Every summer, 200-some artists, mathematicians and technologists gather at the Bridges conference to celebrate connections between mathematics and the arts. It’s five exuberant days of sharing, exploring, puzzling, building, playing and discussing diverse artistic domains, from poetry to sculpture.

The Wolfram Language is essential to many Bridges attendees’ work. It’s used to explore ideas, puzzle out technical details, design prototypes and produce output that controls production machines. It’s applied to sculpture, graphics, origami, painting, weaving, quilting—even baking.

In the many years I’ve attended the Bridges conferences, I’ve enjoyed hearing about these diverse applications of the Wolfram Language in the arts. Here is a selection of Bridges artists’ work.

George Hart

George Hart is well known for his insanely tangled sculptures based on polyhedral symmetries. Two of his recent works, SNO-Ball and Clouds, were puzzled out with the help of the Wolfram Language:

This video includes a Wolfram Language animation that shows how the elements of the Clouds sculpture were transformed to yield the vertically compressed structure.

One of Hart’s earliest Wolfram Language designs was for the Millennium Bookball, a 1998 commission for the Northport Public Library. Sixty wooden books are arranged in icosahedral symmetry, joined by cast bronze rings. Here is the Wolfram Language design for the bookball and a photo of the finished sculpture:

One of my favorite Hart projects was the basis of a paper with Robert Hanson at the 2013 Bridges conference: “Custom 3D-Printed Rollers for Frieze Pattern Cookies.” With a paragraph of Wolfram Language code, George translates images to 3D-printed rollers that emboss the images on, for example, cookie dough:

It’s a brilliant application of the Wolfram Language. I’ve used it myself to make cookie-roller presents and rollers for patterning ceramics. You can download a notebook of Hart’s code. Since Hart wrote this code, we’ve added support for 3D printing to the Wolfram Language. You can now send roller designs directly to a printing service or a local 3D printer using Printout3D.

Christopher Hanusa

Christopher Hanusa has made a business of selling 3D-printed objects created exclusively with the Wolfram Language. His designs take inspiration from mathematical concepts—unsurprising given his position as an associate professor of mathematics at Queens College, City University of New York.

Hanusa’s designs include earrings constructed with mesh and region operations:

… a pendant designed with transformed graphics primitives:

… ornaments designed with ParametricPlot3D:

… and a tea light made with ParametricPlot3D, using the RegionFunction option to punch an interesting pattern of perforations into the cylinder:

Hanusa has written about how he creates his designs with the Wolfram Language on his blog, The Mathematical Zorro. You can see all of Hanusa’s creations in his Shapeways shop.

William F. Duffy

William F. Duffy, an accomplished traditional sculptor, also explores forms derived from parametric equations and cast from large-scale resin 3D prints. Many of his forms result from Wolfram Language explorations.

Here, for example, are some of Duffy’s explorations of a fifth-degree polynomial that describes a Calabi–Yau space, important in string theory:

Duffy plotted one instance of that function in Mathematica, 3D-printed it in resin and made a mold from the print in which the bronze sculpture was cast. On the left is a gypsum cement test cast, and on the right the finished bronze sculpture, patinated with potassium sulfide:

On commission from the Simons Center for Geometry and Physics, Duffy created the object on the left as a bronze-infused, stainless steel 3D print. The object on the right was created from the same source file, but printed in nylon:

Duffy continues to explore functions on the complex plane as sources for sculptural structures:

You will be able to see more of Duffy’s work, both traditional and mathematical, on his forthcoming website.

Robert Fathauer

Robert Fathauer uses the Wolfram Language to explore diverse phenomena, including fractal structures with negative curvature that are reminiscent of natural forms. This print of such a form was exhibited in the Bridges 2013 art gallery:

Fathauer realizes the ideas he explores in meticulously handcrafted ceramic forms reminiscent of corals and sponges:

One of Fathauer’s Mathematica-designed ceramic works consisted of 511 cubic elements (!). Here are shots of the Wolfram Language model and its realization, before firing, as a ceramic sculpture:

Unfortunately, in what Fathauer has confirmed was a painful experience, the sculpture exploded in the kiln during firing. But this structure, as well as several other fractal structures designed with the Wolfram Language, is available in Fathauer’s Shapeways shop.

Martin Levin

Martin Levin makes consummately crafted models that reveal the structure of our world—the distance, angular and topological relationships that govern the possibilities and impossibilities of 3D space:

What you don’t—or barely—see is where the Wolfram Language has had the biggest impact in his work. The tiny connectors that join the tubular parts are 3D printed from models designed with the Wolfram Language:

Levin is currently designing 3D-printed modules that can be assembled to make a lost-plastic bronze casting of a compound of five tetrahedra:

The finished casting should look something like this (but mirror-reversed):

Henry Segerman

Henry Segerman explored some of the topics in his engaging book Visualizing Mathematics with 3D Printing with Wolfram Language code. While the forms in the book are explicitly mathematical, many have an undeniable aesthetic appeal. Here are snapshots from his initial explorations of surfaces with interesting topologies…

… which led to these 3D-printed forms in his Shapeways shop:

His beautiful Archimedean Spire…

… was similarly modeled first with Wolfram Language code:

In addition to mathematical models, Segerman collaborates with Robert Fathauer (above) to produce exotic dice, whose geometry begins as Wolfram Language code—much of it originating from the Wolfram MathWorld entry “Isohedron”:

Elisabetta Matsumoto

In addition to constructing immersive virtual reality hyperbolic spaces, Elisabetta Matsumoto turns high-power mathematics into elegant jewelry using the Wolfram Language. This piece, which requires a full screen of mathematical code to describe, riffs on one of the earliest discovered minimal surfaces, Scherk’s second surface:

Continuing the theme of hyperbolic spaces, here’s one of Matsumoto’s Wolfram Language designs, this one in 2D rather than 3D:

You can see Matsumoto’s jewelry designs in her Shapeways shop.

Koos and Tom Verhoeff

Father and son Koos and Tom Verhoeff have long used the Wolfram Language to explore sculptural forms and understand the intricacies of miter joint geometries and torsion constraints that enable Koos to realize his sculptures. Their work is varied, from tangles to trees to lattices in wood, sheet metal and cast bronze. Here is a representative sample of their work together with the underlying Wolfram Language models, all topics of Bridges conference papers:

Three Families of Mitered Borromean Ring Sculptures

Mitered Fractal Trees: Constructions and Properties

Folded Strips of Rhombuses, and a Plea for the Square Root of 2 : 1 Rhombus

Tom Verhoeff’s YouTube channel has a number of Wolfram Language videos, including one showing how the last of the structures above is developed from a strip of rhombuses.

In 2015, three Verhoeff sculptures were installed in the courtyard of the Mathematikon of Heidelberg University. Each distills one or more mathematical concepts in sculptural form. All were designed with the Wolfram Language:

You can find detailed information about the mathematical concepts in the Mathematikon sculptures in the Bridges 2016 paper “Three Mathematical Sculptures for the Mathematikon.”

Edmund Harriss

Edmund Harriss has published two best-selling thinking person’s coloring books, Patterns of the Universe and Visions of the Universe, in collaboration with Alex Bellos. They’re filled with gorgeous mathematical figures that feed the mind as well as the creative impulse. Edmund created his figures with Mathematica, a tribute to the diversity of phenomena that can be productively explored with the Wolfram Language:

Loe Feijs and Marina Toeters

Loe Feijs and Marina Toetters are applying new technology to traditional weaving patterns: puppytooth and houndstooth, or pied-de-poule. With Wolfram Language code, they’ve implemented cellular automata whose patterns tend toward and preserve houndstooth patterns:

By adding random elements to the automata, they generate woven fabric with semi-random patterns that allude to houndstooth:

This video describes their houndstooth work. You can read the details in their Bridges 2017 paper, “A Cellular Automaton for Pied-de-poule (Houndstooth).”

Caroline Bowen

You can hardly find a more direct translation from mathematical function to artistic expression than Caroline Bowen’s layered Plexiglas works. And yet her craftsmanship and aesthetic choices yield compelling works that transcend mere mathematical models.

The two pieces she exhibited in the 2016 Bridges gallery were inspired by examples in the SliceContourPlot3D documentation (!). All of the pieces pictured here were created using contour-plotting functions in Mathematica:

In 2017, Bowen exhibited a similarly layered piece with colors that indicate the real and imaginary parts of the complex-valued function ArcCsch[z⁴]+Sec[z²] as well as the function’s poles and branch cuts:

Jeannine Mosely

Paper sculptor Jeannine Mosely designs some of her origami crease patterns with the Wolfram Language. In some cases, as with these tessellations whose crease patterns require the numerical solution of integrals, the Wolfram Language is essential:

Mosely created these “bud” variations with a parametric design encapsulated as a Wolfram Language function:

If you’d like to try folding your own bud, Mosely has provided a template and instructions.

Helaman Ferguson

The design and fabrication of Helaman Ferguson’s giant Umbilic Torus SC sculpture was the topic of a Bridges 2012 paper authored with his wife Claire, “Celebrating Mathematics in Stone and Bronze: Umbilic Torus NC vs. SC.”

The paper details the fabrication of the sculpture (below left), an epic project that required building a gantry robot and carving 144 one-ton blocks of sandstone. The surface of the sculpture is textured with a Hilbert curve, a single line that traverses the entire surface, shown here in a photo of an earlier, smaller version of the sculpture (right):

The Hilbert curve is not just surface decoration—it’s also the mark left by the ball-head cutting tool that carved the curved surfaces of the casting molds. The ridges in the surface texture are the peaks left between adjacent sweeps of the cutting tool.

Ferguson attacked the tasks of modeling the Hilbert curve tool path and generating the G-code that controlled the CNC milling machine that carved the molds with Mathematica:

Christopher Carlson

I too participate in the Bridges conferences, and I use the Wolfram Language nearly every day to explore graphical and sculptural ideas. One of the more satisfying projects I undertook was the basis of a paper I presented at the 2015 Bridges conference, “Algorithmic Quilting,” written in collaboration with Theodore Gray and Nina Paley.

The paper describes an algorithmic method we used to generate a wide variety of single-line fills for quilts. Starting with a distribution of points, we make a graph on the points, extract a spanning tree from it and render a fill by tracing around the tree:

We tested the algorithm by generating a variety of backgrounds for a quilt based on frames of Eadweard Muybridge’s horse motion studies:

Here’s an animation of the frames in the quilt:

---

If you’re an artist, designer or architect who uses the Wolfram Language in your work, I’d like to hear about what you do. If you’re looking for a mathematical artist, we know lots of them. In either case, drop me a line at artists@wolfram.com.