Computer Visualization in Mathematics

Indiana UniversityOctober 3, 2002Professor Victor DonnayBryn Mawr College

Math is fun, relevant

and everywhere

“Everyday Math” for K-5 Integrated throughout curriculum Manipulatives

( for kids )

Math and Architecture

Math and Art: Perspective

Math and Sculpture

Math and Crafts: Quilts

Math in Nature

M.C. Escher: Symmetry and Tessellations

Computer: math manipulative

for big kids

Play with ideas Visualize the concepts Experiment with “What if ......”

Goal:

Introduction to some aspects of modern mathematics via the computer.

Geometry - Minimal Surfaces Dynamical Systems and Chaos Theory

Minimal Surface

Fix the boundary wire Dip into soap solution Resulting shape uses minimum

area to span the wire

Schwarz P surface Imagine wires on the 6 ends H. A. Schwarz, 1890

Costa Surface

Discovered by Brazilian Celso Costa, 1980s Torus (?) with 3 holes (punctures)

Video to show relation of Costa Surface to torus

Maryland Science Center

http://www.mdsci.org

Dynamical Systems Something moves according to a rule

Physics: springs, planets Weather Earth’s Ecosystem:

Global Warming, Ozone Hole Economic modeling

Billiards

Rule: One ball Moves in straight line Reflects off wall with angle reflection = angle of incidence

Moves forever - no friction http://serendip.brynmawr.edu/chaos/

Regular Motion Pattern Predictable

Chaotic Motion No pattern Moves “all over the place” Not predictable

Billiard Program

Undergraduate summer research 1996 Team:

Derya Davis, Carin Ewing, Zhenjian He, Tina Shen,

Supervised by: Bogdan Butoi, Math graduate student Deepak Kumar, Professor of Computer Science Victor Donnay, Professor of Mathematics

The Standard Map: 2 Dimensional Dynamics.

Freeware from website of Professor J.D. Meiss: http://amath.colorado.edu/faculty/jdm/programs.html

Phase Space Game athttp://serendip.brynmawr.edu/chaos/

Geodesic Motion on Surfaces

Walk in a “straight line” Path of shortest distance

Round Sphere

Geodesics = great circles Airplane routes Path repeats --> Periodic motion

Question: Does there exist a “deformed” ,

bumpy sphere with chaotic geodesics?

Topology: stretch and bend round sphere - still a “sphere”

But not the normal one!

Motion on this “sphere” is chaotic

K. Burns and V.J. Donnay (1997) ``Embedded surfaces with ergodic geodesic flow'', International Journal of Bifurcation and Chaos, Vol. 7, No. 7,1509-1527.

Schwarz P- surfaceMinimal surface - Surface Evolver

Make caps - Mathematica

Attach caps- Geomview (http://www.geom.umn.edu)

“Torus” With chaotic geodesic motion

Pictures made on Unix workstation•Louisa Winer ‘96•Gina Calderaio ‘01

Another Type of Surface with Chaotic Geodesic Motion

Two surfaces connected by tubes of negative curvatureFinite Horizon configuration

Finite Horizon - Roman Military

The radiolarian Aulonia hexagona, a marine micro-organism, as it appears through an electron microscope

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

Thanks to: Michelle Francl, Chemistry Department Instructional Technology Team:

Susan Turkel Marc Boots-Ebenfield

Gina Calderaio ‘01