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.

"A Frieze Sampler," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 24", 2016

Friezes take their name from an architectural feature: a band of decoration typically along the top of a wall. In the theory of plane symmetry, the first famous classification result says that there exist exactly seven types of frieze patterns. Every pattern ever constructed by anyone, as long as it repeats exactly along one direction, can be classified as belonging to one of these seven types. Mathematicians, like many humans, like to collect exactly one artifact of each type. Here is my sampler of seven friezes, made from source photographs of California wildflowers. I especially like the fourth one down, which exemplifies a glide reflection symmetry—a flip-and-slide motion that leaves the pattern unchanged. --- Frank A. Farris Jul 05, 2017

"Autumn Moths from a Hike in the Hills," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 20" x 20", 2016

The big surprise about wallpaper patterns is that there are exactly 17 types. After making many examples of each type, I still return to the type of this pattern as a favorite; it’s called p31m. Notice the three-fold rotational symmetry, but also the alternation of mirror symmetry with three-way pinwheels. For me, this gives a beautiful balance of sameness and variation. It’s symmetric, yet organic. I took the source photograph on an autumn day, hiking in the golden hills above San Jose, CA. This piece shows that interesting patterns can arise from photographs with rather limited color palettes. --- Frank A. Farris Jul 05, 2017

"The Alchemist's Shelf," by Frank A. Farris, Santa Clara University, CADigital print on aluminum, 51 x 61 cm, 2017

The rich colors of the alchemist's globes come from scenes captured with a camera that records the entire visual sphere. They tell my story of leaving home in San Jose for an autumn sabbatical in Minnesota. At the top left, we see my stairwell at home, where a stained glass window made by Hans Schepker hangs next to a quilt. Moving to the right, we find the glorious Lakewood Chapel and St. John's Abbey. Bright fall colors at the bottom left give way to another view of St. John's Abbey, and then we return home. Texture maps on the spheres use quotients of harmonic polynomials with icosahedral or octahedral symmetry. The scene is composed in Photoshop, with about 14 hours of ray-tracing. --- Frank A. Farris Jul 05, 2017

"Torus," by Jiangmei Wu (Indiana University, Bloomington)Best textile, sculpture, or other medium - 2017 Mathematical Art Exhibition

45 x 45 x 20 cm, Hi-tec Kozo paper, 2014

"Torus" is folded from one single sheet of uncut paper. Gauss’s Theorema Egregium states that the Gaussian curvature of a surface doesn’t change if one bends the surface without stretching it. Therefore, the isometric embedding from a flat square or rectangle to a torus is impossible. The famous Hévéa Torus is the first computerized visualization of Nash Problem: isometric embedding of a flat square to a torus of C1 continuity without cutting and stretching. Interestingly, the solution presented in Hévéa Torus uses fractal hierarchy of corrugations that are similar to pleats in fabric and folds in origami. In my Torus, isometric embedding of a flat rectangle to a torus of C0 continuity is obtained by using periodic waterbomb tessellation. --- Jiangmei Wu

May 09, 2017

"Animal Heads - Wolf," by Chris Watson (Tessellation Art, Prague, Czech Republic)50 x 50 x 5 cm, digital print on canvas, 2016

"Animal Heads - Wolf" is made from hundreds of tessellating wolves. It uses a mosaic technique that I developed specifically to create mathematical artwork. As with traditional square mosaics, each tile is a different colour. In this case, the individual tiles are howling wolves. Look closely to see the detail on each, or take a step backwards and the wolf head is revealed. --- Chris Watson

May 09, 2017

"4-Wheel Decomposition," by Ally Stacey (Oregon State University, Corvallis)45 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 May 09, 2017

"Linked Tetra Frames," by Carlo Séquin (University of California, Berkeley)48 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 May 09, 2017

"Borromean racks," by Saul Schleimer (University of Warwick, Coventry, UK) and Henry Segerman (Oklahoma State University, Stillwater, OK)10 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

May 09, 2017

"Towards Infinite Smallness - 2," by Irene Rousseau (www.irenerousseau.com)47 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 May 09, 2017

"Heighway Dragon Tiling," by Larry Riddle (Agnes Scott College, Decatur, GA)25 x 25 cm, back stitch embroidery on 18 count canvas, 2012

The Heighway Dragon fractal was introduced to mathematicians by Martin Gardner in his Mathematical Games column of Scientific American. One of the amazing properties of the dragon is that despite its boundary being extremely non-smooth, four copies of the dragon fit exactly together around a central point. This back stitch design illustrates that mathematical idea by showing 4 copies of 12 iterations in the construction of the dragon via line segments, each rotated in succession by 90° around the center. The entire image can then be repeated to tile the plane. What might not be obvious from the design is that each dragon can be traced starting at the center point as one continuous alternating sequence of vertical and horizontal stitches. More information. --- Larry Riddle

May 09, 2017

"Capsule," by David Reimann (Albion College Albion, MI) 18 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 May 09, 2017

"Beloved," by Sharol Nau (Northfield, MN)25 x 17 x 18 cm, folded book, 2016

I enjoy folding the pages of a book systematically, especially based on a parabola because of an important option of placing the focus anywhere which leads to surprising shapes, arch curves that spring forward as the book opens. 'Beloved' was created by matching incremental points on an edge of the pages to a common point, the focus of a parabola. The collection of folds makes an envelope to the parabola. In this case I decorated a bit by including a focus near the top complementary to one at the bottom of the page. --- Sharol NauMay 09, 2017

"Symmetric Koch Curve II," by Vincent J. Matsko (University of San Francisco, CA)25 x 25 cm, digital print, 2016

The algorithm used to create this image is the same as that used to recursively define the Koch curve. In this case, the circle is divided into 336 parts (rather than 360 degrees), and the angles used in the algorithm are 283 and 179 of these parts (rather than 60 and 240 degrees). In this case, the curve is bounded, consists of 10,752 segments, and possesses 42-fold symmetry. More information. --- Vincent J. Matsko May 09, 2017

"Being Dis-Functional 00_01e1a / 00_01d1a," by Robert Krawczyk (Illinois Institute of Technology, Chicago, IL)19 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

May 09, 2017

"AAABBB, two juxtapositions: Dots & Blossoms, Windmills & Pinwheels," by Mary Klotz (Forestheart, Frederick MD & Unger WV)Honorable Mention - 2017 Mathematical Art Exhibition

66 x 46 x 3 cm, hand dyed silk ribbon, 2016

This permutated pair of triaxial weavings are exactly identical in weave structure, with identical color sequencing in all three directions. (AAABBB) Only the starting points of the color sequence in the diagonal elements vary between the two. --- Mary Klotz