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 ands, 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.

Mountains in Spring4688 viewsComputers make it possible for me to "see" the beauty of mathematics. The artworks in the gallery of "Mathscapes" were created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector.

--- Anne M. Burns

Five Intersecting Tetrahedra4568 viewsThis is a version of the Ow-Hull "Five Intersecting Tetrahedra." The visually stunning object should be a familiar sight to those who frequent the landscapes of M.C. Escher or like to thumb through geometry textbooks. Read about the object and how it is constructed on the Origami Gallery.

--- Thomas Hull. Photograph by Nancy Rose Marshall.

Spiked Rhombic Enneacontahedron4486 viewsThis structure was conceived by taking a 90-sided polyhedron, whose faces are made from two types of rhombi, and placing a pyramid on each face. The construction uses 180 small squares of paper, all folded and interlocked together without glue. See more models on the Origami Gallery.

--- Thomas Hull. Photograph by Nancy Rose Marshall.

"Different Strokes," by Linda Allison4402 viewsThis 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.

Nested Hexogonal Collapse4400 viewsThis model is a series of concentric hexagons with "zig-zag" creases coming from the center-most hexagon out to the midpoints of the paper's sides. It can be collapsed in many different ways and twisted into interesting shapes, as done here. See more geometrics and tesselations on the Origami Gallery.

--- Thomas Hull. Photograph by Nancy Rose Marshall.

"Hilbert Cube 512"4377 views"Hilbert Cube" is a space-filling recursive curve in 3 dimensions in analogy to the famous Hilbert curve in the plane. Special care has been taken never to place more than 3 coplanar line segments in sequence. At the largest recursion step the geometry has been slightly altered so as to obtain a closed loop. In the proper parallel projection one can see that the 2 halves of this sculpture are connected by only 2 tube segments. This piece of art gives me the association of an abstract, constructivist model of the human brain. See more views of the
"Hilbert Cube 512". --- Carlo Sequin

"Fractal Effervescence," by David April4096 viewsThis 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.

Circle Picture 53959 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

--- Anne M. Burns

Circle Picture 103870 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

--- Anne M. Burns

"Bonhomme de Neige (Snowman)," by Sylvie Gallet3604 viewsSylvie 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.

"Mateko," by Dan Kuzmenka3469 viewsDan 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.

"The Lake," by Harry Benke, Visual Impact Analysis LLC (2007)3337 viewsDigital C-print (laser exposed photographic paper, i.e. Lightjet print), 15" x 12". "'The Lake' is an object rising from ripples in a lake. The object is formed by placing 5 pointed stars on the transparent faces of a dodecahedron. The sine wave and harmonic ripples in the lake as well as the dodecahedron elements are rendered 3D models. The models are digitally composed with a scanned background. The mountains could also be fractal and algorithmically generated, but in this work the mountains are part of the base background scan which gives a better sense of depth to the artwork." --- Harry Benke, freelance artist/mathematician, Novato, CA (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

"And how is your husband Mrs. Escher?" by Nada Kringels3317 viewsNada (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?

"Allosaurus Skeleton, opus 326," by Robert J. Lang. Medium: 16 uncut squares of Wyndstone "Marble" paper, 24". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.3290 viewsThis model was inspired by the brilliant Tyrannosaurus Rex of the late Issei Yoshino.

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.

--- Robert J. Lang

"Poincare FishDish," by Carlo Sequin, University of California, Berkeley3216 viewsA tiling with regular heptagons does not fit into the Euclidean plane, since 3 times the dihedral angle of the heptagon exceeds 360 degrees. But if we are willing to introduce a progressive scale factor, then the whole hyperbolic plane can be fit into the Poincaré disc. Here is a visualization of a {7,3} tessellation where 3 heptagons join at every vertex, using a tiling motif inspired by the famous Dutch artist M.C. Escher. Each heptagon is cut into 7 identical pizza slices with irregular boundaries in the shape of fish that properly interlock with one another. See more tiling patterns on the Poincare disc. --- Carlo Sequin