Art Coursework Year 10 Maths

Mathematician Robert Bosch never intended to become an artist. But 15 years ago, he started looking for ways to engage his students in optimization research, his field of expertise.

Robert Bosch's optimization artworks have included a rendering of the Mona Lisa, which he starts by identifying the points or “cities” (Left) through which the Traveling Salesman route will run (Right). Original image courtesy of Shutterstock/Oleg Golovnev and modifications by Robert Bosch.

Optimization is the mathematical quest for the best way to do something, from finding the shortest distance between two places to figuring out the best way to pack a suitcase. It often involves calculating the highest or lowest value of something. The applications are far-reaching. To Bosch, they also offered a pleasing aesthetic. “I wanted to convince my students that this material I teach is beautiful and incredibly applicable,” says Bosch, who teaches at Oberlin College in Ohio. “My mission was to show them that pretty much any field you could think of has optimization applications.”

So Bosch went looking for examples in areas that seem as far from mathematics as one can get. He settled on visual art. There may not be any obvious overlap between the two pursuits, but Bosch figured that if he could show his students how optimization methods could produce art, then maybe he could convince them the field is applicable almost anywhere. “It became an obsession,” he says.

And it paid off: Bosch is now known among mathematicians—and the math–art subset of that community—for his line drawings, mosaics, and sculptures created using solutions to optimization problems. He’s also organized showings of mathematical art at galleries and conferences, and has given lectures to audiences ranging from elementary school children to the mathematically curious at the Museum of Math in New York.

Finding the Path

Most of Bosch’s artwork is derived from the Traveling Salesman Problem (TSP), the best known and most thoroughly studied example in optimization research. Like all good problems, it’s easy to state and difficult—perhaps impossible—to solve. Imagine a traveler who has to visit each city on a given list only once, returning to the starting point at the end. What’s the shortest route?

If you marked targeted cities on a map, they’d look like a dot-to-dot puzzle that asks you to draw the shortest possible total line and without the numbers provided. For a short list of cities, the TSP is easy to solve: simply measure all possible paths and pick the shortest. As the list grows, so does the difficulty. Finding the shortest journey through just 10 cities requires comparing hundreds of thousands of possibilities; that pick-the-shortest-trip strategy becomes untenable when the city list reaches into the hundreds or thousands. Mathematicians can’t solve the problem in a reasonable amount of time for any given number of cities, and they don’t know such a strategy could ever be made. Mathematicians can, however, check that one solution is better than another; and pretty good solutions suffice in most fields.

The TSP might give mathophobes a headache; to Bosch, it leads to artistic inspiration. To create his pictures, Bosch uses a computer program to convert a drawing (or painting or photograph) into a scattering of dots. Densely packed black dots might be used to represent darker areas of a drawing, whereas lighter areas have dots space farther apart. Gray hues are achieved with a technique called half-toning. Bosch has worked with collaborators—first his student Adrianne Herman, and later Craig Kaplan, a computer graphics expert at the University of Waterloo, in Canada—to refine this approach.

Then comes the math. For that, Bosch uses an algorithm that traces an optimal, nonoverlapping TSP path through the dots. The algorithm doesn’t necessarily find the best route, but it finds a pretty good one. The line segments complete the art, and Bosch’s goal is that the line segments draw a convincing representation of the original. As a rule, geometric optimal TSP routes never cross themselves, which means that Bosch’s pictures comprise a single line that ends where it started, in what mathematicians call a simple closed curve.

One of Bosch’s first pieces was a TSP-derived replica of Leonardo da Vinci’s Mona Lisa. Bosch has also reproduced works by Andy Warhol, including the pop artist’s self-portrait and paintings of a Campbell’s soup can. Bosch is currently challenging himself to create the same quality of TSP art, but using fewer and fewer dots, as a way to do more with less, something like a music composer who limits himself to only a few notes. He wants to create art that’s less reliant on fast computers and more reliant on mathematical modeling. That desire, Bosch says, is universal. “What’s the best we can do with the constraints we have?” he asks. “It’s a fundamental human problem.”

Mathematician Henry Segerman, of Oklahoma State University in Stillwater, says art gives people a reason to engage with the math, especially if they typically shy away from the subject. An artistic objective provides a tangible goal for an abstract quest, says Segerman, who uses math to create art by designing sculptures and puzzles generated on 3D printers. “And if you produce something interesting and beautiful,” he says, “it makes everybody else interested.”

Bosch’s work has implications beyond aesthetics, says Segerman. When computer scientists write algorithms to solve optimization problems, they need big datasets to test out their strategies. Bosch’s method shows how a work of art can be turned into such a dataset. “Take the Mona Lisa,” says Segerman. “Those points are a new set of data to run your algorithms on. There’s some value even there, in having new datasets and problems to attack.”

Here, Bosch used the TSP optimization approach to create a rendering of the eye of his wife, Kathy. Image courtesy of Robert Bosch.

Picture Perfect

Bosch has also used optimization problems to arrange hundreds of dominoes into portraits of subjects, including German mathematician Carl Friedrich Gauss and, more recently, Frankenstein’s monster. Bosch is finishing a book, tentatively titled Optimization Constraints and Design, about the optimization field that introduces readers to problem-solving methods, discusses current applications, and outlines his approach to creating art from math. He has also begun designing TSP-derived sculptures for 3D printers. One recent piece was a sculpture of his wife's eye, visible only if an observer is looking at it directly. Otherwise, it looks like an abstract shape (see figure).

Bosch says his efforts have paid off in the classroom. TSP art provides an access point to start thinking about optimization, and many of his former students have gone on to make their own art. In August, three former students presented original mathematical artwork in Jyväskylä, Finland, at the Bridges conference, an annual celebration of the intersection of mathematics and the arts. Former student Hank Guss, who now works with the Museum of Mathematics, lauds Bosch’s teaching approach. “His optimization work—even without the art—demonstrated the applicability of the coursework,” recalls Guss. “But seeing the art alongside the math gave me the inspiration to create my own.”

The theme Mathematics AND Art may seem strange to those who are more used to thinking Mathematics OR Art, but, in fact, there are many connectors to fill the blank in

Mathematics ___ Art
and its twin
Art___ Mathematics.
Indeed, the many interconnections between mathematics and art provide a wealth of material from which organizers of Math Awareness Month events can select. The list of sources on the Math Awareness web site is a great place to start. In this brief essay, I'll highlight a few of the possible ways to fill in the blanks above. I hope that this will stimulate you to explore many others.

Mathematics produces art

At the most practical level, mathematical tools have always been used in an essential way in the creation of art. Since ancient times, the lowly compass and straightedge, augmented by other simple draftsmen's and craftsmen's tools, have been used to create beautiful designs realized in the architecture and decoration of palaces, cathedrals, and mosques. The intricate Moorish tessellations in tile, brick, and stucco that adorn their buildings and the equally intricate tracery of Gothic windows and interiors are a testament to the imaginative use of ancient geometric knowledge. [R1], [R4], [R21]

During the Renaissance, several artists used simple grids and mathematically-based devices to accurately portray scenes on a flat surface, according to the principles of linear perspective. Several of Dürer's engravings give a glimpse of these techniques. The symbiosis of art and mathematics during these times as linear perspective and projective geometry were developing is one of the most striking examples of art and mathematics evolving almost simultaneously in new directions. [R7]

Today's mathematical tools are more sophisticated, with digital technology fast becoming a primary choice. In the hands of an artist, computers can produce art, powered by unseen complex internal mathematical processes that provide their magical abilities. Mathematical transformations provide the means by which an image or form in one surface or space is represented in another. Art is illusion, and transformations are important in creating illusion. Isometries, similarities, and affine transformations can transform images exactly or with purposeful distortion, projections can represent three (and higher)-dimensional forms on two-dimensional picture surfaces, even curved ones. Special transformations can distort or unscramble a distorted image, producing anamorphic art. All these transformations can be mathematically described, and the use of guiding grids to assist in performing these transformations has been replaced today largely by computer software. Compasses, rulers, grids, mechanical devices, keyboard and mouse are physical tools for the creation of art, but without the power of mathematical relationships and processes these tools would have little creative power.

Mathematics generates art

Pattern is a fundamental concept in both mathematics and art. Mathematical patterns can generate artistic patterns. Often a coloring algorithm can produce "automatic art" that may be as surprising or aesthetically pleasing as that produced by a human hand. Colored versions of the Mandelbrot set and Julia sets are striking examples of this: each is generated by the recursive equation zn = zn -12 + C. In the case of the Mandelbrot set the equation is iterated for each point C in the complex plane, where z0 = 0 and the point C is colored according to rules based on whether the iterated values eventually exceed 2 and the number of iterations after which this occurs. [P15] Other fractals, as well as images based on attractors, are also produced by iteration and coloring according to rules. The intricacy of these images, their symmetries, and the endless (in theory) continuance of the designs on ever-smaller scales, makes them spellbinding. [P10]

Much more mundane mathematical patterns can also provide surprising art. For example, begin with an array of numbers (such as a large data set, a sequence, a modular operation table, or Pascal's triangle) and color the numbers in the array according to some rule. Often surprising patterns -- even art -- emerges. Recursive algorithms applied to geometric figures can generate attractive self-similar patterns. Begin with a curve, a closed figure, or a simple spatial form, apply an algorithm to alter that figure by adding to (or subtracting from) specified parts of that figure, then repeat the algorithm recursively. Many nonperiodic tilings (such as the Penrose tilings) can also be generated automatically, beginning with a small patch of tiles and then applying a recursive "inflation" algorithm.

Transformations and symmetry are also fundamental concepts in both mathematics and art. Mathematicians actually define symmetry of objects (functions, matrices, designs or forms on surfaces or in space) by their invariance under a group of transformations. Conversely, the application of a group of transformations to simple designs or spatial objects automatically generates beautifully symmetric patterns and forms. In 1816, Brewster's newly-invented kaleidoscope demonstrated the power of the laws of reflection in automatically generating eye-catching rosettes from jumbles of colored shards between two mirrors. [P3] Today computer programs use symmetry groups to generate rosettes, borders, wallpaper designs [R11], and Escher-like circle-limit designs such as featured on the 2003 Math Awareness Month poster. [R6] Each of these designs begins with a small fragment or motif (chosen judiciously or randomly) whose transformed images fill out the full design. Periodic tessellations, whether geometric or Escher-like, can be automatically generated by computer programs [R12] or by hand, following recipes that employ isometries.

Art illuminates mathematics

When mathematical patterns or processes automatically generate art, a surprising reverse effect can occur: the art often illuminates the mathematics. Who could have guessed the mathematical nuggets that might otherwise be hidden in a torrent of symbolic or numerical information? The process of coloring allows the information to take on a visual shape that provides identity and recognition. Who could guess the limiting shape or the symmetry of an algorithmically produced fractal? With visual representation, the mathematician can exclaim "now I see!"

Since periodic tessellations can be generated by groups of isometries, they can be used to illuminate abstract mathematical concepts in group theory that many find difficult to grasp in symbolic form: generators, cosets, stabilizer subgroups, normal subgroups, conjugates, orbits, and group extensions, to name a few. [R20]

In the examples above, illumination of mathematics is a serendipitous outcome of art created for other reasons. But there are examples in which the artist's main purpose is to express, even embody mathematics. Several prints by M.C. Escher are the result of his attempts to visually express such mathematical concepts as infinity, duality, dimension, recursion, topological morphing, and self-similarity. [R16] Perhaps the most striking examples of art illuminating mathematics are provided by the paintings of Crockett Johnson and the sculptures of Helaman Ferguson. From 1965 to 1975, Johnson produced over 100 abstract oil paintings, each a representation of a mathematical theorem. [P14] Ferguson's sculptures celebrate mathematical form, and have been termed "theorems in bronze and stone." Each begins with the idea of capturing the essence of a mathematical theorem or relationship, and is executed by harnessing the full power of mathematically-driven and hand-guided tools. [P9]

Mathematics inspires art

Patterns, designs, and forms that are the "automatic" product of purely mathematical processes (such as those described in "Mathematics generates art") are usually too precise, too symmetrical, too mechanical, or too repetitive to hold the art viewer's attention. They can be pleasing and interesting, and are fun to create (and provide much "hobby-art") but are mostly devoid of the subtlety, spontaneity, and deviation from precision that artistic intuition and creativity provide. In the hands of an artist, mathematically-produced art is only a beginning, a skeleton or a template to which the artist brings imagination, training, and a personal vision that can transform the mathematically perfect to an image or form that is truly inspired.

Wallpaper patterns and tessellations can be pleasing from a decorative point of view; few would be viewed as art. [P24] (I feel [P20] is an exception). Escher did not view his tessellations as art, but as fragments to be an integral part of his complex prints. Makoto Nakamura's art also employs this technique. [P17] Jinny Beyer, a designer and quilt artist, uses her artistic intuition and color sense to turn tessellations into art. [P2] Kaleidoscopic designs are the inspiration for quilted art by Paula Nadelstern; her use of color and composition subtly break mathematical rules. [P16]

Dick Termes uses photography and grids to guide his projections of images onto the surface of a sphere, but his "Termespheres" bear his personal interpretation. [P23] Anamorphic artists István Orosz [P18] and Kelly Houle [P13] are guided by mathematical rules of transformation as they create mysterious distortions of images on the picture plane, but also use their intuition and imagination, checking with a mirrored cylinder as the work develops.

Pure mathematical form, often with high symmetry, is the inspiration for several sculptors who create lyrical, breathtaking works. With practiced eye and hand, relying on their experience with wood, stone, bronze, and other tactile materials, the artists deviate, exaggerate, subtract, overlay, surround, or otherwise change the form into something new, often dazzlingly beautiful. With the advent of digital tools to create sculpture, the possibilities of experimentation without destruction of material or of producing otherwise impossible forms infinitely extends the sculptor's abilities. [P4], [P5], [P12], [P19], [P21], [P22]

Mathematics constrains art

We often hear of "artistic freedom" or "artistic license," which imply the rejection of rules in order to have freedom of expression. Yet many mathematical constraints cannot be rejected; artists ignorant of these constraints may labor to realize an idea only to find that its realization is, indeed, impossible. Euler's theorem (v + f = e + 2) and Descartes' theorem (the sum of the vertex defects of every convex polyhedron is 720°) govern the geometry of polyhedra. Other theorems govern the topology of knots and surfaces, aspects of symmetry and periodicity on surfaces and in space, facts of ratio, proportion, and similarity, the necessity for convergence of parallel lines to a point, and so on. Rather than confining art or requiring art to conform to a narrow set of rules, an understanding of essential mathematical constraints frees artists to use their full intuition and creativity within the constraints, even to push the boundaries of those constraints. Constraints need not be negative -- they can show the often limitless realm of the possible.

Voluntary mathematical constraints can serve to guide artistic creation. Proportion has always been fundamental in the aesthetic of art, guiding composition, design, and form. Mathematically, this translates into the observance of ratios. Whether these be canons of human proportion, architectural design, or even symbols and letter fonts, ratios connect parts of a design to the whole, and to each other. Repeated ratios imply self-similarity, hardly a new topic despite its recent mathematical attention. One of the earliest recorded notices of it is in Euclid's Prop. 30, Book VI, the division of a segment in extreme and mean ratio (also known as the golden cut, or golden section). A segment AB is to be divided internally by point E so that the ratio of the whole AB to the part AE equals the ratio of the (larger) part AE to the (smaller) part EB. [P8] This geometric task produces the common ratio AB/AE = (1 + 5)/2, known as the golden ratio, denoted as (or ). The ratio has many unique, almost magical mathematical properties (for example, 2 = + 1, and 1/ = - 1), and it is these properties, as well as connections to the Fibonacci sequence, that have fascinated artists and architects, enabling them to produce designs and compositions with special properties. Other ratios and special geometric constructions (root rectangles, reciprocal rectangles, and grids of similar figures) also guide composition and design. [R10], [P11]

Art engenders mathematics

It is to be expected that in the execution of an artwork, mathematical questions will arise that the artist (or fabricator) must answer. This goes with the territory. In many instances, artists will struggle to answer the questions on their own in order to reach the answer in a way that makes sense to them. Escher did this in seeking to answer the question "How can I create a shape that will tile the plane in such a way that every tile is surrounded in the same way?" [R15] Sometimes these questions need the attention of trained mathematicians, engineers, or software designers and provide interesting practical problems to solve. The intricate textile patterns of designer Jhane Barnes result from close collaboration with mathematician Bill Jones and computer software designer Dana Cartwright of Designer Software. [R14], [P1], [P6]

There are also frequent instances where finished works of art suggest purely mathematical questions, ones that the artist never imagined, nor needed to consider. Folk art from other times and other cultures is a rich source for mathematical questions. Celtic knots and art from African cultures are two examples. [R5], [R8] Modern sculptures can also lead to mathematical questions. [R3] Escher's tessellations and some prints have been the source of several mathematical challenges, most not yet settled. Two of these mathematical questions seek to understand the relationships between local and global symmetry. [R9], [R19].

A most mathematical artist

I want to end this essay with a bit more about the work of the Dutch graphic artist M.C. Escher (1898-1972), who is perhaps the most astonishing recent example of an artist whose work contains a multitude of connections between mathematics and art. [P7] Escher was not mathematically trained, and even struggled with mathematics as a school student. Yet he did not reject mathematics, but instead figured out in his own way, using various (mostly pictorial) sources, the mathematics that he needed in order to realize his ideas and visions. Escher celebrated mathematical forms: polyhedra as decoration, stars, or living structures, mvbius bands, knots, and spatial grids. He used (and sometimes fused) various geometries in his work -- Euclidean in his tessellations, hyperbolic in his Circle Limit series, projective in depicting scenes in linear perspective, spherical in prints and his carved spheres. He employed topological distortions and transformations, strange or multiple perspectives, and visual recursion. He explored the topic of symmetry and tessellation in the plane, on the sphere, and in the Poincari disk, developing his own "layman's theory" of classification of types of planar periodic tilings and symmetric coloring of them, anticipating mathematician's and crystallographer's later studies of these topics. [R15] He asked and answered, in his own way, combinatorial geometric questions. [R17] He depicted abstract mathematical concepts in visual metaphors. And though Escher's work gained him the admiration of mathematicians and scientists, he felt isolated as an artist. Today there are many artists whose work is directly or indirectly inspired by Escher's work. While he has left us his own legacy, others are continuing to explore some of the paths he blazed and also are striking out on new paths from these.[R18]

Resources

Three recent books contain collections of essays on mathematics and art. Ivars Peterson [R13] showcases a wide selection of art and artists of today exemplifying the strong symbiosis between art and mathematics. Two others, [R2] and [R18], contain thoughtful commentaries and discussions as well as essays and art by contemporary artists. Several books and web sites provide text, ideas, problems and projects for courses focused on art and mathematics. Many of these are listed on the 2003 Mathematics Awareness Month web site, which also contains essays by Mark Frantz and Paul Calter, who teach courses on mathematics, art, and architecture. There are several organizations that are dedicated to fostering interaction between art, mathematics, and science. Most hold annual conferences at which artists and mathematicians (and many others) gather to exhibit, lecture, discuss, and mingle; often proceedings (print or electronic) publish the presentations. Web sites for several of these are listed under Organizations below.

Organizations

[O1] Digital Sculpture: http://www.intersculpt.org/

[O2] Dutch Society for Arts and Mathematics: http://www.arsetmathesis.nl (check the galerij)

[O3] Leonardo/ISAST: http://mitpress2.mit.edu/e-journals/Leonardo (International Society for the Arts, Sciences, and Technology; check the Gallery)

[O4] ISAMA: http://www.isama.org/ (International Society of the Arts, Mathematics, and Architecture)

[O5] Nexus Network Journal: http://www.nexusjournal.com/ (Architecture and Mathematics)

[O6] Bridges: http://www.sckans.edu/~bridges (Annual International conference on mathematical connections in art, music, and science; collected papers from its annual conferences are printed)

[O7] Visual Mathematics: http://www.mi.sanu.ac.yu/vismath (electronic journal of ISIS-Symmetry)

[08] Aesthetic Computing http://www.cise.ufl.edu/~fishwick/aescomputing

References: Books, articles, software

[R1] Bourgoin, J., Arabic and Geometrical Pattern and Design (plates), New York: Dover, 1973 (orig. 1879).

[R2] Bruter, Claude P., ed., Mathematics and Art: Mathematical Visualization in Art and Education, Heidelberg: Springer, 2002.
http://www.springer.de/cgi/svcat/search_book.pl?isbn=3-540-43422-4

[R3] Coxeter, H.S.M., "Symmetric Combinations of Three or Four Hollow Triangles," Math. Intelligencer, v. 16 (1994) 25-30. See also Burgiel, H., Franzblau, D.S. and Gutschera, K.R., "The Mystery of the Linked Triangles," Mathematics Magazine, v. 69 (1996) 94-102.

[R4] Critchlow, Keith, Islamic Patterns: An analytical and cosmological approach, New York: Schocken books, 1976. Paperback reprint, London: Thames and Hudson, 1999.

[R5] Cromwell, Peter R., "Celtic Knotwork: Mathematical Art," Math. Intelligencer, v. 15 (1993) 36-47.

[R6] Dunham, Douglas, "Transformation of Escher Hyperbolic Patterns," Visual Mathematics, v. 1, no. 1, 1999.
http://members.tripod.com/vismath/pap.htm - n11 Also see Dunham's essay about the 2003 Mathematics Awareness Month web site.

[R7] Field, J.V., The Invention of Infinity: Mathematics and Art in the Renaissance, Oxford: Oxford University Pr., 1997.

[R8] Gerdes, Paulus, Geometry from Africa: Mathematical and Educational Explorations, MAA, 1999.

[R9] Grünbaum, Branko, "Mathematical Challenges in Escher's Geometry," in M.C. Escher: Art and Science, H.S.M. Coxeter, M. Emmer, R. Penrose, and M.L. Teuber, eds, Amsterdam: North-Holland, 1986, pp. 53-67.

[R10] Kappraff, Jay, Connections, The Geometric Bridge Between Art and Science, New York: McGraw-Hill, 1991. Second ed., Singapore: World Scientific Publ. Co., 2002. http://www.nexusjournal.com/reviews_v4n4-Jablan.html

[R11] Lee, Kevin, Kaleidomania!, Key Curriculum Press.
http://www.keycollege.com/catalog/titles/kaleidomania.html

[R12] Lee, Kevin, Tessellation Exploration, Tom Snyder Productions.
http://www.tomsnyder.com/free_stuff/free_download.asp?PS=TESEXP

[R13] Peterson, Ivars, Fragments of Infinity: A Kaleidoscope of Math and Art, New York: Wiley, 2001.
http://www.isama.org/book/fragments/biblio.html

[R14] Ross, Teri, Math + Technology = Technique: The Jhane Barnes School of Textile Design.
http://www.techexchange.com/thelibrary/jhanebarnes.html

[R15] Schattschneider, Doris, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher, New York: W.H. Freeman, 1990.

[R16] Schattschneider, Doris, "Escher's Metaphors," Scientific American, v. 271 no. 5 (November 1994) 66-71.

[R17] Schattschneider, Doris, "Escher's Combinatorial Patterns," Electronic J. of Combinatorics, 4 (no. 2) (1997), #R17.
http://www.combinatorics.org/Volume_4/wilftoc.html

[R18] Schattschneider, Doris and Emmer, Michele, eds. M.C. Escher's Legacy: A Centennial Celebration (with CD Rom), Heidelberg: Springer, 2003.
http://www.springer.de/cgi-bin/search_book.pl?isbn=3-540-42458-X

[R19] Schattschneider, Doris, and Dolbilin, Nikolai, "One Corona is Enough for the Euclidean Plane," In Quasicrystals and Discrete Geometry (J. Patera, editor). Fields Institute Monographs, Vol. 10, AMS, Providence, RI, 1998, pp. 207-246.

[R20] Senechal, Marjorie, "The Algebraic Escher," Structural Topology, v. 15 (1988) 31-42.

[R21] Sykes, Mabel, Sourcebook of Problems for Geometry, Palo Alto: Dale Seymour Publ, 2000 (orig. 1912).

People and terms

[P1] Jhane Barnes: http://www.jhanebarnes.com/

[P2] Jinny Beyer: http://www.jinnybeyer.com/(check the Quilt Gallery)

[P3] David Brewster: http://www.brewstersociety.com/brewster_bio.html

[P4] Paul Calter: http://www.sover.net/~pcalter (see Calter's essay on the MAM web site)

[P5] Brent Collins: http://www.cs.berkeley.edu/~sequin/SCULPTS/collins.html

[P6] Designer Software: http://www.weavemaker.com/

[P7] M.C. Escher: http://www.mcescher.com/

[P8] Euclid: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

[P9] Helaman Ferguson: http://www.helasculpt.com/

[P10] Fractal art: Googling on "fractal art" produces over 170,000 hits. Many sites have beautiful images.

[P11] Golden ratio: Googling on the words golden ratio, golden section, or divine proportion will produce over 360,000 hits; there is much good, but also much erroneous and fabricated information on this topic.

[P12] George Hart: http://www.georgehart.com/sculpture/sculpture.html (see Hart's essay on the MAM web site)

[P13] Kelly M. Houle: http://www.kellymhoule.com/

[P14] Crockett Johnson: http://www.ksu.edu/english/nelp/purple/art.html

[P15] Mandelbrot and Julia sets: Google produces 8000 hits on the paired names. http://www.geocities.com/fabioc gives an elementary discussion and pictures

[P16] Paula Nadelstern: http://www.paulanadelstern.com/ (check the Quilt Gallery)

[P17] Makoto Nakamura: http://www18.big.or.jp/%7Emnaka/work.html

[P18] István Orosz: http://www.geocities.com/SoHo/Museum/8716/index.html

[P19] Charles O. Perry: http://www.charlesperry.com/ (see Perry's essay on the MAM web site)

[P20] Marjorie Rice: http://members.aol.com/tessellations

[P21] John Robinson: http://www.johnrobinson.com/

[P22] Rinus Roelofs: http://www.rinusroelofs.nl/

[P23] Dick Termes: http://www.termespheres.com/

[P24] Tessellation: Googling on "tessellation" produces over 35,000 hits. Many contain examples of original creations.

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