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.

"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.3662 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, Berkeley3537 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

"The Lake," by Harry Benke, Visual Impact Analysis LLC (2007)3534 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.

"Indra Family," by Jos Leys3509 viewsJos 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.

"Lizard Tetrus," by Carlo Sequin, University of California, Berkeley3414 views24 Lizard tiles, inspired by one of the many planar tilings by M.C. Escher, are mapped around a rounded tetrahedral frame of genus 3. This tiling is a contorted version of the pattern of 24 heptagons displayed on the surface of the marble sculpture "Eight-fold Way" by Helaman Ferguson. That sculpture celebrates Felix Kelin's famous "Quartic Curve" which achieves the maximal symmetry of 168 automorphisms possible on a genus-3 surface. Read more about patterns on the Tetrus surface.. Thanks to Pushkar Joshi and Allen Lee for their help with mapping Escher tiles onto the tetrus. --- Carlo Sequin

Professor Tom Hull3404 viewsTom Hull took his Ph.D. in mathematics at the University of Rhode Island in 1997. His dissertation was on list coloring bipartite graphs, now he mostly studies the mathematics of origami (paper folding).

Tom Hull is an associate professor in the Department of Mathematics at Merrimack College in North Andover, MA.

"Borromean Rings," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.3323 viewsThe Borromean Rings consist of three links. Take one link away and the other links fall apart, but together they are inseparable. Because of this, they are popular as a symbol for strength in unity. Here they are shown from an unusual point of view, and also a Seifert surface is shown. This is an orientable surface, bounded by the links. This image was made with a tool called SeifertView.

--- Jarke J. van Wijk

"Polished," by Heather Lamb3297 viewsHeather 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.

"Warm Glow," by Kerry Mitchell3217 viewsKerry 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.

"Starfruit," by David Makin3193 viewsDavid 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.

"Tree Frog, opus 280," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed in 1993, folded in 2005, 5". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.3159 viewsThe 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

"African Elephant, opus 322," by Robert J. Lang. Medium: One uncut square of watercolor paper, composed and folded in 1996, 8". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.3135 viewsThe 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

"Bull Moose, opus 413," by Robert J. Lang. Medium: One uncut square of Nepalese lokta, composed and folded in 2002, 6". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.3066 viewsThe 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

"Fiddler Crab, opus 446," by Robert J. Lang. Medium: One uncut square of Origamido paper, composed and folded in 2004, 4". Image courtesy of Robert J. Lang. Photograph by Robert J. Lang.3061 viewsThe 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.

I'm especially pleased with this model, which involves a combination of symmetry with one distinctly non-symmetric element. The base is quite irregular, but its asymmetry is mostly concealed. The crease pattern is here.

--- Robert J. Lang

"Spiral Mobius," by Nat Friedman, Professor Emeritus, University of Albany - SUNY (2006)3028 viewsStoneware, 12" x 8" x 12". "This sculpture was made by starting with a cut circular band of clay and then bending and twisting before rejoining the cut ends. Props were used to preserve the shape while drying. The form was then sanded, low fired, sanded, and then high fired." --- Nat Friedman, Professor Emeritus, University of Albany - SUNY