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 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.

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jmm13-bosch~0.jpg
"Truchet from Truchet Tiles," by Robert Bosch (Oberlin College, Oberlin, OH)18" x 18", Digital print on canvas, 2012

Father Sébastien Truchet (1657-1729) was a Carmelite clergyman. He was King Louis XIV's favorite hydraulic engineer. He designed fonts. And in 1704 he published an article, "Mémoire sur les combinaisons," that described his mathematical and artistic investigations into how a simple set of square tiles, each divided by a diagonal into a white half and a black half, can be arranged to form an infinity of pleasing patterns. Today, Truchet's tiles are known as, well, Truchet tiles. To create a Truchet-tile portrait of Truchet, I started with the orientations specified by Pattern D of Plate 1 of Truchet's article. I then allowed the diagonals of the tiles to "flex" or bend at their midpoints. To make a tile darker, I would flex the diagonal into the black half. To make a tile brighter, I would flex the diagonal into the white half. With my Flexible Truchet Tiles, I can approximate any grayscale image, using the image to "warp" any initial pattern of Truchet tiles. -- Robert Bosch
May 16, 2013
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"Knight to C3," by Leo Bleicher (San Diego, CA)30" x 25", Photographic print of 3D model, 2011

My mathematical art forms several large series exploring the creation of complex forms through sequences of simple operations or representations of simple relationships. The operations include geometric transformations, neighbor finding, attraction/repulsion and others. These computational processes attempt to replicate features of both geologic and organic morphogenesis. This image was created by application of a sequence of 3D geometric transformations to a collection of spheres on the faces of a hollow cube. The exact sequence to produce this image was obtained through the use of a genetic algorithm using subjective aesthetic appeal as the fitness function. -- Leo Bleicher
May 16, 2013
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"Cardinal," by Harry Benke (Visual Impact Analysis LLC, Novato, CA)20" x 26.6", Giclee (pigmented archival print), 2012

I'm an artist and mathematician. My art attempts to produce a nexus between abstract mathematical beauty and the natural world to produce a satisfying aesthetic experience. I've been examining Kuen's surface for a very long time. The red shape is Kuen's surface as seen from above, looking down the z axis. Kuen's surface is well known since it has constant negative Gaussian curvature except on sets of measure 0. This surface is virtually never seen from above, which is intriguing and beautiful. -- Harry Benke (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.
May 16, 2013
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"Borromean links cage," by Roger Bagula (Lakeside, CA)3.4" x 3.2" x 0.9", White plastic, 2012

The art work is made using Mathematica and 3d models that can be printed are obtained for most of the work. Since I was a boy I have drawn jet air planes and I have made a number of 3d models what I imagined. I have been studying ruled surfaces determined by torus knots and Möbius connecting surfaces. Here three links in a Borromean configuration are bridged by a cage surface. The bridge surface is not a Seifert surface, but a simple set of ruled Bezier surfaces. --- Roger Bagula
May 16, 2013
125-tetrahedra-VIII.jpg
"125 tetrahedra in 25 projected 5-cells," (VIII) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-VII.jpg
"125 tetrahedra in 25 projected 5-cells," (VII) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-VI.jpg
"125 tetrahedra in 25 projected 5-cells," (VI) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-V.jpg
"125 tetrahedra in 25 projected 5-cells," (V) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-IV.jpg
"125 tetrahedra in 25 projected 5-cells," (IV) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-III.jpg
"125 tetrahedra in 25 projected 5-cells," (III) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-II.jpg
"125 tetrahedra in 25 projected 5-cells," (II) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
125-tetrahedra-I.jpg
"125 tetrahedra in 25 projected 5-cells," (I) by Carlo Séquin, University of California, BerkeleyThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This view (I) is featured on a printed AMS 125th anniversary poster, free upon request.
Mar 18, 2013
sierpinski-sieve-shields.jpg
"Sierpinski Sieve," a pancake by Nathan Shields (www.10minutemath.com)A Sierpinski triangle is a fractal, a structure that displays self-similarity at various scales. This fractal is created by recursively removing triangular pieces from the structure indefinitely - of course, the pancake isn't very hearty if you really do this, but you get the idea. To see about making your own fractal pancakes, as well as other topics I find interesting as a math teacher, check out my blog. --- Nathan Shields (www.10minutemath.com)

Jun 14, 2012
pythagorean-tree-shields.jpg
"Pythagorean Tree," a pancake by Nathan Shields (www.10minutemath.com)This fractal, like many others, is fun to doodle at faculty meetings. Here, each triple of touching squares encloses a right triangle in a traditional visualization of the Pythagorean Theorem. To see about making your own fractal pancakes, as well as other topics I find interesting as a math teacher, check out my blog. --- Nathan Shields (www.10minutemath.com)Jun 14, 2012
mandelbrot-set-shields.jpg
"Mandelbrot Set," a pancake by Nathan Shields (www.10minutemath.com)The Mandelbrot Set, which is one of the tastiest fractals, is the collection of points c on the complex plane which allow the iterated transformation z = z² + c to remain within a given threshold. I've always been awestruck by the infinite complexity that springs from that simple equation. To see about making your own fractal pancakes, as well as other topics I find interesting as a math teacher, check out my blog. --- Nathan Shields (www.10minutemath.com)Jun 14, 2012
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