# Math 4550: Applicable Geometry, Spring 2018

Prof. Robert Connelly

Location: Malott 203
Class time: 10:10-11:00,  MWF
Texts (all available on line through the Cornell Library with your Cornell ID):

• [Zeigler] "Lectures on Polytopes", Graduate Texts in Mathematics, Springer;
• [Dmitry Fuchs and Serge Tabachnikov], "Mathematical Omnibus", AMS
• [Joseph O'Rourke], "How to Fold It", Cambridge.
• [Eric Demaine and Joseph O'Rourke], "Geometric Folding Algorithms", Cambridge.
• Igor Pak's book: https://cornell.box.com/s/r7iqlq8uzz5myc0goh2h9b7pr5n3q47x
• My book with Simon Guest:  https://cornell.box.com/s/44oh9dl348ld6ul6oa658whrzy7q0qxd
• Larry Smith's Linear Algebra used in Math 4310: http://math.cornell.edu/textbooks

They should be available with your Cornell id.  The problems due Friday are below.

Office hours: Tuesday, Thursday 10:00-11:15 in Malott 565, you can make an appointment, or just drop by.

Course description:   Introduction to geometric things that can be applied to other things. For example, there are relations among the vertices, edges, and faces of a convex polytope that always hold, and this can be used to prove their rigidity.  This is a theorem of Cauchy in 1813.  This is why polyhedral domes don't fall down.  The rigidity of frameworks and the tensegrity sculptures of Kenneth Snelson can be understood in terms of quadratic forms and symmetric stress matrices.  These structures can be built and felt.  Flexible surfaces, thought not to exist, can be constructed and held in your hand, and they can be proved to exist.  Jammed packings of circular disks can be shown to exist.  Linkages that can draw a straight line can be constructed, solving a problem of Watt (that he never used).

We will try to tailor the demands of the course to the abilities of the students, with homework assignments every week, and we will try to have everyone give a talk in front of the class on a subject related to the subjects above.  Stay tuned here for the weekly assignments.

Homework will be due Fridays at the beginning of class.

A warm up question to chew on:  A planar quadrilateral has edge lengths 1, 5, 5 7.  What is the maximum area it can contain?

To replace a Final take-home exam here are some ideas for a project and short talk in class.

Homework #1: (Solution): https://cornell.box.com/s/fbnip1m6pxcqkqpd8t8xyqdltxypbsgr

Homework #2 (Do 9/7): 1.  In the following picture label the line segments so that it follows Caucy’s sign rule at each vertex in the figure. (At least 4 sign changes at each vertex.) https://cornell.box.com/s/ctv0nmvo50t2ws93o205to7a3g4dr3sq

2.  Fold a flat piece of paper so that there are four creases at a vertex.   What is the pattern of valley and mountain folds?

Homework #3: (Do 914) https://cornell.box.com/s/esljia6s1jhiv37li7se429rbbcs60rt and read up on symmetric matrices and quadratic forms, as in Larry Smith's book.  Please also read Chapter 5.1 to 5.3 in my book.  See me if you have questions.

Homework #4: (Do 9/21) These are a few exercises in generating stress matrices.  Solutions.

Homework #5: (Do 9/28) The last problem is a contest.  Enjoy. Solutions

Homework #6:  (Do 10/5) Come see me if you would like to see a model for Problem 1. Solutions

Homework #7:  (Do 11/3) Extra Credit for models. Note that there are now 8 problems. Solutions

Homework #8:  (Do 11/9)  It is OK to use computer software to compute things, but you do not need it.  Note the correction to the last problem. Solutions

Homework #9:  (Do 11/16) There was a typo on Problem 3(a), corrected here.  Solutions

Homework #10:  (Do 11/28, Wednesday)  Note some changes since 11/18. Solutions

Last edited: 11/30/18 9:00 PM