Teaching Geometry to Artists

Nexus Network Journal, Apr 2005

Jack Rees discusses his experience teaching geometry to artists. The aim is to introduce scientific ideas to arts students through the visualizations that are such an important part of discourse in science. Described are the intellectual context, define selected concepts using geometry and introduce elementary mathematical formulae—all relying on graphic visualizations to make fundamental ideas clear. The goal is to provide a means by which visually sophisticated persons may think with geometry about culture But it should always be insisted mathematical subject is not to be considered exhausted until it has become intuitively evident... Felix Klein [1893:243]

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Teaching Geometry to Artists

0 It is my privilege to teach geometry to artists. Until recently, I have offered one elective per semester to undergraduates through the liberal arts department of a small art institute. The courses count as science distribution requirements for a bachelor of fine arts degree. Students are drawn from schools of painting , sculpture, ceramics, textiles, illustration, and new media. My best students are bright, which is to say open to being influenced; tenacious, which is to say requiring clear explanations; and tough-minded, which is to say they will not be patronized. Five students like this in a class of twenty is a joy. I had such a class the last semester I taught Jack Rees discusses his experience teaching geometry to artists. The aim is to introduce scientific ideas to arts students through the visualizations that are such an important part of discourse in science. Described are the intellectual context, define selected concepts using geometry and introduce elementary mathematical formulaeall relying on graphic visualizations to make fundamental ideas clear. The goal is to provide a means by which visually sophisticated persons may think with geometry about culture But it should always be insisted mathematical subject is not to be considered exhausted until it has become intuitively evident... - Topology is the geometry of continuity, the last in a series of geometries whose definitions of equivalence become progressively more difficult to describe to students with little formal mathematical education. Topology, conventionally rendered as rubber sheet geometry, is the geometry of stretching, squeezing, or extruding but not of cutting, folding or tearing as long as neighboring points remain neighboring points [Huggett and Jordan 2001]. This course is designed specifically for graphically sophisticated students in the arts2 and is intended, in the main, to introduce geometry as a discipline of great visual and intellectual beauty. (It helps that we can visit the rare book room of The Linda Hall Library of Science and handle a dozen antique books renowned for their scientific and artistic significance; see http://www.lhl.lib.mo.us.) In class, graphic visualizations and geometrical demonstrations (mostly) take the place of a postulational, or, if you will, axiomatic, presentation. In the end it is hoped that students will unite intellectual inquiry and artistic endeavor according to their own interests. This essay offers samples of class content highlighting the visual approach in sections 2 and 3. Section 4 details my assumptions about teaching geometry, by which I mean things being tested in the classroom, and a course outline. Section 5 records some observations based on my experience teaching over the last seven years. Section 6 returns in detail to the content of the topology class. 2 Felix Kleins geometry schema The nineteenth century in mathematics was named by historian Carl Boyer the heroic age in geometry [Boyer 1968: 572-595]. Among the giants of that age stand Felix Klein (1849-1925), who is often praised for his magisterial grasp of the whole of mathematics. This is faint praise among mathematicians. Here is how Constance Reid, author of Hilbert , paraphrasing Richard Courant (who organized Kleins final papers) puts it: And yet Kleins life had not been without its inner tragedy. The power of synthesis had been granted to him to an extraordinary degree. The other great mathematical power of analysis had been to a certain extent withheld. His ability to bring together the most distant, abstract parts of mathematics had been remarkable, but the sense for the formulation of an individual problem and the absorption in it had been lacking. ... Certainty he had perceived that his most splendid scientific creations were fundamentally gigantic sketches, the completion of which he had to leave to other hands [Reid 1970: 178-179].3 What makes for a great mathematician may not be exclusive of what makes for a great teacher and Klein was, by all accounts, a great teacher. He wielded considerable influence over one of the great mathematical schools of the late nineteenth and early twentieth centuries, the University at Gttingen. He established a research center there that was, for a time, a focus of the mathematical universe, attracting David Hilbert from Knigsberg. During his tenure the student body included Hermann Weyl, Richard Courant, and Max Born. The first woman D.Phil. , Grace Chisholm Young, graduated in mathematics from a German university, graduated under his auspices. It is perhaps telling that Klein regarded as his most notable achievement the unification of geometry in what is widely known as his Erlangen Programm of 1872 [Klein 1893].4 Based on the concept of a mapping, Klein showed how the geometries of his age (metrical, projective, line) could be joined into a single geometry using the theory of groups. A group is a set of elements filtered through an operation. To be a group the (...truncated)


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J. M. Rees. Teaching Geometry to Artists, Nexus Network Journal, 2005, pp. 86-98, Volume 7, Issue 1, DOI: 10.1007/s00004-005-0009-z