Algorithmic Mathematical Art ₃

M C Escher

M C Escher (1898 〜 1972) is a Dutch artist whose works are predominated by mathematical themes, many of which are purely algorithmic. Although, Escher is a traditional artist; his algorithmic arts are done manually, not by a program from computer, nor with a explicit algorithmic specification.

It is easy to go from algorithms to visual images, but given a algorithmic artwork, it is difficult to extract the algorithm in the form of a precise runnable program. The following two works by M C Escher illustrates this:

butterfly by M C Escher
“butterfly”, by M C Escher. This image is clearly purely algorithmic. However, it is not trivial to extract the algorithm involved and turn it into a working program. From a pure minimalist point of view, one should be able to start with half a image of a butterfly, and apply a algorithm recursively to get the final image. Note that the butterfly's wings are colored in such a way that mutually orthogonal circles are formed, and this must be encoded in some way into the algorithm. There are different algorithmic approaches. One can do it by describing pen paths as in Turtle Graphics, or by recursive replacement as in L-Systems. Perhaps a least interesting way is by a mapping process with some mathematical model such as hyperbolic geometry on a rotational-symmetric tiling. Once a algorithm is obtained, one should be able to apply it to various motifs, or tweak parameters to get different effects.
Whirl Pool by M C Escher
“Whirlpools”, by M C Escher. This image is also clearly algorithmic. It consists of a strip of a repeated motif (the fish), wrapped into a double spiral. To generate this image algorithmically, one can start with a motif, reflect and glide-reflect to get a long strip, then use a function that map the strip into a spiral, then mirror to get the other spiral. (or use a function that maps a strip into such double-spirals) Finding the function may not be easy. I imagine to get it right it might take several days tweaking it with functional programing languages such as Mathematica, Lisp, Haskell. Perhaps a more elegant but even more difficult approach is to describe it recursively, which would also deal better with singularities at the center of each spiral.
Escher's metamorphosis. This theme of aglorithmtically transforming a tiling is particular type of algorithmic mathematical art. It has been studied notably by Douglas Hofstadter.
Escher's snakes
Another Escher's algorithmic mathematica art: “Snakes”. This artwork is made by inter-linked circles of various sizes to form a weaving, in a specific geometric layout inside a circle. The snake animal drawings are mere decorations that adds spice to the artwork.

To generate this work algorithmically, one needs to encode the knot process. (i.e. when there is a intersection between 2 circles, which one goes on top.). One also needs to encode the geometric layout, namely, the circle's positions and sizes. Note that circles are smallers near the center, but also gets smaller near the edge. This size change is not linear. Also, note that the circles have 3 different colors. The coloring basically cycle thru as they are spread toward the edge. Also, each of the circle ring has a slant (a thin side), either facing the center or facing the edge.

(M C Escher's artworks are copyrighted by M.C. Escher Company B.V.)

What is Algorithmic Mathematical Art

A visual artwork is mathematical if it appeals to mathematicians and the graphics itself encodes a mathematical structure. For example, M C Escher's tiling works are mathematical art, but some of his other works such as Ants Crawling On A Mobius Strip, Illusions, Day And Night, Forever Loop Stairs, are works of art illustrating mathematical ideas, but not the mathematical art discussed here.

A visual artwork is algorithmic if it is recursive, symmetric, or embodies a mathematical equation (such as curves and surfaces). For example, M C Escher's “Butterfly” and “Whirlpools” and his tiling works are algorithmic. All illustrations in this article are algorithmic. Algorithmic art exhibits recursion or symmetry (including quasi-symmetric ones like that of Penrose Tiling ). However, artwork generated by computer programs are not necessarily algorithmic. For example, ray-traced computer generated sceneries, or digitally retouched fractal artworks, are not considered algorithmic, in our context.

The Algorithm in Algorithmic Mathematical Artwork

Algorithmic artwork might not be generated by a computer program. M C Escher's artwork are examples of algorithmic artwork not generated by computer programs. Computer generated algorithmic artworks might not be generated algorithmically either. Here, “algorithmically generated” means a program that distills the inherent algorithmic nature in a artwork. For example, many tilings and patterns artwork today are spitted out by computer programs, but the programs are written in a ad hoc, case-by-case basis, consisting of drawing commands tweaked to match a desired output. Although generated by a program, but the essence of creation is manual. What we really want are programs in a form that embody, capture, or distill the algorithmic nature of the artwork, as a executable specification by means of recursion or symmetry code. The beauty of algorithmic artwork lies in its inherent beauty of algorithmic pattern or symmetry, and its creation process should be done that way. When this is achieved, and its algorithmic essence is captured, and we can then create vast numbers of variations by changing parameters or input. (This does not imply that results will be similar, as we know from chaos theory.). Generating a algorithmic mathematical art by a algorithmic process can be likened to specifying a Sequence by Recursion , or a group by generators and relations , or modeling a physical phenomen by a equation or cellular automata. It is a pursuit of elegance that captures essence, and gives us a precise insight on relations.

Related Websites and References:

Here are some list of people i know who are doing algorithmic art, mathematical art, or whose artwork are heavily math related:

The following is a annotated list of books that have influenced my outlook one way or another.

Books on Islamic geometric patterns.

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