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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"Skelug," by Bradford Hansen-Smith (2007)28 folded circles, 16"x6"x5". "Nine inch paper plate circles are folded and reformed into multiple units that have been arranged in one of many possible combinations of joining. Consistently following the development it began to take on a skeleton-like appearance and by decreasing the diameters of the circles it began to form a twisting conical helix, much like a sea slug, thus the name Skelug. Most all of my explorations with the circle start with folding three diameters, developing the equilateral triangular grid, reforming and joining multiples, which often reveals structural forms observable in nature." --- Bradford Hansen-Smith, Independent consultant, geometer, author, sculptor, Chicago, IL

"Extrapolated Icosahedron," by Bradford Hansen-Smith (2008)52 folded 9" paper plate circles, 13"x13"x13". "Forty circles have been folded, reformed to an in/out variation of a truncated tetrahedron, then octahedronally joined in pairs, and arranged in an icosahedron pattern. This revealed an interesting form of the icosadodecahedron with open pentagon stars. In this case twelve circles were reformed and added to suggest mouth-like openings found in sea anemones or in opening flower buds. This gives function to the open pentagons. Much of what I explore with folding circles are the structural functions of geometry found in life forms that correlate to the movement forms of the folded circle." --- Bradford Hansen-Smith, Independent consultant, geometer, author, sculptor, Chicago, IL

"(10,3)-a," by George Hart, Stony Brook University, Stony Brook, NY (2008)Nylon (selective laser sintering), 3.5" x 3.5" x 3.5". "This is a sculptural interpretation of the (10,3)-a crystal lattice, which has been well known to crystallographers and mathematicians for decades. However I have adapted it by wrapping it in a smooth surface which maintains its high genus topology while giving an organic sensibility. A 4x4x4 block has been selected from the infinite lattice in such a way that it can stand upright on a corner with a 3-fold axis vertical. Viewing the sculpture from different vantage points reveals a rich set of dramatically different tunnels along varying projections. For additional information and images, see http://www.georgehart.com/rp/10-3.html." --- George Hart, Research Professor, Stony Brook University, Stony Brook, NY

"(10,3)-a Twice," by George Hart, Stony Brook University, Stony Brook, NY (2008)Nylon (selective laser sintering), 3.5" x 3.5" x 3.5". "This is a sculptural interpretation, made by selective laser sintering, of two copies of the (10,3)-a lattice. Modern layered fabrication processes allow the construction of two interlocked components which are free to move slightly relative to each other, within the constraints of their being linked. The two copies are congruent, though mirror images. Each interpenetrates the tunnels of the other in a surprisingly complex manner. The 5x5x5 selection from the infinite lattice was made in such a way that the sculpture can stand vertically on a corner. See more works at http://www.georgehart.com." --- George Hart, Research Professor, Stony Brook University, Stony Brook, NY

"Spiral Squares," by Iftikhar Husain, University High School, Newark, NJ (2008)Digital print, 8" x 8". "The artwork 'Spiral Squares' was originally created on a TI-84 Plus graphing calculator. The design was uniquely created by using two equations only of linear function with restricted domain. These two equations have different parameters of the equation of each line segment, such as slope, y-intercept, domain x-left value, and domain x-right value, organized in a table. Each line segment is drawn by picking up its respected parameter value from the table. Once all the values from the table are exhausted the complete spiral square will appear on the calculator screen. The artwork is simple but truly illustrates the mathematical concepts. A single Spiral Square was then created on a computer using Geometer Sketchpad software. The artwork, shown here, is the simple translation effect of the single spiral square horizontally and vertically thrice." --- Iftikhar Husain, Mathematics Teacher and Tech-Coordinator, University High School, Newark, NJ

"A Strange Dream," by Karl Kattchee, University of Wisconsin - La Crosse (2008)Oil crayon on paper, 20" x 24". "This drawing is composed of round shapes, mainly. The arrangement of the round shapes is itself round, so that roundness is occurring on multiple levels. On the other hand, the drawing depicts multiple levels of thinking or being, like a sleeper who dreams within his dream within his dream. A Strange Loop, in the sense of Hofstadter, is evident." --- Karl Kattchee, Associate Professor of Mathematics, University of Wisconsin-La Crosse, La Crosse, WI

"Valley of Serenity," by Ghee Beom Kim (2007)Digital print, 12" x 12". "'Valley of Serenity' has been created using semicircles based on fractal concept with a touch of Op art element. Within a semicircle two smaller semicircles fit in. This process (iteration) continues on until it is visually meaningful. The resulting image conjures up an extraterrestrial terrain of a faraway planet. The smaller semicircles bear a resemblance to a horizon by giving it the effect of perspective." --- Ghee Beom Kim, Freelance artist, Sydney, Australia

"Simple Bowl (32)," by Goran Konjevod, Arizona State University, Tempe, AZ (2006)One folded square sheet of paper, 5" x 5" x 4". "This is one of the simplest pieces in my ongoing series of bowls formed by pleat tessellations. Every fold is a straight line segment parallel to an edge of the square sheet, and no fold is ever undone. The curved surface is purely a result of the intrinsic tension in the sheet of paper trying to unfold itself. This simple bowl is constructed by using alternating pairs of vertical and horizontal pleats, from the edges toward the center." --- Goran Konjevod, Assistant Professor of Computer Science and Engineering, Arizona State University, Tempe, AZ

"Wave (32)," by Goran Konjevod, Arizona State University, Tempe, AZ (2006)First Prize, 2009 Mathematical Art Exhibition. One folded square sheet of paper, 10" x 10" x 5". "The wave is one of the pleat tessellations that continues to amaze me even years after I first folded it. The peculiar symmetry and the tension caused by locking the edges causes two of its corners to bulge in opposite directions, while the remaining two corners remain fairly flat. As in the simple bowl, the pleat sequences all begin at the edges and proceed towards the center of the sheet, but the difference is that all horizontal pleats are oriented the same way, and similarly all the vertical pleats." --- Goran Konjevod, Assistant Professor of Computer Science and Engineering, Arizona State University, Tempe, AZ

"Infinite Curl 7," by Matjuska Teja Krasek in collaboration with Dr. Clifford Pickover (2006)Digital print, 9.9" x 10.1". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.

"The image represents the behavior of mathematical feedback loops, and more particularly the iteration of a complex function. The figure is our rendition of a visually interesting quartic variant of a Ushiki Phoenix Julia set. As with other fractals, the image exhibits a wealth of detail upon successive magnifications. The image �Infinite Curl 7� has been made in collaboration with Dr. Clifford Pickover, the author of more than thirty books about mathematics, art, and science." --- Matjuska Teja Krasek, Freelance artist, Ljubljana, Slovenia, EU

"Interlaced Hearts," by Matjuska Teja Krasek (2001)Digital print, 12.6" x 11.9". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.

"The implicit decagon constituted of five smaller decagons expresses tenfold and fivefold rotational symmetry. The image where golden heart-like shapes are exposed shows self-similarity, the richness of relations between the decagons, pentagonal stars, Penrose rhombs, kites and darts with the golden ratio used several times as a scale factor." --- Matjuska Teja Krasek, Freelance artist, Ljubljana, Slovenia, EU

"Five Left Tetrahedral Cosets," by Francisco Lara-Dammer, Indiana University, Bloomington (2008)Digital print, 20" x 20". "This is a Klein diagram (named after the nineteenth-century German mathematician Felix Klein) that represents A5, the group of symmetries of the icosahedron. Another way of describing A5 is as the alternating group on five elements, namely, the group of all even permutations of five entities. This diagram emphasizes A5's tetrahedral subgroup A4 (the group of symmetries of the tetrahedron, also the group of even permutations of four entities), which has twelve elements, plus the four left cosets of A4. The general diagram is obtained by centrally projecting an icosahedron onto a sphere (with the center of one face projected onto the north pole) and then making a stereographic projection of the sphere down onto a horizontal plane. Each coset has been identified with one color. The circle contains a hundred and twenty regions from which sixty correspond to the dark blue background, and the other sixty are split with the five left cosets. The reason I have realized Klein diagrams is to understand more clearly the beauty of Group Theory." --- Francisco Lara-Dammer, Research assistant. Center for Research on Concepts and Cognition, Indiana University, Bloomington, IN

"Cornrow," Stephen Luecking, DePaul University, Chicago, IL (2008)Giclee print, 13" x 13". "Images begin as super ellipses constructed from bezier curves in which the weight and position of the control points are randomized, using a random number generator to induce eccentricity. The eccentric curves are then layered subjected to various improvised Boolean and path edits. The results are not intended to be read as mathematical objects, thus the randomizing and improvising procedures. Rather the goal is to seek out visual tensions implicit in the relationship between the curves and the tondo format, between the wholeness of the circle and the fragmentation in its interior." --- Stephen Luecking, Professor of Computer Graphics, School of Computing and Digital Media, DePaul University, Chicago, IL

"Cob," by Stephen Luecking, DePaul University, Chicago, IL (2008)Giclee print, 13" x 13". "Images begin as super ellipses constructed from bezier curves in which the weight and position of the control points are randomized, using a random number generator to induce eccentricity. The eccentric curves are then layered subjected to various improvised Boolean and path edits. The results are not intended to be read as mathematical objects, thus the randomizing and improvising procedures. Rather the goal is to seek out visual tensions implicit in the relationship between the curves and the tondo format, between the wholeness of the circle and the fragmentation in its interior." --- Stephen Luecking, Professor of Computer Graphics, School of Computing and Digital Media, DePaul University, Chicago, Illinois

"The Empty Paradox," by Kaz Maslanka, D3 Technologies (2008)Digital print, 16" x 20". "The equation is the familiar function of x equal to 1/x which yields a hyperbolic curve when graphed and results an asymptote when x = 0. Compassion multiplied by Wisdom is equal to 1 over X as the limit of X approaches Buddha's mind. Buddhist philosophy tells us that Buddha's mind is emptiness yet the philosophy also tells us that emptiness is different than nothingness or zero. In fact it is quite paradoxical for we are told that emptiness is very much something. This piece also uses visual imagery for poetic expression with Buddhist symbolism of flexibility and eternity represented by bamboo and pine trees respectively. As an artist, my interest in correlating experience through language spawned my desire to study mathematics and physics. I am currently pursuing my interest in using mathematics as a language for art." --- Kaz Maslanka, Mathematical poet and Polyartist, D3 Technologies, San Diego, CA