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.

"Möbius Shorts," by Anduriel Widmark (Denver, CO)6 x 6 x 6 cm, flame worked borosilicate glass, 2016

Art is a good excuse to play. Exploring patterns and symmetry through abstraction presents an opportunity to look outside of a regular pattern of seeing. "Möbius Shorts" is similar, but distinct from the Möbius strip. Here, "Möbius Shorts" is outlined with glass rods that hold an invisible surface that is topologically equivalent to a klein bottle with a hole in it. This one-sided surface seems to hold rings that float in glass thus revealing its simple and elegant shape. -- Anduriel Widmark Apr 27, 2018

"165Q - Light Painting from the Pendulum Project," by Paul Wainwright (Atkinson, NH)50 x 50 cm, silver gelatin photographic print from 4" x 5" negative, 2017

Light painting of an LED attached to a Blackburn pendulum strung to produce orthogonal periods of motion in a 4-to-3 ratio. Musically, this would represent a major fourth interval. Exposure was made looking up in total darkness using a 4 x 5 inch sheet film camera. Total exposure time was 10 minutes 48 seconds, during which time the camera was rotated 90 degrees on a very slow turntable. Because the formula for the period of motion of a pendulum contains an angular dependence, the Blackburn pendulum goes in and out of resonance as the amplitudes decrease, resulting in interesting patterns that, to me, are quite beautiful. -- Paul Wainwright Apr 27, 2018

"Skeletal model of cubic close packing lattice structures," by Chia-Chin Tsoo (National Applied Research Laboratories, Hsinchu, Taiwan) and Bih-Yaw Jin (National Taiwan University, Taipei)14 x 14 x 14 cm, 3cm glass bugle beads, 2014

An octet-truss lattice structure with face centered cubic (fcc) crystal symmetry is a truss-like space frame based on the stacking of tetrahedra and octahedra in a ratio of 2:1. The octet-truss space frame is strong and lightweight because of the inherent rigidity of their fundamental building motifs. The bugle bead model presented represents a finite skeletal structure according to the fcc close packing and consists of a central smaller cuboctahedron (brown bugle beads) and an outer layer of frequency-two cuboctahedron (yellow bugle beads). --- Chia-Chin Tsoo and Bih-Yaw JinApr 27, 2018

"Beaded Pentagonal Hyperbolic Surface," by Stacy Speyer (Alameda, CA)18 x 18 x 5 cm, wooden beads on copper wire, 2017

Wooden beads are made into 32 pentagons: 8 in the inner ring and 24 in the outer ring. Four pentagons meet at every vertex creating the folds in this {5, 4} hyperbolic surface. The undulating surface shares an organic beauty with the variations of color in the wooden beads. While the copper wire's ability to hold the beads in this irregular space adds a sense of motion to the piece. The visual simplicity of the 8 pentagons in the inner ring contrasts with the density of the outer ring, which has 3 times as many pentagons. -- Stacy Speyer Apr 27, 2018

"Order and Chaos," by John Shier and Doug Dunham (University of Minnesota Duluth)One of our ideas concerns transitions from order to disorder. Going linearly from one to the other can seem jarring, so we use modifying functions that are 0 for some interval, then sweep up to 1 for an equal interval. This is the form of all the modifying functions in this image. The orientations of the squares start out upright and the positions are regular, then they become more random on the right. Similarly, the edges start out straight then become curved. There are two sets of paths through 3-dimensional RGB color space, the paths of one set start on the left at white and the other at black. Then they proceed toward random mid-range colors, so that at the right the color distributions of the two sets become the same. -- John Shier and Doug Dunham Apr 27, 2018

"Dodecahedral-Cluster of 25 Klein-Bottles," by Carlo Séquin (University of California, Berkeley)24 x 24 x 24 cm, 3D-print, PLA plastic, 2017

In this sculpture, 24 of the "4-stub Dyck funnels" have been aligned with the 24 edges of a rhombic dodecahedron, and their stubs have been connected with 48 tunnels. This yields a surface of genus 50 -- the equivalent of the connected sum of 25 Klein bottles with 24 punctures, exhibiting the 24-fold symmetry of the oriented cube. It took 132 hours to build this model on a LULZBOT 3D-printer. Support removal took several more hours. -- Carlo Séquin

Apr 27, 2018

"Trefoil spine," by Saul Schleimer (University of Warwick, Coventry, UK) and Henry Segerman (Oklahoma State University Stillwater, OK)9 x 9 x 8 cm, 3D printed nylon plastic, 2015

A spine of a three-dimensional manifold with boundary is a two-dimensional complex that the manifold deformation retracts to. Here, we show the trefoil knot, together with a spine of its complement in the three-sphere, stereographically projected to euclidean space. The windows form a distorted rectangular grid, with all angles 90 degrees. In one grid direction the windows lie along semicircles, each with both ends on the vertical axis. In the other grid direction, the windows trace out trefoil knots. The only exception is the windows meeting the dual circle to the vertical axis. This design was suggested to us by Dylan Thurston. -- Saul Schleimer and Henry Segerman Apr 27, 2018

"Perijove," by Kerry Mitchell (Phoenix, AZ)41 x 51 cm, digital print onto aluminum panel, 2017

This image is a manipulation of a photograph of Jupiter's red spot, taken by the Juno spacecraft in July 2017. The image was taken at perijove, the point in the orbit when the spacecraft is closest to the surface of the planet, which gives the image its title. The manipulation was achieved using an iterated discrete form of a complex nonlinear partial differential equation. The form of the equation was chosen for aesthetics, yet the resulting structures suggest the intricate fluid dynamics present in the planet's giant storm. -- Kerry Mitchell Apr 27, 2018

"Seed pod shaped lamp with spiraling hyperbolic band," by Gabriele Meyer (University of Wisconsin, Madison)60 x 40 x 40 cm, yarn and shaped line, 2017

This piece started as an ellipsoid. I then crocheted a band spiraling around it from top to bottom. Eventually this band became hyperbolic. Solutions to differential equations can look like this. Or the object could look like a seed pod. -- Gabriele Meyer Apr 27, 2018

"Search Lights. The Hilbert Metric Tiling the 2 Simplex," by Chris McCarthy (Borough of Manhattan Community College, New York, NY)28 x 43 x 2 cm, computer generated image printed on glossy paper, 2014

This artwork came out of my dissertation which involved proving theorems related to the Hilbert Metric. The Hilbert metric is a special way to define the distance between points. The Hilbert metric applied to the interiors of these two triangles results in a hyperbolic non-Euclidean geometry. With respect to this hyperbolic geometry the lines coming out of these vertices are parallel (non-intersecting). To our eyes, those lines do not look parallel, unless perhaps we see them as being parallel lines going off into the distance, drawn in perspective. As we look at the various patterns formed by the lines and the coloring, our mind tries to make sense of the complexity; our perception shifts between seeing hexagons (which are circles with respect to the Hilbert metric), parallel lines, and the occasional parallelepiped. -- Chris McCarthy Apr 27, 2018