The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

"Different Strokes," by Linda AllisonThis image, like most of those selected for this exhibition, is generated with Ultra Fractal, designed by Frederik Slijkerman. "Different Strokes" consists of 10 layers, using Julia and Mandelbrot fractal formulas with other formulas and algorithms for coloring. The layers are merged into a unique image using different techniques and transparencies for each layer in the composition. The author, Linda Allison, is a disabled housewife living in Florida. Since 1994, Linda has dedicated part of her free time to designing fractal images. Having no formal mathematical training, Linda possesses an incredible ability to represent the concept of infinity in images with smooth and delicate color palettes. Her shapes blend and separate in absolute harmony, with balanced framing that combines the classicism of the first fractals with the latest advances of fractal art.

"Fractal Effervescence," by David AprilThis image comes from the fusion of the three separate image files generated with the software Apophysis, designed by Mark Townsend. Each one of the files contains different types of transformations-linear, polar, and spherical-that produce a curious dialog between the vertical lines, the sinuous curves with the appearance of smoke, and the bubbling circular shapes. In this type of fractal there is only one method of coloring. Sometimes a tiny change to the color gradient can distort the image dramatically. Fractal artists, hunting for treasure, can tease out unexplored forms, but a slight difference in this or that parameter can make them pass by that secret treasure without seeing its hidden beauty. David April lives in Illinois (USA). His interest in fractals is relatively recent, but he compensates for that with an enormous enthusiasm and fascination for finding new forms.

"Ebony and Ivory," by Bill BeathBill Beath is an Australian photographer specializing in depicting nature, the countryside and architecture. His photographic work combines raditional film processes with the most modern digital techniques. His first contact with fractals was a photograph of a Nautilus shell, which led to the Fibonacci sequence, which led to his discovery of fractals and fractal art. Since then, Bill Beath has been permanently immersed in fractal art, as much as an art form as an integral part of his photographic work. For that reason this image is based on the "Fibonacci Julia" algorithm, developed by Kerry Mitchell. It shows a fascinating shape somewhere between a natural design and an exquisite man-made design. The name of the image, "Ebony and Ivory," refers to the palette used, based on elegant tones of black and white.

"Encore," by Paul DecellePaul DeCelle is a mechanical engineer in Michigan (USA). His image for this exhibition is a very handsome composition based on a portion of the Mandelbrot set (magnified approximately 10 to the 13th times). The artist has used techniques known for more than 10 years, but can still surprise the viewer by its majesty, especially in large-scale reproductions. If we imagine the Mandelbrot set as an extensive mountain range, the composition relies on two basic principles. The "Slope" algorithm assigns the same color to those regions with the same height, like in a topographical map. The "Lighting" algorithm colors towards white those regions of the surface illuminated by an imaginary sun sitting on the horizon, while the shadows partially obscure the surface. The result is a three-dimensional effect that enriches and enhances the detail in the original fractal.

"Bonhomme de Neige (Snowman)," by Sylvie GalletSylvie Gallet is a mathematics professor at a secondary school near Paris. With 20 years of experience in writing fractal formulas and algorithms, she is an expert in the handling of color gradients. In fact, Sylvie avoids complex and postprocessed images, in preference to designs with little elaboration, whose value resides in the intelligent and creative use of color. "Bonhomme de Neige" is a good example of Sylvie's art. It is a conceptually simple image, but the careful use of color transports us immediately to an image of Christmas and winter countryside. Few fractal artists are capable of transmitting such direct visions and sensations.

"Overwrought," by Damien JonesDamien Jones is a respected artist and fractal expert. His Internet domain fractalus is one of the most complete sources to start with for fractal art. Through years of explorations of the mathematics for aesthetic reasons, Damien’s work has appeared in numerous books, magazines, posters, calendars, and international exhibitions. Born in the United Kingdom, he currently resides in Florida (USA) with his wife Michelle, whom he married while collaborating on the organization of this exhibition. The image "Overwrought" belongs to the Mandelbrot set, although it is difficult to see because of the use of "turbulence," which distorts the calculations before the application of the fractal coloring. After the image is colored, the turbulence is removed and the calculation continues. The process produces a cloudy texture but keeps the underlying shapes unaltered. The coloring—austere, mournful, and at times apocalyptic—often produces an emotional response in the viewer of the art.

"And how is your husband Mrs. Escher?" by Nada KringelsNada (Brigitte) Kringels is a German expatriate who has been living in Spain for 14 years, where she learned to use Ultra Fractal. This image consists of 25 layers using basically two algorithms designed by Kerry Mitchell, "Gaussian Integer" for the background and "Rose Range Lite" for the top layers. During the composition phase of the image, Nada Kringels discovered various shapes that immediately resembled some of the work of M.C. Escher, so she decided to introduce geometric impossibilities into the design. To finish the background, in marked feminine character according to the author, she began to imagine that it had been made by Mrs. Escher. Fascinated with this possibility, Nada Kringels began to consider in her image the idea of Mrs. Escher as an artist, without even knowing if this Mrs. Escher existed—in fact she did, Jetta Umiker, with whom Maurits Cornelius Escher had three children. Ah, by the way, how is your husband, Mrs. Escher?

"Grey Moon Rising," by Klaus-Peter KubikMany fractal formulas and algorithms produce conventional geometric figures with certain parameters. For example, the Julia set iterated using the origin as its parameter produces a circle. The style of Klaus-Peter Kubik is focused on producing conventional geometric figures using fractal techniques. He likes to explore the combinations of the simple figures of circles and squares with attractive shapes for the viewer. He also exploits the possibilities of fractal geometry to create textures. The rough, grey texture of the circle symbolizes the surface of the moon while the vertical and horizontal lines, similar to those made with a pencil, emphasize the geometric structure of the image. Klaus-Peter Kubik works for the German government in the public health field and has participated in nearly a dozen exhibitions since 1994.

"Mateko," by Dan KuzmenkaDan Kuzmenka is a North American researcher in the field of chemistry. Like many other scientists, Dan discovered fractal geometry in 1985 reading an article in the magazine Scientific American, although it wasn’t until 1999 that he began to create his first fractal images. Mateko is a word invented by its author, who maintains a personal challenge to find new ways of expressing spirals—the most important fractal icon—without showing the same shape time and time again. For this image he experimented with different color palettes and ways to combine them before the colors we now see appeared; these colors are unusual for Dan Kuzmenka, who usually uses warmer colors and earth tones.

"Polished," by Heather LambHeather Lamb was born and lives in Scotland. From an early age she has developed a strong interest for mathematics that strengthened by her studies at the Open University, where she became familiar with fractal geometry and the Mandelbrot set. A strong association exists between nature and fractal geometry and Heather Lamb exploits this, creating images that evoke the real world while at the same time transforming mathematics into something that can be understood and visualized. For this image she was inspired by her childhood experiences with polished stones, in which the true beauty of their colors is only discovered during the process of polishing. The colors were chosen to reproduce the appearance of stone, but also to be harmonious with each other and produce a balanced image. Masks with black and white gradients were used to precisely place the shadows and lights and provide a realistic sensation of polish and a tangible, three-dimensional effect that accentuates the image.

"Indra Family," by Jos LeysJos Leys is a Belgian mechanical engineer who has always shown a special interest for mathematics in general and fractal art in particular since he programmed his first fractal image 25 years ago. "Indra Family" is a tribute to the professors David Mumford, Caroline Series and David Wright, the authors of the book "Indra’s Pearls: The Vision of Felix Klein." The iterative calculation techniques of the Kleinian Groups described in this book reveal new fractal images that until then had remained unexplored. The name "Indra's Pearls" is a Hindu and Buddhist concept that represents a network of silk strings that extend to infinity in all directions, and contains at each intersection a very bright and luminous pearl that reflects each of the pearls of the network, that then reflect the others and so on, without end, like mirrors reflecting to infinity.

"Spiral with opaque lines," by Andreas LoberThis image belongs to a simple Julia set, but the refined technique of Andreas Lober, who graduated from the University of Heidelberg with a degree in mathematics, converted it entirely into a creative prodigy. The coloring algorithm is simple: find the minimum value of │z│ during the iteration, deflecting lightly the values pseudo-randomly; this produces the sine waves that heighten the composition. The values are trapped during the calculation in discrete intervals; this produces the peculiar coloring that appears to be done with colored pencils. Other preferences of Andreas Lober include designing tilings that cover the plane with squares containing geometric shapes, so that they fit perfectly with the adjacent eight squares. These experiments produce tesselations of great visual impact and, in this case, variations have been used to obtain the frames contained in the image.

"Starfruit," by David MakinDavid Makin is a British computer programmer born in North Wales, who loves fractal geometry and science fiction. The majority of his work comes from his investigations into the use of coloring algorithms. In this case he employed three algorithms applied to a Julia set. The first of his algorithms, named "MMF3-Turning Points," generated the starred forms that characterize the image and suggested the title of the shape immediately to him (the starfruit is a tropical fruit whose cross section produces a five-pointed star). With the second algorithm, "MMF3-Orbital Waves," he used the idea of complementing the first layer with the handsome curved lines that accentuate the set. At this point he proceeded to include the third algorithm, "MMF3-Alternative fBm II," which provides a more organic texture. Finally, David Makin took considerable time in combining the three layers with color palettes and the algorithms described that produced the final result.

"Warm Glow," by Kerry MitchellKerry Mitchell is an aeronautical engineer born in Iowa (USA) who since 1984 has occupied diverse positions related to NASA. At the same time he is a computational artist of great technical resources that he uses to represent fractal images and visualize mathematical relationships. A subject that always accompanies the work of Kerry Mitchell is to show the complexity and beauty that flows through extremely simple mathematical rules. The metaphorical idea of the complexity of nature associated with the simplicity of deterministic mathematical formulas is a constant in his work. For this image Kerry Mitchell has applied to a zoom of the Mandelbrot set a coloring algorithm named "Buddhabrot," invented by Melinda Green (see
"The Buddhabrot Technique" at www.superliminal.com/fractals/bbrot/bbrot.htm). The result is an image of mystical character that suggests a seated Buddha at different scales.

"20040402," by Samuel MonnierThe title of this picture does not involve any mathematical riddle, but is simply the reference number by which Samuel Monnier identifies his pictures. This young Swiss man, who is preparing for his Ph.D. in Theoretical Physics, does not like to put titles on his pictures as he feels it interferes with the sensations his work can produce in the viewer. The basic concept on which this image rests is to begin with a more or less repetitive initial design and superimpose various layers with this design at different scales. This procedure generates an image that shows structures with a wide range of scales, although from a strict point of view one cannot consider it to be fractal.