rolling wheels investigating curves with dynamic software
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Rolling Wheels Investigating Curves with Dynamic Software. Effective Use of Dynamic Mathematical Software in the Classroom. David A. Brown – Ithaca College JMM 2012 – Boston, MA Wednesday January 4. Rolling Wheels and Revolving Planets. Brachistochrone Problem - PowerPoint PPT PresentationTRANSCRIPT
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 PLANETSEpicyclesWhat 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!!