmic knot/weaving-pattern generations. (There have been systematic mathematical studies of weavings/fabrics by B Grünbaum and G C Shephard).
There are many varieties of pattern designs by different cultures. The decorative Celtic knots and Islamic geometric tilings are particularly mathematical. Although there are great number of existing designs throughout the world and history, but with respect to mathematical or algorithmic classifications, there are not many varieties. This is not surprising when we consider that these artistic patterns are developed culturally by artisans, not as a result of mathematical analysis of plane symmetry or systematic algorithmic exploration.
For plane patterns, it is known since the last century that there are 17 types of symmetries. According to Branko Grünbaum et al [Symmetry in Moorish and Other Ornaments, Comp. and Maths. with appls., Vol. 12B, Nos. 3/4. pp. 641-653, 1986], not all types are found in Alhambra, contrary to popular belief. The mathematical understanding of tilings and patterns is relatively poor even today. It is essentially a unexplored area. The mathematical analysis of weavings is almost completely untouched. (Weaving, as the mathematical content of Celtic knots and Islamic patterns. It can be considered topologically as knots and braids, or as certain uniform lattice network with marked nodes and edges with a geometric layout.)
Penrose tiling is a aperiodic tiling. The discovery of Penrose tiling was a break-thru in mathematics and crystallography with great ramifications (⁖ material science). The beauty of aperiodic tilings is that their orderliness is subtle. Such type of beauty was not possible in the past.
Modern math knowledge adds incredible amount of new possibilities at decorative designs. The math understanding of symmetry, and the discovery of aperiodic tilings, and also designs based on hyperbolic geometry, were all not possible before.
More example of tilings:
Fractals are plots of mathematical objects. They began as a visual aid of a mathematical process. Essentially, each point in the plane is colored according to how that value (x,y) behaves when fed into a formula recursively.
Fractals, by its very nature of garishness and bizarreness, has been extremely popular among computer artists — even those who are not interested in mathematics or computer science. Fractals as a visual art have been explored extensively, and great many varieties and galleries have been created on the web.
The exploration of fractals as a visual art form has somewhat limited expression. Artistic creativity is limited to concocting equations and coloring schemes. Some people have started to mix computer generated fractals with manual manipulations, such as mixing in digitally modified photos. Such artwork ceases being algorithmic or mathematical.
For more fractal images, please see:
As we understand higher dimensional spaces, it opens a great gate of algorithmic art thru the process of projection or slicing of higher dimensional objects to 3-dimensional space or the plane.
Non-Euclidean geometry and higher dimensional geometry and topology are understood by only a few mathematicians in the world. (perhaps a few thousand people, or less if we are specific in the field) Consequently, artistic exploits thru their understanding are basically non-existent. Almost all ideas discussed in this page can be thought about in higher dimensions and or non-Euclidean space. In higher dimensions, there are totally new concepts that are non-existent in lower dimensions. (orientability, embedding, isotropism, …) As math and technology march on, we may see more visual art explorations thru higher dimensions.
As a example, one idea of creating tilings and patterns is to slice thru some regular lattices of some higher-dimensional manifold. And, we may ask whether it is possible to encode some higher-dimensional manifold properties (such as orientability or the geometry it admits) by a visual exhibition. (i do not quite understand this, but for example, we can illustrate the angle-invariant property of inversion by showing the before and after images of a rectangular grid. We can also show linear and affine transformations by the way they look when applied to a grid. We can show properties of projective transformation and Conic Sections by shadowing. (think of beautiful Stained glass in great Cathedrals) We can show homeomorphism or continuous transformation by animation, especially on tilings. We can color surfaces by their curvature, and draw gridlines along constant curvature lines, and show invariance of curvature such as the helicoid-catenoid surface family.
Evolutionary programing is a emerging method to generate visual art. The method is often done as genetic programing with human arbiter for judging the survival of the fittest. However, most of the investigations done so far over the 1990s tend to be insults to eye with the excuse of being abstract “art”. They are not mathematical.
Cellular Automata (CA) methods can also be employed for mathematical visual art. Stephen Wolfram's book A New Kind of Science, for example, contains many pleasing images of cellular automata, although not for the express purpose of visual art. For example, it is frequently cited that the patterns on sea shells are results of cellular automata. (seashell photo) This is a example of using CA for non-mathematical art. (That is, the image does not contain some inherent math structure or appeal.). However, CA probably can be exploited for mathematical art. For example, consider the great varieties of seashell shapes. (See: Mathematics of Seashell Shapes) Some has horns, spikes, ribs. Some are long, some are flat, and there are many types of overall spiraling structure. Seashells are grown from the calcium deposits secreted by the mollusks, and over the years many different shapes emerged thru evolution. Therefore, it is conceivable that a 3-D cellular automata can be set up with genetic programing to simulate the evolutionary process, as to obtain the many shapes of seashells as pure art of mathematical spiraling surfaces.