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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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chadbourne14-inclusion2.jpg
"Enigmatic Plan of Inclusion II, " by Conan Chadbourne (San Antonio, TX)24" x 24", archival inkjet print, 2013
Best photograph, painting, or print

"My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the frequently mystical or cosmological significance that can be attributed to such imagery. "

These images are investigations of the subgroup structure of the icosahedral group (A5). At the center of each image is a graphical representation of A5, as formed by orientation-preserving pairs of reflections in circles and lines in the plane. This is surrounded by similar graphical representations of the seven conjugacy classes of (proper, non-trivial) subgroups of A5, with the trivial group depicted as the space outside of the large circular frame. The interstices between the group images indicate the relationships of inclusion between the different groups, with colors being used to distinguish maximal subgroup relationships, and small graphical markers used to indicate the particular numbers of conjugates involved in each relationship. --- Conan chadbourne (http://www.conanchadbourne.com)
May 05, 2014
chadbourne14-inclusion1.jpg
"Enigmatic Plan of Inclusion I, " by Conan Chadbourne (San Antonio, TX)24" x 24", archival inkjet print, 2013
Best photograph, painting, or print
2014 Mathematical Art Exhibition

"My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the frequently mystical or cosmological significance that can be attributed to such imagery. "

These images are investigations of the subgroup structure of the icosahedral group (A5). At the center of each image is a graphical representation of A5, as formed by orientation-preserving pairs of reflections in circles and lines in the plane. This is surrounded by similar graphical representations of the seven conjugacy classes of (proper, non-trivial) subgroups of A5, with the trivial group depicted as the space outside of the large circular frame. The interstices between the group images indicate the relationships of inclusion between the different groups, with colors being used to distinguish maximal subgroup relationships, and small graphical markers used to indicate the particular numbers of conjugates involved in each relationship. --- Conan Chadbourne (http://www.conanchadbourne.com)
May 05, 2014
burns14-visualproof.jpg
"Visual Proof," by Anne Burns (professor emerita, Long Island University, Brookville, NY)13" x 13", digital print, 2013

"Visualization is an important aid in the study of mathematics."

Each of the disks in the 3X3 matrix of disks is a picture of the first five backward iterations of f(z)=z^n+c/z^m where c is a small positive real number. The rows represent n=2,3,4 and the columns represent m=2,3,4. The black disks in the center consist of the set of points z such that |f(z)|>1.1. The second largest sets of disks are blue; they are the inverse images of the black disks under f; ochre disks are the inverse images of blue disks; red disks are the inverse images of ochre disks, etc. First notice the n+m symmetry in each disk. Next, can you identify n and m by this pattern? Hint: choose one blue disk in each entry and count the number of pre-images closer to the center and the number of pre-images further away from the center. -- Anne Burns (http://anneburns.net)
May 05, 2014
bodner14-11n9stars.jpg
"Eleven and Nine Stars," by B Lynn Bodner (Monmouth University, West Long Branch, NJ)20" x 16", paper and ink computer-generated image, 2013

"Geometric Islamic patterns, especially those consisting predominantly of star polygons, fascinate me. "

This pattern was created using Euclidean construction techniques and the electronic equivalent of the compass and straightedge. It was based on a rectangular repeat unit sketch known as Catalog Number 42 (CN42) of the Topkapi Scroll, which contains 114 Islamic architectural and ornamental designs dating from the 15th or 16th century. CN42 has half 11-stars along its vertical edges and half 9-stars along its horizontal edges. To achieve a complete pattern, this repeat unit must be replicated by reflection across its edges. The image displayed here contains four colored copies of the CN42 repeat unit. Since CN42 is the only sketch in the Topkapi Scroll containing 11-pointed star polygons, it was one of the more challenging ones to recreate. --- B. Lynn Bodner (http://bluehawk.monmouth.edu/~bodner/papers/index.htm)
May 05, 2014
belcastro14-spring.jpg
"Spring Forest (5,3)," by sarah-marie belcastro (Hadley, MA)Embedded, unembedded, and cowl; 12" x 11" x 9", Knitted wool (Dream in Color Classy, in colors Happy Forest and Spring Tickle), 2009 and 2013

"I am a mathematician who knits as well as a knitter who does mathematics."

A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Here are three instantiations of a (5,3) torus knot:
(a, middle) The knot embedded on a torus. A (p,q) torus knot may be drawn on a standard flat torus as a line of slope q/p. The challenge is to design a thickened line with constant slope on a curved surface. (b, top) The knot projection knitted with a neighborhood of the embedding torus. The knitting proceeds meridianwise, as opposed to the embedded knot, which is knitted longitudinally. Here, one must form the knitting needle into a (5,3) torus knot prior to working rounds. (c, bottom) The knot projection knitted into a cowl. The result looks like a skinny knotted torus. --- sara-marie belcastro (http://www.toroidalsnark.net)
May 05, 2014
farris-cmMpls.jpg
"A cm Pattern from a Minneapolis Skyline, (vertical format)," by Frank A. Farris, Santa Clara University, CAInk jet on paper.

Fantastical samurai appeared when I used a photograph of the Minneapolis skyline on an autumn day in conjunction with wave functions adapted for cm patterns. --- Frank A. Farris
Sep 18, 2013
Z_J-water.jpg
"Land and the Sea," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013
mod1-orig.jpg
"Manifold II (Bird's Eye View) in memory of Bill Thurston," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013
manifold5b.jpg
"Manifold 5," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013
crochet-in-tree-orig.jpg
"Manifold II (in tree) in memory of Bill Thurston," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013
autumn.jpg
Autumn (Thanksgiving), by Daina Taimina (Cornell University, Ithaca, NY), photo @Daina Taimina44x44x29 cm, 2006-2010, wool, silk

Inspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)
Jun 21, 2013
7shades1.jpg
"Seven Shades of Purple," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures --- Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013
farris-turtles-hi-res.jpg
"Turtles with Local Symmetry," Frank A. Farris, Santa Clara University, CAInk jet on paper, 2012.

There is no mirror symmetry in this image, only 6-fold rotational symmetry. And yet our eye cannot help seeing symmetry in the turtle shapes. This "local symmetry" is the topic of an article, "Local symmetry in wallpaper," in preparation. --- Frank A. Farris
Jun 21, 2013
farris-pepper-hi-res.jpg
"A cmm Pattern from Peppers on a Cutting Board," by Frank A. Farris, Santa Clara University, CAInk jet on paper.

The bright oranges in this cmm pattern come from a photograph of assorted chopped peppers, with collard greens and the glint of the knife as nice contrasts. --- Frank A. Farris
Jun 21, 2013
farris-tiffany-hi-res.jpg
"Tiffany Glass from a Mountain Gentian and its Negative," by Frank A. Farris, Santa Clara University, CAInk jet on paper, 2012.

Using a composite photograph of a mountain gentian juxtaposed with its negative to produce an intense contrast of purple with the original green, I used just the right waves to make wallpaper with horizontal mirrors, vertical glides, and 2-fold rotational symmetry. In the notation of the International Union of Crystallographers, the symmetry group of this pattern is pmg, but if we allow color-swapping transformations as generalized symmetries, the larger group is cmm, so this pattern type is called cmm/pmg. The "Seeing Symmetry" virtual exhibition at includes more works and details. --- Frank A. Farris
Jun 21, 2013
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