This is an introduction to the geometry of points and distances with applications to and from the theory of rigid and non-rigid structures. A basic role of geometry in science and mathematics is to determine when distance constraints on a configuration of points determine the configuration itself. This is connected to the theory of frameworks as used in engineering and well as distance geometry in mathematics. A brief list of topics that I hope to cover during the semester is described below. (The original content of this course as Geometric Topology has been changed for just the Fall of 1998. We will still do geometry, and a little topology may creep in but that is all.)

Prerequisites: A good background in linear algebra (including matrices, determinants, symmetric matrices, eigen vectors, etc.) and some basics of calculus. A little abstract algebra incuding the definintion of a finite group would help, but it is not necessary.

Topics: (The unfamiliar words below will be defined in the course. The following is meant to suggest the flavor of what is to be covered.)

1. A classification of the congruences of Euclidean space.
2. Infinitesimal and static rigidity of frameworks and tensegrities
3. Infinitesimal rigidity implies rigidity
4. Stresses and spider webs
5. Applications to glasses, protein structure, and rigid membranes with holes
6. Cauchy's Theorem abut the rigidity of convex polyhedra
7. The stress-energy quadratic form/mathix
8. Super stability and global rigidity
9. Applications to the packing of congruent spherical balls
10. The calculation of highly symmetric tensegrities using representations of finite groups

More information and links: The word tensegrity was coined by R. Buckminster Fuller to describe a structure that was created by Kenneth Snelson, a sculptor. These are structures made of sticks that are suspended in mid-air with cables attached at the ends or the sticks. The questions as to what geometric properties determine its stability are a major subject of this course. See the brief introduction to tensegrities. For a catalogue of pictures of several hundred different examples of tensegrities that are stable enough to be built, see the catalogue constructed with Allen Back, and a general introduction, "Mathematics and Tensegrity", in the March-April 1998 issue of the American Scientist. For an application of the idea of a tensegrity to biology as a way of understanding the structure of the cell, see the article "The Architecture of Life", by Donald Ingber, in the January 1998 issue of the Scientific American.

Instructor:

Meeting times and room:

Tuesday-Thursday, 2:55 to 4:10 PM, in White B29

Homework:

There will be regular weekly homework assignments, a take-home final and at least one take-home prelim. People will be expected to work on their own, but they may work in groups for specific projects as long as it is worked out beforehand.

Grading:

This will be determined mostly by homework problems and there will be a few in-class presentations.

Link to Cornell Mathematics Home Page

Link to Bob Connelly's Home Page

Last update: August 3,1998