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Computer Visualization in Mathematics

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.

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Computer Visualization in Mathematics

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  1. Computer Visualization in Mathematics Indiana University October 3, 2002 Professor Victor Donnay Bryn Mawr College

  2. Math is fun, relevant and everywhere • “Everyday Math” for K-5 • Integrated throughout curriculum • Manipulatives ( for kids )

  3. Math and Architecture

  4. Perspective Math and Art:

  5. Math and Sculpture

  6. Math and Crafts: Quilts

  7. Math in Nature

  8. Symmetry and Tessellations M.C. Escher:

  9. Computer: math manipulative for big kids • Play with ideas • Visualize the concepts • Experiment with “What if ......”

  10. Goal: • Introduction to some aspects of modern mathematics via the computer. • Geometry - Minimal Surfaces • Dynamical Systems and Chaos Theory

  11. Minimal Surface • Fix the boundary wire • Dip into soap solution • Resulting shape uses minimum area to span the wire

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

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

  14. Maryland Science Center http://www.mdsci.org Video to show relation of Costa Surface to torus

  15. Dynamical Systems • Something moves according to a rule • Physics: springs, planets • Weather • Earth’s Ecosystem: • Global Warming, Ozone Hole • Economic modeling

  16. 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/

  17. Regular Motion • Pattern • Predictable Chaotic Motion • No pattern • Moves “all over the place” • Not predictable

  18. 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

  19. 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 at http://serendip.brynmawr.edu/chaos/

  20. Geodesic Motion on Surfaces • Walk in a “straight line” • Path of shortest distance

  21. Round Sphere • Geodesics = great circles • Airplane routes • Path repeats --> Periodic motion

  22. 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!

  23. 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.

  24. Schwarz P- surface Minimal surface - Surface Evolver Make caps - Mathematica Attach caps- Geomview (http://www.geom.umn.edu)

  25. “Torus” • With chaotic geodesic motion

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

  27. Another Type of Surface with Chaotic Geodesic Motion Two surfaces connected by tubes of negative curvature Finite Horizon configuration

  28. Finite Horizon - Roman Military

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

  30. Thanks to: • Michelle Francl, Chemistry Department • Instructional Technology Team: • Susan Turkel • Marc Boots-Ebenfield • Gina Calderaio ‘01

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