Fundamental Regions: The Math/Art of Susan Goldstine

Fundamental Regions: The Math/Art of Susan Goldstine BOYDEN GALLERY ST. MARY'S COLLEGE OF MARYLAND JANUARY 18 - MARCH 12, 2022

We invite you to experience the visual arts through the intellectual practice of mathematics in “Fundamental Regions: The Math/Art of Susan Goldstine.” As the premier exhibition of Spring 2022, we celebrate Professor Goldstine’s distinguished career as an award-winning mathematician, artist, and gifted teacher with her first solo show. Goldstine joined the faculty of the College’s Department of Mathematics and Computer Sciences in 2004, and her artistic scholarship has been featured in numerous group shows, publications, lectures, presentations, and collaborations. Her accomplishments in the sciences undeniably establish her expertise in a field of research relevant to our understanding of the world we inhabit, for which she was named a Steven Muller Professor of the Sciences 2019-2022.

We co-created “Fundamental Regions” to showcase Goldstine’s skill as an artist-scholar-teacher in sharing complex ideas through making artwork and highlighting the process of discovery in creating mathematical ideas in physical form. By engaging through dialectic experience in the exhibition through first-person video labels and mathematical fiber arts displayed in a “traditional” art gallery, we dismantle binary tendencies to oppose math and art, or idea and form. We investigate, interrogate, and (re)interpret the category of “fine art” in recognizing that ideas are as much a part of the art-making as technique in multiple media. Goldstine’s art is conceptual and challenges us to see beyond the visual to intangible and experiential excellence.

Both the exhibition and catalogue are organized with mathematical themes over artistic media – we encourage you to seek out the essence of pattern, geometric procedures, and infinite possibilities in “Growth,” “Symmetry,” and “Topology.” Goldstine’s essays and mathematical patterns accompany images of each of the artworks featured in the exhibition to contextualize these themes in the catalogue. While in the exhibition, we welcome you to search for the overarching theme of distilling complex systems down to their essences in textures, shapes, shadows, and movement.

Erin Peters, PhD Director, Boyden Gallery and Collection

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Artist Statement There are two things I have done for as long as I can remember: explore mathematics and create things with my hands. As a girl (!) who showed a notable aptitude for mathematics in school, I worked out fairly early that I was likely to become a mathematician. Being an artist, on the other hand, snuck up on me later in life.

Sharing the beauty of mathematics with other people has always been as important to me as the mathematics itself. This is not always an easy sell.

Many people actively dislike or fear math, and I know I am lucky to have had parents and teachers who nurtured my interest. As a student and a young professor, I created a steady stream of mathematical models, toys, and curiosities to show to my classmates, colleagues, and students. These had a critical purpose besides my own amusement and edification: to let people who might be petrified by equations or homework problems see profound mathematics in an attractive and alluring form.

Under the generous influence of the mathematical arts community, my work has evolved over the past decade into the body of artworks featured in this exhibition. My central goal as an artist is to invite viewers into mathematical insights regardless of their personal relationship to math. I am particularly devoted to exploring handcrafts, especially those in the fiber arts traditionally viewed as feminine. The interplay between mathematics and fiber arts is endlessly fascinating, both in the ways that mathematics allows for a deeper understanding of knitting, crochet, beading, and so forth, and in the ways that these crafts can illuminate complex concepts in mathematics.

Fundamental Regions Mathematics is overflowing with complex systems generated by simple objects and rules. For example, if we take the blue and white trapezoid below and repeatedly copy it with the mirror reflections and rotations diagrammed to its right, we discover an endlessly propagating design that covers as much flat surface as we wish. To create the infinitely extensible pattern on the right, we only need the trapezoid on the left, and no smaller motif will do. Mathematically speaking, this makes the trapezoid a fundamental region of the design.

Literally, a fundamental region for a repeating geometric pattern is a smallest piece of the pattern from which we can reconstitute the whole. Indeed, many of my artworks based on symmetry explicitly reference this concept, and the design pictured here is tucked away in a corner of this exhibit.

Metaphorically, the notion of taking a small seed and using it to grow an intricate structure permeates large swaths of mathematics. Each of the pieces here is an attempt to illustrate some elemental mathematical principle, to revel in its simplicity and its complexity.

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+ reflection rotation by 120º

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Topology In my transition from math professor who makes elegant models to practicing mathematical artist, many of my pre-exhibition pieces were topological: twisty one-sided surfaces I sewed out of cloth napkins, coffee mugs I painted with intricate networks of colors, and so forth. The pieces here are the eventual outgrowth of that early work.

Except for the twisty surface in “Fortunatus X,” these artworks are inspired by a set of famous theorems about maps. According to the Four-Color Theorem, any map of regions on a plane or a sphere can be colored in with four or fewer colors so that neighboring regions always use different colors. As it turns out, more complicated surfaces require substantially more colors. On the surface of a donut —in formal math language, a torus—there are maps that require seven colors. For a double torus—a twoholed donut, or a ball with two handles—the magic number is eight. The image here is a schematic from a research paper in which I developed a practical system for artists to make a map with eight mutually neighboring countries on a double torus.

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“Tea for Eight,” 2010 Glaze on commercial ceramic 6 x 8 x 5 in

“Eight-Color 8,” 2014 Glass beads, sterling silver bead, cotton crochet thread 10 x 6 x 41/2 in

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“Map Coloring Jewelry Set,” 2014 Glass beads, gold-filled beads, thread, ear wires 12 x 12 x 6 in

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“Granny's Double Torus,” 2020 Wool alpaca blend yarn 13 x 10 x 10 in

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“Fortunatus X,” 2012 Fabric, batting, thread 6 x 6 x 8 in

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Symmetry Mathematicians have a very granular view of symmetry. For two-dimensional designs that repeat in one direction (frieze patterns ) or two directions (wallpaper patterns ), mathematicians categorize their symmetry types by considering what symmetries they have and how those symmetries interact. For example, the three black-and-white wallpaper designs below all have the same symmetry structure. The second row of figures illustrates this common structure by marking the axis lines for reflections in red, the axis lines for glide reflections (which reflect across the line and then glide parallel to the line) in orange, and the centers of rotation in purple and blue. The third row shows only these symmetry markings; observe that except for the scale and the orientation, the patterns of symmetries are the same in all three cases.

These artworks catalog symmetry types compatible with different artforms. They include results from joint research with Ellie Baker (“Serpentine Symmetries”) and Carolyn Yackel (“Float Free, Bumblebee,” “Fourteen Ciphers”). The two knitted shawls are not exhibition pieces: “Linear Lace Shawl” is a test knit for a knitting pattern I published on ravelry.com, and “Fourteen Ciphers” appeared in the Bridges 2019 Math + Fashion Show in Linz, Austria.

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“The Double Knitting Groups,” 2016 Silk/merino yarn 161/2 x 171/2 in

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“Coded Symmetries,” 2018 Merino/alpaca yarn, glass beads 143/4 x 11 in

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“The Symmetries Diagram Themselves,” 2018 Merino/silk yarn, glass beads 193/4 x 11 in

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“Fundamental Frieze Scroll I,” 2018 Merino/alpaca yarn, glass beads 13 x 71/2 in

“Fundamental Frieze Scroll II,” 2018 Merino/alpaca yarn, glass beads 171/2 x 91/2 in

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“Linear Lace in Burgundy,” 2017 Merino/cotton yarn, wooden dowels 103/4 x 83/4 in

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“Symmetry Flow,” 2020 Watercolor paper, watercolor pencil, stainless steel wire, electrical tape 22 x 65 x 65 in

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“Frieze Frame,” 2015 Glass beads, crochet cotton thread 91/4 x 91/4 in

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“Makeri Mosaic,” 2019 Merino/cotton yarn, wooden dowels 103/4 x 83/4 in

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“Float Free, Bumblebee,” 2018 Merino/alpaca yarn, wooden dowels 181/2 x 11 in

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“Fourteen Ciphers,” 2019 Berroco Remix Light Yarn 23 x 82 in

“Linear Lace Shawl,” 2017 Merino/silk yarn 66 x 32 in

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“Serpentine Symmetries,” 2017 Glass and crystal beads, thread, clasp, ear wires 12 x 12 x 6 in

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Growth In the late 1990’s, Daina Taimina and David Henderson showed that if you crochet or knit a fabric in which the number of stitches per row grows exponentially, the fabric has constant negative curvature, making it an excellent approximation of a piece of the hyperbolic or non-Euclidean plane. This gives the cloth a distinctive flared appearance, like a skirt that is entirely made of ruffles. My attending a talk by Taimina and Henderson in 2007 is what inspired me to learn to crochet, and subsequently to knit.

The ubiquitous Fibonacci sequence is formed by starting with 1 and 1 and then forming each new number by adding the two previous numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

The further out we follow the sequence, the closer it gets to exponential growth, so a fabric in which the number of stitches per row follows the Fibonacci pattern will be very roughly hyperbolic. I have named my most recent works on this theme after Virahanka, the Sanskrit poet who described this sequence half a millennium before Fibonacci was born.

As it happens, the Virahanka/Fibonacci numbers appear in the seed spirals in many plants, including sunflowers and daisies. The mechanism is illustrated in “Seed Values I” and “Seed Values II,” which track the literal growth of a daisy.

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“Hyperbolic Constellation” 2015 Glass beads, crochet cotton thread 2 x 4 1/2 x 4 in

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"Knit Bifurcation," 2015 Merino yarn 24 1/2 x 12 1/2 in

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“Fibonacci Downpour,” 2015 Merino yarn, cotton thread, embroidery hoop 13 1/4 x 10 x 10 in

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“A Tree for Virahanka,” 2021 Berroco Remix Light yarn, copper wire 13 x 22 in

“Virahanka's Thoughts Overflow,” 2021 Berroco Remix Light yarn, copper wire 14 x 11 x 8 in

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“Seed Values I,” 2021 Print 24 x 24 in

“Seed Values II,” 2021 Print 24 x 24 in

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Bibliography Baker, Ellie, and Susan Goldstine. Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist. A K Peters/ CRC Press, 2014.

Goldstine, Susan. "Capturing Eight-Color Double-Torus Maps."

Bridges Seoul: Mathematics, Music, Art, Architecture, Culture: Conference Proceedings, Tessellations Publishing, July 2014, pp. 377–380.

Goldstine, Susan. “Crystalline.” Knitty, no. 57, Sept. 2016, knitty.com/ISSUEdf16/PATTcrystalline/PATTcrystalline.php.

Goldstine, Susan. “Eight Heptagons: The Double Torus Revisited.” Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture: Conference Proceedings, Tessellations Publishing, July 2020, pp. 413–416.

Goldstine, Susan. "A Recursion in Knitting." Bridges Finland: Mathematics, Music, Art, Architecture, Culture: Conference Proceedings, Tessellations Publishing, July 2016, pp. 395-398.

Goldstine, Susan.

“Self-Diagramming Lace.”

Bridges Stockholm: Mathematics, Art, Music, Architecture, Education, Culture: 2018 Conference Proceedings, Tessellations Publishing, July 2018, pp. 519–522.

Goldstine, Susan. "A Survey of Symmetry Samplers." Bridges Waterloo: Mathematics, Art, Music, Architecture, Education, Culture: Conference Proceedings. Tessellations Publishing, July 2017, pp. 103–110.

Goldstine, Susan, and Ellie Baker. "Building a better bracelet: wallpaper patterns in bead crochet." Journal of Mathematics and the Arts, vol. 6, no. 1, 2012, pp. 5-17.

Goldstine, Susan, Sophie Sommer, and Ellie Baker. "Beading the Seven-Color Map Theorem." Math Horizons, vol. 21, no. 1, Sept 2013, pp. 22-24.

Goldstine, Susan, and Carolyn Yackel. " A Mathematical Analysis of Mosaic Knitting: Constraints, Combinatorics, and Color-Swapping Symmetries." To appear in Journal of Mathematics and the Arts.

Taimina, Daina. Crocheting adventures with hyperbolic planes.

A K Peters, 2009.

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Boyden Gallery and Collection Staff: Erin Peters, PhD, Director Kaidyn (KD) Sexton, Gallery Co-Manager & Graphic Designer Daniel Mixson, Gallery Co-Manager Emily Smith, Consulting Collections Manager

A Steven Muller Distinguished Professor in the Sciences Event The Steven Muller Distinguished Professorship in the Sciences honors faculty whose accomplishments in the sciences establishes their expertise in a field of research relevant to our understanding of the world we inhabit. The Muller Distinguished Professor in the Sciences contributes to a vital dialog among scientists which is enhanced by the laboratory and field research contributions of St. Mary's College students.

-----SUPPORTED BY

-----Thank you MSAC! To discover more about the Maryland State Arts Council and how they impact Maryland, visit msac.org.

Artist’s Acknowledgments Thank you to all of the people and organizations recognized here who directly supported this exhibition. I am also grateful to my principle coauthor, Ellie Baker; to my faculty colleagues, particularly Katherine Socha, Carrie Patterson, and Emek Köse; to the Knitting Circle at the Joint Mathematics Meetings, especially founders sarahmarie belcastro and Carolyn Yackel; to the Bridges Organization and the vibrant community of mathematical artists it supports; to the Journal of Mathematics and the Arts; and to Ingrid Daubechies, Dominique Ehrmann, and the rest of the Mathemalchemy team.

Want see more?

You can find Susan Want to see to more?

You can find Susan here orathere! http://faculty.smcm.edu/sgoldstine/

-----Thank you

Art and Art History Department Arts Alliance of St. Mary’s College of Maryland Lee Capristo Jennifer Falkowski Katie Gantz

-----Katia Meisinger

Office of Academic Affairs

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Boyden Gallery is located on the 2nd floor of Montgomery Hall on the St. Mary's College of Maryland campus Gallery Hours: T-F 11-6, Sat 11-4 www.smcm.edu/boyden-gallery