ROLLING WHEELSINVESTIGATING CURVES WITH DYNAMIC SOFTWARE

Effective Use of Dynamic Mathematical Software in the Classroom

David A. Brown – Ithaca CollegeJMM 2012 – Boston, MAWednesday January 4

ROLLING WHEELS ANDREVOLVING PLANETSBrachistochrone 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.

ROLLING WHEELS ANDREVOLVING PLANETSBrachistochrone 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.

ROLLING WHEELS ANDREVOLVING PLANETS Brachistochrone Problem

Curve of fastest descent is the inverted cycloid

ROLLING WHEELS ANDREVOLVING PLANETS

EpicyclesWhat is the path traced out by the moon as it

revolves around Earth, which is revolving around the Sun?

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

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

ROLLING WHEELS - GEOGEBRA

Topic is introduced with GeoGebra worksheet.

ROLLING WHEELS - MATHEMATICA

Students can also play using Mathematica

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

ROLLING WHEELS – WANKEL ENGINE

The Mazda Rotary Engine

Credit: http://en.wikipedia.org/wiki/Wankel_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.

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.

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.

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.

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:

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 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. 185-189.

WHEELS ON WHEELS ON WHEELS

This presentation and all files are available at

http://faculty.ithaca.edu/dabrown/wheels

Thank You and Happy New Year!!