computer visualization in mathematics
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Computer Visualization in Mathematics. Indiana University October 3, 2002 Professor Victor Donnay Bryn Mawr College. Math is fun, relevant and everywhere. “Everyday Math” for K-5 Integrated throughout curriculum Manipulatives. ( for kids ). Math and Architecture. Perspective. - PowerPoint PPT PresentationTRANSCRIPT
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