Art and Mathematics  –  Martin Strauss (Session 2) (FULL)
With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics. Conversely, Mathematics gives techniques for analyzing, appreciating, and even creating Art, as well as the basis for gallery design, digital cameras, and processing of images. In this class we will explore the Mathematics in great works of Art as well as folk art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we’ll make our own art, by drawing, painting, folding origami papers, and more. Major topics include Projection, Symmetry, Wave Behavior, and Distortion. Projection includes the depiction of three-dimensional objects in two dimensions. What mathematical properties must be lost, and what can be preserved? How does an artwork evoke the feeling of three-dimensional space? We’ll study perspective, depictions of globes by maps, and the role of curvature. Turning to symmetry, we’ll study rotational and reflective symmetry that arise in tiling and other art and math. We’ll study more generalized symmetry like scaling and self-similarity that occurs in fractals as well as every self-portrait, and is central to mathematical concepts of dimension and un very different from the work at coarser scales—it is not self-similar. Describing light as waves and color as wavelength at once explains how mirrors, lenses, and prisms work and explains some uses of light and color in art. Finally, we ask about distorting fabrics and strings, and ask about the roles of cutting, gluing, and of stretching without cutting or gluing. Is a distorted human figure still recognizable, as long as it has the right number of organs and limbs, connected properly? Background in Math and interest in Art suggested. No artistic talent is necessary, though artistically talented students are encouraged to bring art supplies if they are inexpensive and easily transportable.

Mathematics and Music Theory –  Lon Mitchell (Session 3) (FULL)
Mathematicians can create complex and beautiful theorems from relatively basic assumptions, while Music Theorists often try to identify basic patterns and “rules” in complex and beautiful music. In this course, we will explore some of the recent attempts to meet in the middle, connecting mathematical patterns and structures to music from the ancient to the modern. In Mathematics, we will explore topics such as group theory, graph theory, geometry, and metric spaces, encountering some of the most important structures in the modern discipline. Fundamental results of these areas will be discussed, and students will construct and explore examples and related patterns. In Music Theory, we will take existing music by composers such as Bach and Beethoven and use mathematical structures to provide a possible explanation of what they were thinking as they composed. In addition, we will investigate the techniques of modern composers such as Arnold Schoenberg who advocated composition based on prescribed axioms. Students will be given the chance to write music using these different techniques. Although we will use the modern (Western) twelve-tone scale as a reference, our explorations will take us into discussions of tuning, temperament, and the physics of sound. We will investigate mathematical theories of what makes the best scale, how some of those scales occur in the music of other cultures, and how modern composers have engineered exotic scales to suit their aesthetics. Software allowing students to experiment with creating their own musical systems will be provided. Prospective students should have a good command of (high-school) algebra and experience with reading music in some form.

The Geometry of Music – Alessandro Danelon (Session 1) (FULL)
Math and music are related in multiple ways. From the Western historical point of view we find connections in the math of Pythagoras, in the development of the equal temperament, and in the theoretical and artistic work of Iannis Xenakis. One can use group theory to phrase music structures, and both mathematicians and musicians claim the creative process as the moving power of their discipline. Composers used symmetries in their compositions, and scientists tried to associate sounds to their objects of study according to their inner mathematical structure. The aim of this course is to highlight some geometric and algebraic structures in the theory of music. We will start reviewing musical notions like scales, intervals, triads, harmonic progressions, tonality, modality and harmonic rhythm together with the physics of the sound (pitch and frequencies). We will then study the geometry of pitch organization and transposition and move on to explore the harmonic structure and harmonic structure of a phrase, together with the geometry of chromatic inversions. At this point we revise musical scales and intervals with geometric tools and move on to discover the geometry of harmony: Riemann’s Chromatic inversions, and Euler’s Tonnetz. On the algebraic side, we will introduce groups, their theory, and use their language in the theory of music. We will also discuss how a deeper understanding of the underlying music can help us in improving our playing and performances. Performers are encouraged to bring their instruments.