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Math ImageryThe 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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"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, INApr 14, 2009
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"The Path Crumpled Paper Takes," by Jeanette Powers, Rockhurst University, Kansas City, MO (2008)Ink and paper, 11" x 15". "A classic example to explain fractal dimension is the piece of crumpled paper. In this example, one takes a sheet of paper to be 2 dimensional (ignoring the very thin thickness). This then is a good representation of the mathematical plane. However, if we crumple the paper into a ball, as seen below the frame, it seems to take on a volume, or third dimension. Now, there is a meta-level to the inter-dimensionality of this system. If one flattens the paper back into the two dimensional sheet of paper, then one can draw a continuous line ( in blue) of all the folds that happened during the crumpling process. Now a line is considered to be one dimensional, but is the space this line takes up really best described with only one dimension?" --- Jeanette Powers, Student, Physics and Math Department, Rockhurst University, Kansas City, MOApr 14, 2009
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"Uniqueness--and Infinity," by Jo Niemeyer (2008)Prints, 20" x 20". "To realize the concept of 'uniqueness' in art is a true challenge. And no easy task for an artist. This could be said for the concept 'infinity' as well. As a visual experiment with both of them, this problem is shown systematically in two steps in the following two graphics: The top image shows the overlapping of two geometrical grids. The size of the mesh corresponds to the relation 1 : 0.625. Or the Fibonacci numbers 5 and 8. There are nine grid elements, which overlap accurately. Furthermore the constellations of overlappings reiterate themselves. The two grids behave periodically. There is infinity - but no uniqueness. The bottom image shows the overlapping of two grids as well. Very similar to the top image, but the size of their meshes correspond here exactly to the relation of the golden section. 1 : 0.6180339... As the last number is an irrational number, the two grids behave aperiodically. Only the upper left two grid elements overlap accurately. Each overlapping constellation of the elements is unique, even if the size of the grid would be extended to infinity!" --- Jo Niemeyer, Freelance artist, Schluchsee, GermanyApr 14, 2009
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"Pulse," by Jeanette Powers, Rockhurst University, Kansas City, MO (2008)Acrylic, 6" x 24". "This piece explores Hausdorff Dimension. Chaos and dynamical systems collapse in ordered ways. A nebula coalescing into a galaxy, a frozen molecule tossing through the tumult and falling as a six-sided crystal, the Mandelbrot Set. As an artist, I've tried to use chaotic interactions as a tool to express the limitations of our control and the beauty of chaos. This painting uses cellophane crushed into wet pigment to create the random patterning of the surface. The result is a chaotic landscape reminiscent of leaves, cells, rivulets, the cracked dirt of arid lands. All chaotic processes which leave a recognizable mark. The pattern is not exact, but exhibits self-similarity." --- Jeanette Powers, Student, Physics and Math Department, Rockhurst University, Kansas City, MOApr 14, 2009
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"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
Apr 14, 2009
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"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, CAApr 14, 2009
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"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
Apr 14, 2009
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"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, AZApr 14, 2009
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"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, NJApr 14, 2009
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"(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, NYApr 14, 2009
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"(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, NYApr 14, 2009
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"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, AZApr 14, 2009
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"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, WIApr 14, 2009
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"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, MOApr 14, 2009
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"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, MDApr 14, 2009
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