T C 310 • Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics
2:00 PM-3:30 PM
"Equations are just the boring part of mathematics. I attempt to see things in terms of geometry." Stephen Hawking
In many cases, nature finds optimal solutions to geometric problems. For example, among all possible shapes spanning a loop of wire, a soap film always has least area. The hexagonal packing system used by bees in their honeycombs is the most efficient (i.e. least perimeter) way to divide a plane into equal areas. Remarkably, these and other shapes found in physical or biological systems also appear in abstract mathematics as solutions to certain optimization problems. Many of these forms also frequently appear in art and architecture and are regarded as beautiful by many cultures.
These phenomena raise several interesting questions. What does it mean to have a best form? Indeed, is the concept of best even well defined in this context? Can mathematical reasoning prove that something is optimal in a precise sense? What roles do evolution and physical principles play in finding such forms? How and why do various cultures integrate such forms into their patterns of artistic expression?
This course will examine some of the attempts of science to provide rigorous explanations of optimal geometry. Historically, such attempts had both empirical and synthetic (i.e. philosophical or theological) motivations. Some even incorporated ideas that today might provocatively be labeled intelligent design. The course will examine how mathematics can contribute to understanding and explaining optimal forms. In accord with the Modes of Reasoning rubric, we will devote particular attention to the roles of geometric reasoning and mathematical proof in making our explanations intellectually rigorous.
Our studies will be divided into eight units. Each unit will focus on the overarching questions of the course as seen from one topical perspective. These are as follows:
1. What are some possible meanings of optimal geometry?
2. Case study: examples seen in soap films
3. Explanations of optimality: synthetic versus empirical principles
4. Can optimality be proved with mathematical rigor?
5. Examples from biology: patterns and similarity in growth and form
6. Optimal geometry in art and architecture
7. Symmetry: why we see it and why it breaks
8. Topology versus geometry: what is the shape of space?
The course grade will be based on three components.
o Class participation (20%). This component will assess participation in class discussions and completion of daily "minute papers" brief impressionistic paragraphs written in response to a question raised during class discussion.
o Unit projects (50%). Each topical unit in the course will be accompanied by a project. For example, one project might be collecting digital photographs of optimal shapes found in nature or architecture and arranging these in a poster presentation. Another project could involve research into historical explanations of optimality in addition to those discussed in class. Several will require the students to engage in mathematical reasoning and proof. Although this is not a Substantial Writing Component course, most projects will involve writing short essays (2-3 pages).
o Final exam (30%). This component will assess the students' ability to discuss the concepts and questions encountered in the course in a precise and reasoned manner.
The course will draw on the following sources. (This list may be revised and expanded as the course is further developed. Some readings will be excerpted and included in a photocopy packet.)
o Stefan Hildebrandt and Anthony Tromba, The Parsimonious Universe: Shape and Form in the Natural World.
o Ian Stewart and Martin Golubitsky, Fearful Symmetry: Is God a geometer?
o D'Arcy Wentworth Thompson, On growth and form.
o Ian Stewart, Lifes Other Secret: The New Mathematics of the Living World.
o Kimberley Elam, Geometry of Design: Studies in Proportion and Composition.
o Jay Kappraff, Connections: The Geometric Bridge between Art and Science.
o George L. Hersey, Architecture and Geometry in the Age of the Baroque.
o Jeffrey R. Weeks, The shape of space
About the Professor
Dan Knopf is an Associate Professor of Mathematics. He joined the University of Texas at Austin in fall of 2004 and received a National Science Foundation CAREER award in 2006. He is an active researcher in geometric analysis, in particular, the use of geometric evolution equations to find and classify canonical or optimal geometries. He is author or coauthor of over twenty scholarly publications, including four books. His nonacademic interests include running, gardening, and rooting for the Longhorns.