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. 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.“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.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.
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:
Sandor Kabai has created many fantastic algorithmic mathematical art. He is one of the most productive on the subject. His gallery is at:
http://www.kabai.hu/
Russell Towle. He does mostly polytope and zonotope studies of higher dimensions:
polytope,
zonohedra.
Douglas Hofstadter. He is most famous for his
Godel, Escher, Bach, Pulitzer-prize winning book. He has long been interested in AI and its relation to art. He has published article in Scientific American on Parquet Deformations. That is, the gradual transformation from one tilings to another in a strip. I don't know if he has done much artwork per se as discussed here.
Ed Pegg Jr's, a recreational mathematician. Creator of the site mathpuzzle.com. The site contains a huge amount of high-quality mathematical illustrations. However, most of them are not done programmatically. Most of them are simple illustrations, but some are artistic. For example, his “Mitre System” tilings:
http://www.mathpuzzle.com/Mitre2.html
The following is a annotated list of books that have influenced my outlook one way or another.
Tiling and Patterns (1987) By B Grunbaum and G C Shephard.
amazon
The most authoritive and informative opus on the subject. Special thanks to professor B Grunbaum for sending me his various publications on the subject.
A New Kind Of Science (2002) By Stephen Wolfram.
amazon
Aspects on Cellular Automata and its beauty.
The Algorithmic Beauty of Sea Shells By Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R Fowler.
amazon
There is also a similar book on plants.
Visions of Symmetry (1990) By Doris Schattschneider.
amazon
A coffee-table book of M C Escher's tiling work. Most beautiful.
Books on Islamic geometric patterns.
Islamic Art and Architecture from Isfahan to Taj Mahal (2002) By Thames and Hudson.
amazon
The Art of Islam (UNESCO Collection of Representative Works: Art Album Series) (1992) By Nurhan Atasoy, Afif Bahnassi, Michael Rogers. (ISBN: 2080135104)
amazon
Excellent, squarish large book. Mostly just photos with captions. Covers mostly architectures, and wall decorations.
Splendors of Islam “Architecture, Decorations and design” (2000) By Dominique Clevenot and Gerard Degeorge.
amazon
Coffee-table book. Contains the most beautiful photos of Islamic architecture and decorations.
The Art and Architecture of Islam 1250 〜 1800 By by Sheila S Blair, Jonathan M Bloom.
amazon
Largish Coffee-table book. Lots of text with lots of photos.
Islamic arts By by Sheila S Blair, Jonathan M Bloom. Not a coffee-table book. Scholarly book, with mainly text with excellent photo or image about other page. Color all types of artifacts. Very good.
Symmetries of Islamic Geometrical Patterns (1995) By Syed Jan Abas, Amer Shaker Salman.
amazon
Contains black and white drawings of about 250 Islamic wallpaper patterns. Perhaps the most complete collection in print. The author Syed Jan Abas has a home page at:
http://www.bangor.ac.uk/~mas009/islampat.htm.
