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.

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

"Totem," Harry Benke, Visual Impact Analysis LLC (2008)Archival digital print, 19" x 13.6". "'Totem' represents the frontier, the uncharted, the often surprising and almost mystic nature of mathematical discovery. The totem is composed of ellipsoids ((x2/a2)+(y2/b2)+(z2/c2)) = 1, ray-trace rendered over an algorithmically generated fractal skyscape. Atmospheric effects were calculated as well such as scattering, moisture etc. The totem signifying the last guidepost to the unknown." --- Harry Benke, freelance artist/mathematician, Novato, CA (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

"Rhombic Dodecahedron I," by Vladimir Bulatov (2008)Metal sculpture, 4.5" diameter. "The base of this sculpture is rhombic dodecahedron (polyhedron with 12 rhombic faces with cubical symmetry). Each of the 12 faces was transformed into a curved shape with 4 twisted arms, which connects to other shapes at vertices of valence 3 and 4. The boundary of the resulting body forms quite a complex knot. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life." --- Vladimir Bulatov, Independent Artist, Corvallis, OR

"Rhombic Triacontahedron III," by Vladimir Bulatov (2007)Metal sculpture, 4.0" diameter. "Stellation of rhombic triacontahedron with 30 identical rhombic faces makes base for this sculpture. All internal intersections of rhombic faces were carefully eliminated by cutting away parts of rhombuses. The resulting 3D body was given organic shape by replacing straight faces with smooth subdivided surface. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life." --- Vladimir Bulatov, Independent Artist, Corvallis, OR

"Overlapping Circles I," by Anne Burns, Long Island University, Brookville, NY (2008)Digital print, 13" x 12". "This is an iterated function system made up of Mobius Transformations, programmed in ActionScript. I began my studies as an art major; later I switched to mathematics. In the 1980's I bought my first computer and found that I loved programming and could combine my all of my interests: art, mathematics, computer programming and nature." --- Anne Burns, Professor of Mathematics, Long Island University, Brookville, NY

"Recursive Construction for Sliding Disks," Adrian Dumitrescu, University of Wisconsin, Milwaukee (2008)Digital print, 11" x 5". "Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. One can easily show that 2n moves always suffice, while the above construction shows pairs of configurations that require 2n-o(n) moves for this task, for every sufficiently large n. Disks in the start configuration are white, and disks in the target configuration are shaded. " --- Adrian Dumitrescu, University of Wisconsin, Milwaukee

"Twice Iterated Knot No. 1," by Robert Fathauer, Tessellations Company (2008)Third Prize, 2009 Mathematical Art Exhibition. Digital print, 19" x 12". Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity.

"The starting point for this iterated knot is a nine-crossing knot that has been carefully arranged to allow seamless iteration. Four regions of this starting knot are replaced with a scaled-down copy of the full starting knot, incorporated in such a way that the iterated knot is still unicursal. These same four regions are then replaced with a scaled-down copy of the iterated knot, resulting in a complex knot possessing self similarity." --- Robert Fathauer, Small business owner, puzzle designer and artist, Tessellations Company, Phoenix, AZ

"The PowerStar: Synergetic Sacred Geometry, " Warren Scott Fentress (2008)MagneBlocks, 20" x 20" x 20". "Platonic fundamental shapes like the tetrahedron and pyramid, which are culminated recursively into 'powered tetrahedrons & pyramids', are arranged into pentagonal forms that mimic the 5-fold geometry of flowers. I invented MagneBlocks because of the bicameral mind resonating with the fundamental consciousness waveforms that permeate spacetime." --- Warren Scott Fentress, Imaginatuer, Brookfield, CT

"Spiral Mobius," by Nat Friedman, Professor Emeritus, University of Albany - SUNY (2006)Stoneware, 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

"Trefoil Knot Minimal Surface," by Nat Friedman, Professor Emeritus, University of Albany - SUNY (2006)Limestone, 9" diameter by 4" depth. "This sculpture was carved from a circular piece of limestone. The form is based on the shape of the soap film minimal surface on a configuration of a wire trefoil knot. There is a nice interaction of the form and space with light and shadow." --- Nat Friedman, Professor Emeritus, University of Albany - SUNY

"The Net," by Mehrdad Garousi (2008)Digital art print, 24" x 18.5". "This image exhibits a very complex, yet ordered series of lonely fibers that are woven in each other. This generated lacy net is not flat and goes to infinity at the center and also many times in each of its main arms. Another wonderful mathematical and artistic representation is where hexaploid weaving is modified into a triple one without cutting or deleting any fibers. Self similarity is the main property of this work, as any small hole in the main arms is nearly similar to the whole image. Having experimented with other media, I chose mathematical fractal image making as one of the newest and most wonderful common areas between mathematics and art." --- Mehrdad Garousi, Freelance fractal artist, painter and photographer, Hamadan, Iran

"Aristolochia Grandiflora," by S. Louise Gould, Central Connecticut State University, New Britain (2008)Inkjet print on treated silk, quilted and sparsely beaded to emphasize symmetries, 20" x 21.5". "My artwork usually connects textiles or paper with mathematical, specifically geometric ideas. 'Aristolochia Grandiflora' is a floral fractal. When I first saw the plant at Frederik Meijer Gardens in Grand Rapids in full bloom in May, it seemed a natural subject for exploring the seventeen wallpaper patterns in the plane. Starting with a photograph that I had taken in the garden, I sampled sections of the plant image and used KaleidoMania to generate samples of each of the seventeen wallpaper patterns. These were printed on 8.5 by 11 inch treated silk pages and folded, cut, pieced, quilted and beaded to create mathematical art to wear." --- S. Louise Gould, Associate Professor, Department of Mathematical Sciences, Central Connecticut State University, New Britain, CT

"Black and Blue Ricochet Trio," by Gary R. Greenfield, University of Richmond, VA (2008)Digital print, 14" x 24". "Many of my computer generated algorithmic art works are based on visualizations that are inspired by mathematical models of physical and biological processes. These three side-by-side black and blue "ricochet compositions" were generated by placing particles on each of the sides of a 16-gon, assigning them starting angles, and then letting each move in a straight line until it encounters an existing line segment at which point it is reflected--the ricochet--and then paused so that the next particle may take its turn. Further, if a particle ricochets off its own path, then the area it has just enclosed is filled using the requisite black or blue drawing color that particles were alternately assigned." --- Gary R. Greenfield, Associate Professor of Mathematics and Computer Science, University of Richmond, Richmond, VA

"A Mathematician's Nightmare," by JoAnne Growney (2008)Laser print on paper, 15 1/2" x 17 1/2" . "The poem, 'A Mathematician's Nightmare,' introduces a version of the unsolved Collatz Conjecture which asserts that when prescribed operations are iterated on any positive integer, the sequence produced will eventually reach 1. The prescribed operations are these, for any starting positive integer n: if n is even, replace n by n/2 (i.e., decrease n by half); if n is odd, replace n by (3n+1)/2 (i.e., increase n by half and round up to the next integer); my exhibit-entry displays both the poem and a graph of the sequence of iterations applied to the integer 27." --- JoAnne Growney, Poet, Professor Emerita, Department of Mathematical Sciences, Bloomsburg University. Residence: Silver Spring, MD

"Crane," by Zdenka Guadarrama, Rockhurst University, Kansas City, MO (2008)Mobile--Gauze, papyrus, silver and wood, 10" x 10" x 15". "'Crane' represents the continuous dimensional transition from a point, represented by a silver sphere, to a line, a plane and finally a crane. This transition is depicted in parallel to the evolution of the creative process which starts with an idea, represented by the same silver sphere, and which through refinements and trials culminates in the bird as well. [My] projects consist in artistic explorations that happen in parallel to the teaching/learning of mathematics (measure theory or complex analysis, for example). I search to generate art using mathematics and art inspired in the mathematics that I share with my students in order to motivate them to learn more mathematics, to make some extra connections, and to create some art of their own." --- Zdenka Guadarrama, Assistant Professor of Mathematics, Department of Mathematics and Physics, Rockhurst University, Kansas City, MO