Geometry and the Imagination Yana Mohanty Palomar College

A point of view Geometry is a way of analyzing and classifying objects around us. Examples: –Wallpaper or floor tile patterns –Knots –Surfaces of 3-dimensional objects Change perceptions of what math is and feelings about math. Mathophobes and mathophiles can coexist…in one classroom!

History of the course Title taken from Geometry and the Imagination by Hilbert and Cohn Vossen. Developed at the Geometry Center at the University of Minnesota by John H. Conway, Peter Doyle, Jane Gilman and Bill Thurston. Peter Doyle taught it at UCSD, Dartmouth College to classes consisting of math majors and non-math majors. I taught parts of it to students in grades 5-12 at the San Diego Math Circle

Course materials and methods of evaluation Texts: –The Shape of Space by Jeff Weeks –The Knot Book by Colin Adams –Course notes Grading: –Homework problems from the above –Short, low pressure quizzes testing students on the basics –Journal assignments –Projects vary in extent depending on student level ideas can be found the course notes

A possible topic: classify all wallpaper or tile patterns. Example: brick wall Rough idea : find smallest repeating piece and described how such pieces are glued together 2 mirrors cross Gyration of order 2 2*22 mirrors

A vendor of tiles… Specialists in the Installation, Restoration and Supply of Decorative Victorian Tiles and Period Mosaics

Pattern 1

Detail of pattern 1 *442 mirrors

Pattern 2

Detail of pattern 2 3*3 order 3 gyration point Identical points where 3 mirrors meet

17 patterns possible

Another topic: study of knots Associate a polynomial to a knot in a clever way: and thisPolynomial of thismust be equal Polynomial of this must be 1and this

Example of what you can get out of this Polynomial of this is Polynomial of this is Moral: you cant make the green knot into the pink one without tearing it open and regluing!

Features of this topic Practice manipulating polynomials and rational functions Associate above with tangible objects Introduce a question that has stumped some of the smartest people in the world: Suppose the polynomial of something like this is 1. Does that mean it is not tangled? Exercise in logic: untangled implies polynomial is 1 polynomial is 1 does NOT imply that knot is untangled

Surface: the skin of a 3-D object without any breaks Examples of surfaces

Classifying surfaces Terminology: VERTEX EDGE FACE

We are also permitted to have… bi-gon VERTEX EDGE FACE

Define the Euler Characteristic, =V-E+F Where V = number of vertices E = number of edges F = number of faces Leohard Euler, 1700s

Examples

Euler numbers of some surfaces.

What the Euler number can tell you As long as your surface is not too strange (does not include Mobius bands) the Euler number tells you exactly what it is! The Euler number is insensitive to any deformations (as long as there are no tears) or the cell decomposition you choose.