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Rolling Wheels Investigating Curves with Dynamic Software

Learn how to use dynamic mathematical software to investigate the brachistochrone problem and the path traced by revolving planets. This interactive presentation explores various constructions, simulations, and equations to explain the motions and symmetries of these curves. Discover the applications of dynamic software in the classroom and gain a deeper understanding of mathematical concepts.

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Rolling Wheels Investigating Curves with Dynamic Software

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  1. Rolling WheelsInvestigating Curves with Dynamic Software Effective Use of Dynamic Mathematical Software in the Classroom David A. Brown – Ithaca College JMM 2012 – Boston, MAWednesday January 4

  2. Rolling Wheels andRevolving Planets Brachistochrone Problem I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

  3. Rolling Wheels andRevolving Planets Brachistochrone Problem Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

  4. Rolling Wheels andRevolving Planets • Brachistochrone Problem • Curve of fastest descent is the inverted cycloid

  5. Rolling Wheels andRevolving Planets Epicycles What is the path traced out by the moon as it revolves around Earth, which is revolving around the Sun?

  6. I Have Used This in • Calculus II - worked okay as a project • Calculus III – worked better than in Calc II • Mathematical Experimentation – works well • Inquiry-based course in experimental mathematics • Dynamic software • Two week lab • Expectations

  7. The Lab • The Lab Assignment • Expectations • Investigate the various constructions • Use technology to simulate and explore curves • Explain WHY the equations explain the motions • Explain the symmetries • Be artistic

  8. Rolling Wheels - GeoGebra • Topic is introduced with GeoGebra worksheet.

  9. Rolling Wheels - Mathematica • Students can also play using Mathematica

  10. Rolling Wheels – Software • Students learn to use dynamic software by manipulating some premade sheets. • Examples for student use • Mathematica – parametric curves • GeoGebra • Cycloids • Trochoids • Epicylces • Epicycles – Dynamic Worksheet

  11. Rolling Wheels – Wankel Engine • The Mazda Rotary Engine • Firing Chamber is an Epitrochoid • Hard to ignore the Releaux triangle • This set-up minimizes compression volume, thereby maximizing compression ratio. • Back to Lab. Credit: http://en.wikipedia.org/wiki/Wankel_engine

  12. Rolling Wheels Student Take-Aways • Cycloids – good motivator; easy to understand and predict • Trochoids – Circles rotating inside and outside of stationary circle • Hypocycloids: If ratio of radius rotating circle to stationary radius is p/q (rational, in lowest terms), then there are |p-q| cusps. • Epicycloids: If ratio of radius rotating circle to stationary radius is p/q (rational, in lowest terms), then there are |p|+|q| cusps.

  13. Wheels on Wheels on WheelsEpicycles • A Bit of Number Theory – refer to Lab and Epicycles worksheet. • The curve generated by a=-2, b=5, and c=19 has 7-fold symmetry. • The curve generated by a=1, b=7, and c=-17 has 6-fold symmetry. • WHY? • Note that -2, 5, and 19 are all congruent to 5 mod 7 • 1, 7, and -17 are all congruent to 1 mod 6 • Look at these in complex variable notation.

  14. Wheels on Wheels on WheelsEpicycles • A Bit of Number Theory – refer to Lab • The curve generated by a=1, b=7, and c=-17 has 6-fold symmetry. • As t advances by one-sixth of 2π, each wheel has completed some number of turns, plus one-sixth of another turn: • This is the heart of the symmetry.

  15. Wheels on Wheels on WheelsEpicycles • Motivates: f has m-fold symmetry if, for some integer k, • We can add any number of terms, and then, we see that we are dealing with terms in a Fourier Series:

  16. Wheels on Wheels on WheelsEpicycles • Theorem: A (non-zero) continuous function f has m-fold symmetry if and only if the nonzero coefficients of the Fourier Series for fhas frequencies nwhich are all congruent to the same number modulo m (and is relatively prime to m.) • Reference: SurprisisngSymmetry, F.Farris. Mathematics Magaize. Vol 69, Number 3, Jun 1996; p. 185-189.

  17. Wheels on Wheels on Wheels • This presentation and all files are available athttp://faculty.ithaca.edu/dabrown/wheels • Thank You and Happy New Year!!

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