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 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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"Hyperbolic Afghan {3, 7}," by Heidi Burgiel (Bridgewater State University, Bridgewater, MA)KnitPicks Shine Sport yarn: 60% cotton 40% modal, 7 x 44 x 44 cm, 2015

"Hyperbolic Afghan {3, 7}" illustrates a tiling of the hyperbolic plane by triangles, 3 at a vertex, in crocheted cotton. Adapting techniques developed by Joshua and Lana Holden, the piece is not assembled from flat triangles but instead approximates constant curvature over its entire surface. Its coloration, inspired by William Thurston's rendition of the heptagon tiling underlying the Klein quartic, suggests the identifications required to construct that surface as a quotient of the hyperbolic plane. --- Heidi Burgiel Mar 09, 2016

"Dragon Curve Double Knit Scarf," by Rachelle Bouchat (Indiana University of Pennsylvania, Indiana, PA)Merino Wool Yarn, 137 x 18 cm, 2015

This double knit scarf brings together the recursive construction of a fractal, the dragon fractal, as well as the recursive construction of an integer sequence, the Fibonacci sequence. The main panels of the scarf are based on a pattern developed from the eleventh iteration of the dragon fractal. Moreover, the striping pattern in between the main panels is illustrative of the Fibonacci sequence with color changes after 1 row, after another 1 row, after 2 rows, after 3 rows, after 5 rows, and with another color change after 8 rows. As this is a double knit pattern, the back side of the scarf is shown in the reverse color pattern. --- Rachelle Bouchat Mar 09, 2016

"The Jordan Curve Theorem," by Robert Bosch (Oberlin College Oberlin, OH)Lasercut woods, 15 x 45 cm, 2015

The Jordan Curve Theorem states that when a simple closed curve is drawn in the plane, it will cut the plane into two regions: the part lies inside the curve (here, the slightly darker-colored inset piece of wood), and the part that lies outside it (here, the slightly brighter and thicker frame). --- Robert Bosch Mar 09, 2016

"Rainbow Brunnian Link Cowl," by sarah-marie belcastro (MathILy, Holyoke, MA)Knitted wool (various sources) and printed photographs, 30 x 30 x 7 cm, 2015

The central property of the Borromean rings--that removing any component unlinks the remaining components, which collectively form the unlink--generalizes to the class of Brunnian links. The Rainbow Brunnian Link Cowl has seven components rather than the three components of the Borromean rings. All linking is intrinsic, rather than introduced post-construction via grafting. The Rainbow Brunnian Link Cowl is also a garment that can be worn two different ways, which are pictured alongside that cowl in the exhibit. --- sarah-marie belcastro Mar 09, 2016

From "Serenity to Monkey-Mind and Back (Two Twisted Tessellated Transforming Tori)," by Ellie Baker (Lexington, MA) Printed polyester crepe de chine, bead crochet (glass beads and thread), 70 x 50 cm, 2015

This infinity scarf and bead crochet necklace are twin tori. The fabric design is (an elongated version of) the infinitely repeating planar pattern that a tiny explorer could map by charting the surface of the necklace in all directions (the universal cover of the beaded rope). The two colors, identical tessellated wave motifs, gradually transform from "calm" to "busy." The pattern at each step has an increasing "busyness" quotient (a measure of how much the individual beads in a fundamental tile differ in color from neighboring beads). The scarf, sewn from a parallelogram to create a mobius-like twisted torus, has a small hole in one seam so that it can be turned inside out to explore the puzzling behavior of torus inversions. --- Ellie Baker Mar 09, 2016

"Coordinate Axis, Highly Unlikely Square and Highly Unlikely Triangle," by Gwen Fisher (www.beadinfinitum.com)Materials: seed beads, Nymo nylon thread

These three pieces are woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. The Coordinate Axis shows the basic structure of box stitch, and is also suitable for a game of children's Jacks. The Highly Unlikely Square and Triangle are beaded versions of the Impossible Triangle of Roger Penrose that was made famous by M.C. Escher. Compared with a regular square frame or triangular frame like you might hang on your wall, these frames have one quarter turn on each side. To see the effect of these twists, imagine painting a regular square frame with four colors to identify four paths: inside, outside, front and back. A similar coloring on the Highly Unlikely Square identifies four paths or faces, one of which is outlined with gold seed beads. Starting at the corner closest to the camera traveling clockwise, the golden face is outside, back, inside, front. In fact, all four faces are congruent. The effect of the quarter turns on the Highly Unlikely Triangle is different; there is only one face that travels around the triangle four times. -- Gwen Fisher (www.beadinfinitum.com) Feb 19, 2016

"Mobius Frame with 2 Holes (View II)," by Gwen Fisher (www.beadinfinitum.com)Materials: seed beads, Nymo nylon thread

This Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen Fisher (www.beadinfinitum.com)Feb 19, 2016

"Mobius Frame with 2 Holes (View I)" by Gwen Fisher (www.beadinfinitum.com)Materials: seed beads, Nymo nylon thread

This Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen Fisher (www.beadinfinitum.com)Feb 19, 2016

"Octahedral Cluster," by Gwen Fisher (www.beadinfinitum.com) Copyright 2005 by Gwen L. Fisher.Materials: white opalite glass, seed beads, Nymo nylon thread

The regular octahedron has 8 triangular faces, 6 vertices each of valence 4, and 12 edges. These 12 edges correspond to the 12 largest (white) beads in the Octahedral Cluster beaded bead. The 6 vertices of the octahedron appear as 6 stars with 4 points each. The 8 triangular faces correspond to where the points of the stars meet. This beaded bead is hollow, yet structurally stable. The stability comes from the way the small beads fit snugly into the spaces between the larger beads. The beaded bead shows virtually no thread; only beads are visible. It is springy between the fingertips and reforms its shape remarkably well. When free from compression, it is round from every angle. -- Gwen Fisher (www.beadinfinitum.com)Feb 19, 2016

"Sierpinski Tetrahedron (View V)," by Gwen Fisher (www.beadnfinitum.com) Materials: glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread

There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen Fisher (www.beadfinitum.com)Feb 19, 2016

There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen Fisher (www.beadfinitum.com) Feb 19, 2016

There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen Fisher (www.beadfinitum.com) Feb 19, 2016

There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen Fisher (www.beadinfinitum.com)Feb 19, 2016

"Sierpinski Tetrahedron (View I)," by Gwen Fisher (www.beadinfinitum.com) Materials: glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread

There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen Fisher (www.beadinfinitum.com)Feb 19, 2016

"Flow of complex valued functions (4)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. BurnsOct 22, 2015