Applet for discrete dynamical systems

I’m teaching the last two weeks of one of my colleagues’ Differential Equations course. I’m leading the class through a chapter on discrete dynamical systems. In preparation for the first lecture I created a couple of java applets using Geogebra. I thought others might be interested in them, so I’m linking to them here. The…

An applet for teaching the limit of a sequence

I’m currently teaching real analysis. Right now we’re discussing limits of sequences. The definition is: The limit of a sequence is (or converges to ) if, given any , there exists a natural number such that for all . I used GeoGebra to create the following applet, which illustrates the definition of a limit. (Clicking…

Applets for multivariable calculus

When I was in Portland a few weeks ago for MathFest 2009 I attended the minicourse Creating Demonstrations and Guided Explorations for Multivariable Calculus using CalcPlot3D by Paul Seeburger. Paul has created some applets to be used in calculus classes (mostly, but not exclusively, multivariable calculus). They can be found on his Exploring Multivariable Calculus…

Carnot’s Theorem

Here’s a neat theorem from geometry. Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c…

Parabolas and focal points II

Yesterday I wrote about parabolic mirrors. I pointed out that parallel rays hitting a parabola typically do not reflect back to a focal point. That happens only when the rays are parallel to the axis of the parabola. Then my friend Dan sent me an email encouraging me to look at the envelope of the…

Parabolas and focal points

A colleague from another department stopped by my office this morning with a question about parabolas and parabolic mirrors. In the simplest terms, his question was the following: When parallel rays reflect off a parabola, do they always converge to a focal point? As we can see in the diagram below, rays that are parallel…

The Japanese Theorem

[Update: I’ve written quite a bit more about this theorem since 2009. See this page for more details.] I’ve been playing with GeoGebra for the last few days. As an exercise I decided to create applets to demonstrate the extremely beautiful Japanese Theorem. The first appearance of the Japanese theorem was as a Sangaku problem….

Four color theorem applets

The four color theorem is a beloved result with a long and fascinating history. The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in 1852 by Francis Guthrie. After many, many failed proofs, the conjecture was finally put to…