R OLLING W HEELS I NVESTIGATING C URVES WITH D YNAMIC S OFTWARE Effective Use of Dynamic Mathematical Software in the Classroom David A. Brown – Ithaca College JMM 2012 – Boston, MA Wednesday January 4

R OLLING W HEELS AND R EVOLVING P LANETS 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.

R OLLING W HEELS AND R EVOLVING P LANETS 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.

R OLLING W HEELS AND R EVOLVING P LANETS Brachistochrone Problem Curve of fastest descent is the inverted cycloid

R OLLING W HEELS AND R EVOLVING P LANETS Epicycles What is the path traced out by the moon as it revolves around Earth, which is revolving around the Sun?

I H AVE U SED T HIS 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

T HE L AB 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

R OLLING W HEELS - G EO G EBRA Topic is introduced with GeoGebra worksheet.

R OLLING W HEELS - M ATHEMATICA Students can also play using Mathematica

R OLLING W HEELS – S OFTWARE Students learn to use dynamic software by manipulating some premade sheets. Examples for student use Mathematica – parametric curvesparametric curves GeoGebra Cycloids Trochoids Epicylces Epicycles – Dynamic WorksheetDynamic Worksheet

R OLLING W HEELS – W ANKEL E NGINE The Mazda Rotary Engine Credit: Firing Chamber is an Epitrochoid Hard to ignore the Releaux triangle This set-up minimizes compression volume, thereby maximizing compression ratio. Back to Lab.

R OLLING W HEELS S TUDENT T AKE -A WAYS 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.

W HEELS ON W HEELS ON W HEELS E PICYCLES 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.

W HEELS ON W HEELS ON W HEELS E PICYCLES 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 2 π, each wheel has completed some number of turns, plus one-sixth of another turn: This is the heart of the symmetry.

W HEELS ON W HEELS ON W HEELS E PICYCLES 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:

W HEELS ON W HEELS ON W HEELS E PICYCLES Theorem: A (non-zero) continuous function f has m- fold symmetry if and only if the nonzero coefficients of the Fourier Series for f has frequencies n which are all congruent to the same number modulo m (and is relatively prime to m.) Reference: Surprisisng Symmetry, F.Farris. Mathematics Magaize. Vol 69, Number 3, Jun 1996; p

W HEELS ON W HEELS ON W HEELS This presentation and all files are available at Thank You and Happy New Year!!