arwen
Uploaded by
33 SLIDES
484 VUES
330LIKES

Computer Visualization in Mathematics

DESCRIPTION

Join Professor Victor Donnay from Bryn Mawr College as he demonstrates the fun and relevance of mathematics in our daily lives. This presentation emphasizes the integrated nature of "Everyday Math" in K-5 education, utilizing manipulatives and visual arts. Dive into modern mathematical concepts like minimal surfaces, dynamical systems, and chaos theory, while discovering relationships between math, architecture, and nature. Experience the engaging interplay of visualization and experimentation in mathematics, from geometric shapes to chaotic motion, all through computer technology.

1 / 33

Télécharger la présentation

Computer Visualization in Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related