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.

"Capsule," by David Reimann (Albion College Albion, MI) 351 views18 x 45 x 18 cm, paper backed cherry veneer, metal fasteners, 2016

This is a capped cylinder was created using 75 squares that are 5 cm on a side and connected at their corners using split-pin fasteners. The end-caps are based on dodecahedral hemispheres and the cylinder is based on a planar hexagonal tessellation, which results in a polyhedral form in which every vertex has order 3. The edges in the underlying form have been replaced by squares, resulting in an open lattice form. Every opening is either triangular (throughout), pentagonal (on the endcaps), or hexagonal (on the cylinder). --- David Reimann

"4-Wheel Decomposition," by Ally Stacey (Oregon State University, Corvallis)350 views45 x 60 cm, Crayola crayon and Sharpie, 2015

Learning math inspires me to make art which in turn helps me better understand the math. This work is essentially a change-of-basis calculation going from the space of Closed Jacobi diagrams (trivalent graphs with an outer circle) to the space of Chord Diagrams, which are trivalent graphs with all vertices on the outer circle. These algebras are important to the field of Vassiliev Knot Invariants. This is done via what's the called the STU relation which is a way of resolving internal vertices. All algebraic steps are shown in the foreground. Making these helps me keep track of calculations in my research in an aesthetically pleasing way. --- Ally Stacey

"Being Dis-Functional 00_01e1a / 00_01d1a," by Robert Krawczyk (Illinois Institute of Technology, Chicago, IL)347 views19 x 8 x 12 cm, sandstone, 2016

Lemniscate of Bernoulli, included angle of 90 and 290 degrees with a 225 degree twist; 90 and 270 degrees with a 180 degree twist. This series investigates a simple form based on approximately one-half of a Lemniscate of Bernoulli curve, developed about 1694. It is very similar to a common eight curve, except the loops are more elliptical. For the enclosure a simple solid surface was incorporated. Custom software has been developed to generate these containers. These containers are computationally generated by custom developed software, a method which can express a consistency of a developing concept and enable a variety of designs that focus on the idea and its variations. Then the digital model is 3D printed in sandstone. More information.--- Robert Krawczyk

"Towards Infinite Smallness - 2," by Irene Rousseau (www.irenerousseau.com)347 views47 x 47 x 8 cm, Venetian glass on wood hand cut by artist, 2016

This mosaic wall sculpture consists of tessellated (tightly fitted together) patterns and each sculpture is a 3D disc with 18.5" diameter. The hyperbolic sculpture is composed of planes, each suggests negative curvature and represents the concept of infinite smallness contained within the bounding edge. Hyperbolic geometry can be represented as the points in a circular disc with hyperbolic distance defined. A hyperbolic plane has infinite structures within boundaries.In my hyperbolic sculpture repeating patterns which decrease in size at the bounding edge metaphorically represent infinity. Symmetry is a transformation that preserves the distance as shown on the surface tiling patterns of the sculpture. --- Irene Rousseau

"Linked Tetra Frames," by Carlo Séquin (University of California, Berkeley)346 views48 x 40 x 30 cm, pvc-pipes, painted, 2016

With Walt van Ballegooijen we have studied the number of topologically different ways in which two identical tetrahedral wire-frames could be interlinked. Not counting mirror images separately, we have found eleven different configurations. For each topological configuration there is a maximal clearance between the infinitely thin mathematical wire frames. If the edges are expanded to a diameter identical to this clearance, then a rigid structure results. This particular configuration has been scaled up to accommodate fattened edges of 22mm diameter, the size of the PVC pipes employed. --- Carlo Séquin

"Borromean racks," by Saul Schleimer (University of Warwick, Coventry, UK) and Henry Segerman (Oklahoma State University, Stillwater, OK)345 views10 x 10 x 10 cm, 3D printed nylon plastic, 2013

This sculpture consists of three identical pieces. Each has two identical rods; each rod is a rectangular prism with racks on three of its four sides. These racks mesh with one or two racks on the other pieces, for a total of 12 meshings. The pieces interlink in the fashion of the Borromean rings; their long axes form a standard orthogonal frame. When we take the directions of the racks into account, the sculpture has a handedness. At its middle position it realizes its maximal symmetry, a dihedral group of order six. Curiously, the Borromean Racks give an example of a triple of gears that mesh pairwise, but are not frozen. Challenge: Does this pattern of racks extend to tile three-space? If so, how many degrees of freedom does it have? --- Saul Schleimer and Henry Segerman

"Coordinate Axis, Highly Unlikely Square and Highly Unlikely Triangle," by Gwen Fisher (www.beadinfinitum.com)284 viewsMaterials: 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)

"Mobius Frame with 2 Holes (View II)," by Gwen Fisher (www.beadinfinitum.com)279 viewsMaterials: 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)

"Mazzocchio by Leonardo da Vinci," by Nedeljko and Milka Adžić (Novi Sad, Serbia)271 views40 x 40 x 10 cm, plastic, 2017

I was inspired by the Leonardo da Vinci codex, and I realized the polyhedra illustrated by Leonardo da Vinci. The polyhedra represent both Renaissance elements of surprise and modern sculpture, highlighting the relevance of Leonardo and his artistic-intellectual context in the present time. Filippo Brunelleschi invented perspective during the Renaissance period, when the use of polyhedrons and other geometric forms became more frequent, and Mazzocchio, the popular Florentine headwear of the time, also became an icon of perspective. -- Nedeljko and Milka Adžić

"A Butterfly Through Time," by Linda Beverly (California State University East Bay, Hayward)263 views61 x 51 cm, ink on paper, 2017

I enjoy seeking out interesting intersections between mathematics, computer science, and art. This image is an embedding of natural imagery video of flowers and a butterfly, using Locally Linear Embedding (LLE) a nonlinear dimensionality reduction technique, introduced by Saul & Roweis in 2000. LLE, an unsupervised machine learning algorithm, was applied to five frames of creative commons video of a butterfly flapping it wings surrounded by flowers utilizing open source software. -- Linda Beverly

"Slinky Spheres," by Dan Bach (Oakland, CA)239 views40 x 45 cm, inkjet print on canvas, 2017

Twenty colored spheres are surrounded by ten greenish, slinky-like toroidal helices. The helices also follow paths traced out by linear combinations of the normal and binormal vectors to the curve joining the centers of the spheres. This makes a kind of a symbiotic-geometric relationship between the solid objects and the surrounding safety net. Do you see a pattern for the colors of the spheres? -- Dan Bach

"Bounded by the Trefoil," by Elliott A. Best (University of California Riverside)238 views24 x 12 x 12 cm, copper mesh, paper mâché, 2016

This Seifert Surface is an orientable surface whose boundary is the trefoil knot. The sculpture is based on a copper mesh frame with paper mâché and acrylic paint. Contrasting black and blue colors are used to highlight the distinct sides of this orientable surface. In creating a physical model, I found a deeper appreciation for the particular character of the trefoil knot. -- Elliott A. Best

"Chaos Game Cut 6," by Tom Bates (Santa Barbara, CA)218 views50 x 70 cm, digital print, custom software, 2017

This piece derives from work I did to generalize the so-called chaos game, which yields a Sierpinski gasket, to any regular polygon. It is a montage of fractal and near-fractal images that are the natural product of such a generalization, especially when that is cast into code. The stained-glass-like background, an "off-resonance" hexagonal analog to the stochastically produced Sierpinski gasket, is behind an "inverse" pentagonal analog, and a flare from the paths taken between chosen points of an inverse and off-resonance heptagonal analog. -- Tom Bates

"Temari Permutation Ball," by Debra K. Borkovitz (Wheelock College, Boston, MA)215 views17 x 17 x 17 cm, styrofoam, yarn, thread, 2017

I was surprised and delighted to discover a truncated octahedron when I first drew the graph whose vertices are permutations on four elements and whose edges connect permutations that swap adjacent elements (the Cayley graph of S_4, generated by (12), (23), and (34)). I had it in the back of my mind for years that I'd like to find a visual representation of this graph, and when I started learning temari -- a Chinese/Japanese craft of embroidery on yarn/thread balls -- I realized this medium would work well. This ball is my second effort, and I am currently working on a third -- experimenting with different color combinations. -- Debra K. Borkovitz

"The Symmetric Four-Color Simple Imperfect Squared Square," by Regina Bittencourt (Santiago, Chile)211 views50 x 50 cm, acrylics on canvas, 2017

This artwork mixes three math problems: It is a Simple Imperfect Squared Square of order 21, colored using the Four-Color Map Theorem and design and painted with Symmetry; the 21 squares are not of different sizes which makes it imperfect, and simple because no subset of the squares forms a rectangle or a square; and the Four-Color Map Theorem states that any map in a plane can be painted using four colors, so that regions sharing a common boundary do not share the same color. The main square has rotational Symmetry of order 4, since the tiling is invariant when rotated by 90 degrees. -- Regina Bittencourt