Art and Mathematics – Martin Strauss
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
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.