T C 310 • Modes Of Reasoning
TTH 930am-1100am GEA 127
Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics
“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 (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 also explore how artists and architects incorporate geometric principles in their designs.
The main focus of the course will investigate what insights might be gained by looking at nature and art through the lens of mathematics — specifically, by investigating how our ways of understanding mathematics and the natural world inform and interact with each other. In accord with the Modes of Reasoning rubric, we will devote particular attention to the roles of geometric reasoning and mathematical epistemology in making our explanations intellectually rigorous. We will do this through four “case studies,” organized around the following topics:
- Optimality and Minimal Surfaces (can optimality be proved with mathematical rigor?);
- Pattern and Abstraction (why does nature repeat itself?);
- Evidence and Proof (what convinces a mathematician that something is true?); and
- Symmetries and their Structure (why do symmetries form and break?).
The course will draw on multiple sources, which will be excerpted and included in a custom course packet. Some of the sources currently under consideration are as follows:
- The Parsimonious Universe: Shape and Form in the Natural World, by Stefan Hildebrandt and Anthony Tromba;
- Architecture and Geometry in the Age of the Baroque, by George L. Hersey;
- Connections: The Geometric Bridge between Art and Science, by Jay Kappraff;
- Fearful Symmetry: Is God a geometer?, by Ian Stewart and Martin Golubitsky;
- Geometry of Design: Studies in Proportion and Composition, by Kimberley Elam;
- Life's Other Secret: The New Mathematics of the Living World, by Ian Stewart;
- On growth and form, by D'Arcy Wentworth Thompson; and
- The shape of space, by Jeffrey R. Weeks.
The course grade will be based on the following components:
- Exams (64%). There will be four in-class exams, each worth 16% of the total grade.
- Homework (20%). Each topical unit in the course will be accompanied by a small project.
- Class participation (16%). This component will assess participation in class discussions and completion of daily “minute papers” — brief impressionistic paragraphs written in response to questions raised during class discussion.
About the professor
Dan Knopf is an Associate Professor and the Graduate Adviser in the Department of Mathematics. He joined the University of Texas at Austin in fall of 2004. He received a National Science Foundation CAREER award in 2006, and a College of Natural Sciences Teaching Excellence Award in 2012. 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-five scholarly publications, including five books. His nonacademic interests include running, gardening, spoiling two cats, and rooting for the Longhorns.
T C 310 • Modes Of Reasoning
TTH 200pm-330pm WEL 3.402
Optimal Geometry in Nature, Art, and Mathematics
TC 310: Modes of Reasoning — Unique Number 43550
“Equations are just the boring part of mathematics. I attempt to see things
in terms of geometry.” — Stephen Hawking
The ob jectives of this course are to help students explore some of the
ways in which mathematicians think about the natural world — speci?cally,
how their ways of understanding mathematics and the natural world inform
and interact with each other. We will do this through four “case studies”,
organized around the following topics:
(1) Optimality and Minimal Surfaces;
(2) Pattern and Abstraction;
(3) Evidence and Proof; and
(4) Symmetries and their Structure.
Class location: WEL 3.402
Class times: 2:00–3:30 Tuesdays and Thursdays
Instructor: Associate Professor Dan Knopf
O?ce: RLM 9.152
O?ce hours: 10:00–12:00 Mondays, and by appointment
Syllabus: This syllabus/?rst-day handout will be updated during the
semester. A current version will always be available on Black-
board as well as through a link from the home page above.
Accommodations: Students with disabilities should request appro-
priate academic accommodations from the Division of Diversity and
Community Engagement, Services for Students with Disabilities.
Visit http://www.utexas.edu/diversity/ddce/ssd/ or else call
471.6259. (Please inform me of any approved accommodations as
early in the semester as possible.)
Doing the readings is a vital component of success in this course.
Our main source will be The Parsimonious Universe: Shape and Form
in the Natural World, by Stefan Hildebrandt and Anthony Tromba. It is
available (used) from the Co-op.
Various supplemental readings are collected in a custom course packet,
also available from the Co-op. Its readings are excerpted from the following:
• Architecture and Geometry in the Age of the Baroque, by George L.
• Connections: The Geometric Bridge between Art and Science, by
• Fearful Symmetry: Is God a geometer?, by Ian Stewart and Martin
• Geometry of Design: Studies in Proportion and Composition, by
• Life’s Other Secret: The New Mathematics of the Living World, by
• On growth and form, by D’Arcy Wentworth Thompson; and
• The shape of space, by Je?rey R. Weeks.
You should not expect to succeed in this course without regular attendance
and active participation.
Your ?nal grade will be based on the following components:
• Four in-class exams, each worth 20% of the total grade. (See below.)
• Regular homework assignments, collectively worth 10% of your grade.
• Class participation, worth 10% of your grade. (You will complete
daily “minute papers” — brief paragraphs written in response to
questions raised during class discussion.)
There will be no ?nal exam.
Your overall grade will be computed using the following scale:
D- D D+ C- C C+ B- B B+ A- A
51–55 56–63 64–65 66–67 68–75 76–77 78–79 80–87 88–89 90–91 92–100
The following is only a broad outline of the course schedule. Exact dates
and contents of assigned readings, homework due dates, et cetera, will be
announced as the course progresses. The schedule may also be changed for
pedagogical reasons. It is your responsibility to be aware of any changes
announced in class.
Optimal Geometry in Nature, Art, and Mathematics 3
Prologue: Weeks 1–2
(January 19, 21, 26, 28)
• Motivation and introduction
• Read Prologue and Chapter 3 (excerpts) from Parsimonious.
Optimality and Minimal Surfaces: Weeks 3–5
(February 2, 4, 9, 11, 16, 18)
• Read Chapters 5 and 6 (excerpts) from Parsimonious.
• Read excerpts from The shape of space.
• Exam I: February 18
Pattern and Abstraction: Weeks 6–8
(February 23, 25; March 2, 4, 9, 11)
• Read excerpts from Geometry of Design, Life’s Other Secret,
and On growth and form.
• Exam II: March 11
Spring Break: (March 15–20)
Evidence and Proof: Weeks 9–11
(March 23, 25, 30; April 1, 6, 8)
• Read Chapters 1–2 from Parsimonious.
• Read excerpts from Connections.
• Exam III: April 8
Symmetries and their Structure: Weeks 12–14
(April 13, 15, 20, 22, 27, 29)
• Read excerpts from Architecture and Geometry and Fearful Sym-
• Exam IV: April 29
Epilogue: Week 15
(May 4, 6)
• Preparation of ?nal pro jects
Other important dates: February 15 is the last day to drop without pos-
sible academic penalty; March 29 is the last day to drop except for urgent