Bridges: A World Community for Mathematical Art

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T T his is not the first time the Mathematical Communities column has featured the Bridges Organization: the 2005 conference, 1 in the breathtaking Canadian Rocky Mountains at Banff, was described in these pages by Doris Schattschneider (Schattschneider 2006), a regular Bridges participant and Escher specialist. The 2005 conference saw the debut of Delicious Rivers, Ellen Maddow's play on the life of Robert Ammann, a postal worker who discovered a number of aperiodic tilings. 2 Marjorie Senechal, The Mathematical Intelligencer's current editorin-chief, served as Maddow's consultant. 3 A theatre première at a conference on mathematics? A production performed by mathematicians, moonlighting as actors? But this is Bridges.
A quick look at the 2005 conference relays the essence of this scientific and artistic ''happening'' resembling a firstrate festival of the arts. True to its title, Renaissance Banff, the 2005 Bridges gave all members of its community, whether based in the sciences or the arts, the feeling that they had helped bring about a genuine rebirth. I use ''community'' in its most complete sense-including adults, children, artists, university professors, art lovers, and local people-for the wealth of conference activities could only be accomplished through the participation of each and every individual present.
In addition to formal conference lectures and the theatre performance, the program included an international mathematical art exhibit, a mathematical music night, and a mathart workshop series developed for teachers by teachers. Groups constructed enormous mathematical art installations and models. Following the musical performance by professional musicians, many mathematicians and artists grabbed an instrument and continued playing, just as anyone could lend a hand in constructing a colossal installation. As members of a collective undertaking, Bridges participants experienced the joy of creating a colorful and unique mathematical art community in which layperson and expert worked side-by-side as equal partners. 4 And, as a result of negotiations initiated in Banff, the Journal of Mathematics and the Arts 5 -a periodical dedicated to examining connections between mathematics and the arts-was started in 2007, published by Taylor & Francis, with the professional support of the Bridges community.
Schattschneider noted that ''Mathematics creates art''; ''Mathematics is art''; ''Mathematics renders artistic images''; ''Hidden mathematics can be discovered in art''; ''Mathematics analyzes art''; ''Mathematical ideas can be taught through art.'' After eighteen consecutive years of Bridges gatherings, we can say that the inverses are also true: Art creates mathematics; Art is mathematics; Artistic images render mathematics; Hidden art can be discovered in mathematics; Art analyzes mathematics; Artistic ideas can be taught through mathematics. Together with the organization supporting these events, the Bridges conferences have established a two-way bridge, aiding transfer between mathematics and the arts that has a significant amount of traffic.
From its beginnings in 1998, Bridges has advocated for mathematics as a core component of STEM (Science, Technology, Engineering, Mathematics) education. Years before the STEM acronym was even created (Christenson 2011) and spread, Bridges had humanized it. The Bridges community has never had to expand its approach from STEM to STEAM (Science, Technology, Engineering, Arts, and Mathematics): it has always included those aspects of the arts, design, creative thinking, and artistic imagination so very necessary to, yet still so very lacking from many STEM projects today. From its inception Bridges has given the STEAM movement inspiration for a transdisciplinary and intercultural platform. Figure 1. A colorful group of Bridges participants in front of the Gossamer Zometool model, a massive sculptural tribute to the late architectural visionary, Jean Christoph Kling, at the Bridges 2009 ''Renaissance Banff II'' conference. Five meters in diameter and assembled from 50,000 Tinkertoy-like parts, the gossamer model was the largest of its type ever attempted. One hundred fifty mathematicians and artists from all over the world and several of their children assembled the small plastic Zometool components into superstructures that became the final sculpture. The work took more than 250 person-hours, performed during breaks between presenting and attending talks. Although the model is built entirely from points and straight lines (i.e., nodes and struts) the model looks like a 3-dimensional ''spirograph'' drawing, with organic curves that mimic life forms. The underlying structure is derived from a shadow of a 6-dimensional cube. (Photo: Carlo H. Séquin.) From the ''Persian Paradigm'' to STEAM and Back Again: The Inception of Bridges The first few Bridges conferences were hosted by Southwestern College in Winfield, Kansas. How did this smalltown college, not known for its contributions to the art and science discourse, assume this role?
As is often the case with active and successful communities and networks, Bridges was begun by a many-sided individual with contacts in both science and culture. When Reza Sarhangi immigrated to the United States from Iran in 1986, at the start of his mathematical career, he brought with him the qualities associated with STEAM integration today. His broad range of personal interests as well as his research into ancient Persia's mathematical past helped him direct attention to mathematics' complex cultural roots. His work as a university teacher in education was defined by these goals as well. For his integrational approach to mathematics and art, Sarhangi looked far beyond the wellknown works of artists such as M. C. Escher (Sarhangi and Martin 1998) to the joint efforts of mathematic and artistic communities in ancient times.
Mathematics, arts, and crafts coexisted side-by-side during the medieval period of Persian history. 6 Sarhangi has continually emphasized the work of Abul Wafa al-Buzjani (940-997/998), one of the most famous mathematicians of his time. Al-Buzjani's treatise, On Those Parts of Geometry Needed by Craftsmen, was written to educate craftspeople in geometry. For medieval craftspeople, creating the decorative motifs common to the Persian art of this era demanded not only constant training, but also regular consultation with mathematicians. Indeed, decorating the inner as well as the outer, spherical surface of a cupola with tiles featuring highly regular, yet still extremely complex geometric patterns would have required advanced knowledge of geometry. The sophisticated, mathematical nature of Persian decorative arts not only makes them interesting from a historical perspective, but also provides a fascinating area of research for mathematicians today. 7 Before immigrating to the United States, Sarhangi was more than a teacher of mathematics interested in Persian traditions. He was a graphic artist, teacher of drama, playwright, theatre director, and props designer. When added to his background in mathematics and history, his firsthand experience of complex and collective artistic processessuch as creating and performing a theatrical play-gave him deep insight into the equally complex processes involved in designing and producing medieval Persian tilings. Sarhangi made good use of his many areas of expertise, first as an innovative, young university professor open to new experiments, then later as an educator of math instructors. As department chair of Southwestern College's Department of Mathematics, he already introduced creative study modules or theatrical plays on mathematics to change how mathematicians were educated.
In his new country, Sarhangi looked for the kind of academic community capable of supporting his broad range of interests. In the early 1990s, the college implemented a new Integrative Studies Program that drew together faculty from all other traditional and professional programs. As director of the Integrative Studies Program, Daniel F. Daniel, Sarhangi's close friend and mentor at Southwestern, suggested that Sarhangi establish a new course for this program. His course on connections between mathematics and the arts became very popular among students.
Sarhangi also attended the Art and Mathematics (AM) conferences organized by Nat Friedman at the State University of New York at Albany from 1992 to 1998, which opened their doors to artists, architects, and other experts applying mathematics creatively. The spirit of cooperation engendered by these AM gatherings led to the publication of several interdisciplinary papers uniting different perspectives to form a kaleidoscope-like vision of the given topic. With some exaggeration, it can be said that Sarhangi felt he was witnessing the rebirth of a long-forgotten paradigm from Abul Wafa al-Buzjani's time. He could see firsthand how a new form of art arises from the dialogue between the mathematician creating the theories for solving complex artistic or architectural problems, and the master putting theories into practice. This art possesses a unique, aesthetic quality all its own, whose analysis demands a new approach, a kind of ''interdisciplinary aesthetics'' both mathematical and artistic in nature. 8 Known as ISAMA (International Society of the Arts, Mathematics, and Architecture) 9 as of 1998, the AM movement was the direct, American antecedent to the later Bridges conferences. Indeed, three of Bridges' first four directors, George Hart, 10 Carlo Séquin, 11 and Reza Sarhangi, were ISAMA veterans. The fourth director, Craig S. Kaplan, 12 started his career in mathematical art in 1999 after joining the ISAMA and Bridges communties and co-organizing the MOSAIC 2000 conference, which examined connections between computer programming and the arts. 6 On artistic consequences of connections between the European renaissance and the medieval Arabian science's visual investigations, see Belting (2011). 7 See Reza Sarhangi's numerous articles on Abul Wafa al-Buzjani, e.g., Sarhangi (2006 10 Research professor of Computer Science at Stony Brook University. Hart is also a sculptor whose work is recognized around the world for its mathematical depth and creative use of materials. 11 Professor of Computer Science at the University of California, Berkeley. His works in computer graphics and in geometric design have provided a bridge to the world of art. In collaboration with several sculptors of abstract geometric art, Sé quin has found a new interest and yet another domain in which the use of computer-aided tools can be explored and where new frontiers can be opened through the use of such tools. 12 Professor at University of Waterloo, editor of Journal of Mathematics and the Arts. The main focus of Kaplan's research is on the relationships between computer graphics, art, and design, with an emphasis on applications to graphic design, illustration, and architecture. Robert W. Fathauer 13 soon assumed responsibility for organizing the Bridges art exhibits. 14 Many groups and ''schools'' have connected to form the background for the American, European, and Asian science and art communities currently involved in the Bridges Organization. These include the Mathematics and Culture conferences (Emmer 2004(Emmer -2012(Emmer , 2012(Emmer -2014(Emmer , 2015, the Nexus conferences, 15  Inter-and Transdisciplinarity: Bridges as a ''World View'' The nearly 300-page proceedings from the first Bridges conference 19 was reviewed in Nexus Network Journal by the mathematician Solomon Marcus, a pioneer in numerous interdisciplinary as well as transdisciplinary areas related to mathematics (Marcus 1999). He called for the further broadening and deepening of artistic analysis within the newly emerging Bridges discourse to include, for example, connections between mathematics and poetry. The Bridges Organization has followed his suggestions.
Year by year, as exhibits increasingly put new mediums of mathematical art on display, conference lectures open these works to further analysis. The movement's tendency to refine its themes inspires new program elements, such as the Bridges Poetry Afternoons organized by Sarah Glaz. 20 Marcus noted that ''[at the Bridges conferences] artists enlarge their creative horizon by looking at the achievements of modern science, while scientists have a chance to see, in a new light, their own results'' (Marcus 1999: 156). And in fact many of the experiences and moments reflected in this conference series demonstrate art's ability to broaden the horizons of science as well. 21 Probably the best evidence for the two-way traffic on Bridges' bridges is the many transdisciplinary collaborations that have grown from encounters between mathematicians and artists at Bridges-related events. For example, Ars Geometrica Symposiums (2007-2009, Hungary)-which prepared the ground for Bridges 2010 in Hungary-led to the international recognition of the mathematical artworks of sculptor István Böszörményi (Gailiunas 2007) and also sparked Böszörményi's still ongoing collaboration with the renowned mathematician Lajos Szilassi, whose ''Szilassi-polyhedron'' was noted and popularized by Martin Gardner (Gardner 1978). Szilassi's mathematical theories and models have inspired the sculptor Böszörményi's art (Böszörményi 2013). Artists also have fruitfully inspired mathematical research, as the case of the mathematician László Vörös and the artist Tamás F. Farkas shows (Vörös 011). 13 A former researcher of Jet Propulsion Laboratory, and the founder of Tessellations, a company that specializes in products that combine art and mathematics. He is an internationally renowned author of activity books on art and mathematics and a mathematical artist, whose work has been shown in numerous exhibits in the United States, Canada, and Europe. 14  international team first as an art project in Bridges Art Exhibits. Because of its geometrically interesting properties, the Spidron-system has also been featured later on a number of scientific forums, including as a cover-story of Science News (Peterson 2009) or in The Math Book by Clifford A. Pickover (Pickover 2009) and elsewhere. From its beginning, Bridges has broken with the monotony of traditional conference lectures. The goal was to bring research alive, have it on display-or even put it on stage. Mathematician Mike Field-coauthor of Symmetry in Chaos (Field and Golubitsky 2009)-described his experiences at the second and third Kansas conferences: ''At the Winfield conferences, the international virtuoso violinist Corey Cerovsek played for the audience after plenary sessions in the morning, often preceding his performance with an extempore talk on a topic from physics or mathematics (concurrently with his musical studies, Corey completed all the coursework for a Ph.D. in mathematics when he was about sixteen). Aside from the music, there would usually be theatre shows as well as teacher workshops, held at the end of the conference. ''[…] I […] can testify to the great atmosphere of these meetings, where artists, mathematicians, computer scientists, and educators would talk into the early hours'' (Field 2006: 730). Reza Sarhangi draws on his past in theatre to evoke the personable atmosphere and wealth of experiences at these first gatherings: ''Theatre involves making connections with the audience that go beyond just the script […] So at Bridges, I-and the other three board members-want the conference attendees to get more than just the content of the papers, but to have an enjoyable experience that integrates art, dance, and other performances'' (Crease 2014: 17).

Bridges Around the World
Barely sixty participants attended the first Bridges conferences; today they attract annually around 250-300 conference participants from around the globe and thousands of people in the audience. Since Sarhangi moved from Winfield to Towson, Maryland, in 2002, the Bridges Organization-a Scientific and Educational Non-Profit Corporation established in the State of Maryland in 2006-has handled the administrative tasks related to these events.
The resounding success of Bridges 2003 in Granada, Spain (organized together with ISAMA)-in the shadow of the Alhambra, one of the greatest math-art works in history-prompted the organizers to hold future conferences in tourist destinations, and to include thematic daytrips. Bridges conferences have been held in Winfield, USA (1998-2001 . In 2016, Bridges will embark on its first Nordic conference, held at the University of Jyväskylä in Finland, the most northern location ever to host a Bridges event. 22 Figure 5. (a-c) ''Worldsand'' (Sammlung Weltensand) composition at Bridges 2010 by Elvira Wersche, who collects different types and colors of sand from all over the world and uses it to construct complex mosaics, composed of geometrical patterns, on the floors of museums, churches, and synagogues. When the artwork was finally ready, a dancer literally erased the carefully constructed pattern. By doing so, the artist emphasized that everything is in constant flux and everything is only temporary. (Photos: László Mihály.) After its community took to globe-trotting, the Bridges conferences began generating a form of organized math-art tourism. But the intellectual benefits extend beyond the conference participants. Thanks to international media, the attention paid to Bridges events-hosted by resident scientific and cultural institutions-strengthens local math-art communities. I can attest that the Experience Workshop Math-Art Movement, 23 an independent community of  mathematicians, artists, and educators, established in Hungary in 2008 preceding the Bridges Pécs 2010 conference, is still growing. Through this movement tens of thousands of Hungarian students and thousands of teachers have been exposed to the Bridges philosophy of experience-oriented mathematics education through the arts.

Participants, Goals, and Events at Bridges Conferences
The sheer diversity of topics and areas addressed at Bridges conferences draws a wide audience of scholars and artists. 24 In addition to mathematicians, scientists, and art and education experts, they attract painters, teachers, musicians, architects, literary scholars, computer programmers, sculptors, dancers, craftspeople, and model-builders.
Each conference reflects many different aspects. Bridges presents a platform for scholars, experts, and artists intent on pushing boundaries and exchanging experiences. Beyond supplying professional support, it encourages mathematics teachers to utilize creative, artistic processes and tools in passing on mathematical knowledge, and art teachers to reveal the mathematics involved in certain artworks or artistic processes. The educational relevance of math-art approaches has been demonstrated in interactive, experience-oriented workshops since Bridges' early days. Works of art are displayed in math-art exhibits, not just discussed in conference lectures. The Bridges collection has since grown to be the largest exhibit of mathematical art in the world. Throughout the years, creative programs have become increasingly structured and have now evolved into separate areas of expertise directed by skilled professionals. The key elements, which form the backbone of any Bridges event (such as the plenary and section lectures, education and fun workshops, mathematics and art short movie festival, the mathematical art exhibit, the mathematical theatre show, music night, family day, poetry afternoon, and public events) can be studied on the Bridges Organization's website, where each year's program is archived. 25 Bridges' transdisciplinary program has elaborated new transdisciplinary standards and has productively solved a high number of unforeseeable and unprecedented challenges. These are risks run by any truly transdisciplinary venture; they may never emerge in relatively homogeneous scientific or artistic communities with established traditions and history.
For example, Bridges has struggled to balance the openness of the conference with academic legitimacy. In the beginning, Bridges conferences were highly inclusive; everybody who wanted to would have an opportunity to share his or her work. This model works in disciplines in which journals are the true coin of the realm and conferences are just a chance to present work in progress to colleagues. But it can be problematic for participants from other disciplines-such as Bridges director Craig Kaplan's own field, Computer Science-who want to assign significant weight to conference papers: ''I think one of my influences has been to fight in favour of higher standards for regular papers, stricter peer review practices,'' Kaplan says. ''Of course, it helped to set up online tools that streamlined the mechanics of reviewing so that we could focus on the quality. I think we've done a decent job of compensating for these changing standards by ensuring there are lots of other forums in which participants can present their work at the conference.'' The polyphonic nature of Bridges conferences requires an organic and open universe of polyphonic standards to ensure that each contribution will be judged according to the highest standards of its own field. If there is a new kind of need in the community, then a new platform is created for it. Different juries review the submissions for the art exhibits and the short-film festival, and there is a transdisciplinary program committee for reviewing the conference papers representing a multitude of scientific and artistic research fields.
Bridges has always had a core community of participants who make up the ''choir.'' ''And sure, we preach to them,'' admits Kaplan. He believes that designing a conference to appeal to its long-term supporters will make it better overall. But, like Bridges' president Reza Sarhangi, Kaplan thinks that Bridges has done a good job of bringing in new faces, new voices, and new ideas every year. Bridges is increasingly offering itself as a public face for mathematics, a way to make it accessible to the general public. This was perhaps most successful in 2014, when Bridges was invited to Seoul by the International Congress of Mathematicians (ICM) to serve as a public counterpoint to the ICM happening across the city. Yet many mathematicians still don't see the special value in Bridges' polyphonic difference. ''We need a stronger voice outside of Bridges itself. The Joint Mathematics Bridges to the Future: The Next Generation Today, as changes in the world bring about unseen alterations in the structure of knowledge, any form of research, learning, or creativity capable of heightening awareness toward interlocking systems possesses untold value. First and foremost, such knowledge allows one to preserve a sense of exploration and inquiry, elements essential to all rational and creative activities. Without these, the ability to recognize those patterns and trends dictating current changes is lost.
The first mathematical museum in the United States, MoMath, 26 was established in New York City to connect seemingly disparate phenomena while simultaneously encouraging creative and imaginative opportunities in math education for children and young adults. As one of the founding members of this internationally unique institute, George W. Hart not only adapted the Bridges community's STEAM-approach, but also the work of many distinguished Bridges members to this special, interactive setting.
The Bridges Organization has combined forces in organizing the MoSAIC (Mathematics of Science, Art, Industry and Culture) 27 event series, sponsored by the Mathematical Sciences Research Institute (MSRI) to spur the STEAM approach among young people. As a part of this program, popular aspects of Bridges events are held at university campuses throughout the United States. MoSAIC's astounding success and a new generation in the Bridges community suggest we have every reason for confidence.

ACKNOWLEDGMENTS
Thanks to the Bridges Board of Directors for their information, to Maya Tóth, to Osmo Pekonen, and to my colleagues in the Bridges Finland 2016 Local Organizing Committee at the University of Jyväskylä.