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  • Educator Victor J. Donnay
  • Animator Mark Storer, Henry Chung, Ricky Earl, Noel Wong, Michael Field
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Additional Resources for you to Explore
The best way to develop an understanding of billiard motion and the difference between regular and chaotic motion is to observe billiard motion first hand. While one can do this in a billiard hall, there one is restricted to billiard motion on rectangular tables. With computer simulation programs, one can observe billiard motion on a wide range of different table shapes.

My Bryn Mawr College students have created a series of interactive web-based applets that allow the user to play billiards inside a circle and the half-circles connected by straight edges, which is called the Stadium billiard, as well as within polygons and ellipses.

There are a nice set of interactive billiard programs written in the software system Mathematica.

To explore billiard motion in a circle, use this program. By moving the locators, the user can adjust the starting position and starting angle of the trajectory. By adjusting the slider, one can control the number of bounces the ball will make.

The circle is a special case of an ellipse in which the length of the horizontal and vertical axis of the ellipse are equal.
The billiard motion in an ellipse presents some captivating new twists to what one sees in a circular billiard.
To explore billiard motion in an ellipse, use this program or this program.

To produce the circular billiard, adjust the slider to set the axis of the ellipse to 1 and then the two focal points converge to the same point. Rather than using the slider, one can also click on the little box to the right of the slider and then type in the value 1.

It is also fun to explore billiard motion inside regular polygons: triangles, squares, pentagons and so forth, which one can do using this program.

At present, there is not a Mathematica simulation for the chaotic billiard motion inside the Stadium billiard (two half circles connected by straight lines). However, a pair of YouTube videos provides one way of seeing the difference between the regular motion of the circle billiard and the chaotic motion of the Stadium billiard. In these videos, 500 non-interacting particles (billiard balls) are started with slightly different initial positions and allowed to move. In the circle billiard, these trajectories retain some structure while in the Stadium billiard, the trajectories lose all cohesion due to the chaotic motion.

Have we reached the tipping point discussed in this lesson? Read the article summary published in Nature and answer that question for yourself. See  the future predicted tipping point year for various cities in the world. How will this affect your lifestyle? For more information, read the Smithsonian article about the same topic. Watch this video from Berkley Initiative in Global Change Biology and this one from the Weather Channel regarding other tipping points in nature.

Climate scientists are concerned about the 400 ppm of carbon dioxide that have accumulated in our atmosphere. Why such concern? Read the article from the National Oceanic and Atmospheric Administration and find out. Then visit the Global Climate Change page at NASA and read individual scientists thoughts about the topic. Global Warming 101 by National Geographic summarizes the topic in a quick video summary.

Bill McKibben, author and environmentalist has an organization dedicated to building a grassroots global climate movement. Interested? Peruse the website and find out more information about global climate change, the goals of, and what you can do to make a difference.

Ready to contribute to cutting carbon emissions? How do you start? Visit the Michigan State University site on decreasing carbon emissions and find some answers. Read the New York Times article: Reducing Your Carbon Footprint for some more background information. Then, visit the Nature Conservancy and use their online calculator to find your personal carbon footprint. How does your footprint compare to the rest of the world?

Bill Nye the Climate Guy? Watch as Bill Nye, the Science Guy addresses climate change on the Smithsonian website. Scroll down the site for more articles about the topic!

For more information, check out these TED-Ed Lessons on climate change!

Why I must speak out about climate change - James Hansen

The science behind a climate headline - Rachel Pike

Climate change: Earth’s giant game of Tetris - Joss Fong

Cloudy climate change: How clouds affect Earth’s temperature - Jasper Kirkby
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Lesson Creator
New York, NY
What do you suggest we do? Is it too late to reverse these climate changes? Are humans capable of such drastic change? What would you be willing to give up or change to make this possible?
10/23/2014 • 
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