**This lesson plan developed under the sponsorship of NASA Glenn
Research Center's K-12 Program.**

Before you attempt to use Orbifold Pinball as an educator to teach scientific or mathematical principles, you should read this page and understand it. We recommend that you print this page out and read it once or twice before experimenting with Orbifold Pinball. Once you do, go to:

The general principle being taught by Orbifold Pinball is sensitive dependence on initial conditions. Simply put, this is the idea that very small changes (or differences) in the starting conditions of a complex system can produce very large changes in the system.

The most common example of this phenomenon is the weather: the slightest change to atmospheric conditions can result in extremely large changes in weather. Another way to say this is: a very small difference in the initial measurements of a complex system can produce a vastly different prediction from the prediction based upon the original measurement.

This example of weather is used so commonly to explain sensitive dependence because the man most responsible for discovering or developing this idea was a meteorologist: Edward Lorenz. Lorenz supposedly named the principle "The Butterfly Effect" because of an example he used to help explain the principle: a butterfly flapping its wings over Brazil can produce a tornado in Oklahoma.

Part of the charm of the idea is its highly counterintuitive nature. While it is hard to imagine that a tornado might have been caused by a butterfly, it is so. But perhaps a better way of expressing this thought would be to say that if you fail to account for the small changes in atmospheric conditions produced by an object so tiny as a butterfly, you may fail to predict something as big as a tornado.

Despite the fact that sensitive dependence is fairly easy to understand (probably due to the homely examples used to explain it), it is quite uncommon to find anything like it in high school or college science or math texts. Most texts tend to rely heavily upon equations and concepts that can be written as a fundamentally proportional relationship, such as Newton’s:

Nature, however, does not share the modern science
curriculum’s enthusiasm for solvability and linearity. In fact,
the vast majority of systems in nature (be they biological, chemical,
purely physical, or even neurological) are neither linear nor
solvable. And a good number of them are *chaotic*. And the ones
that are chaotic always are sensitively dependent on initial
conditions.

Before we can play the game, however, there is one more principle that needs to be understood. This revolves around two terms: deterministic and predictable. The word "deterministic" means simply that if you know the initial conditions of any system (such as its mass and the force supplied instantaneously to the object), you can know that outcome (such as the acceleration produce in the body within a frictionless environment). The term "predictable" means virtually the same thing. Then why two terms? Because in chaotic systems, something can be deterministic without being predictable.

How is this possible? Think of the butterfly and the weather. The weather is a deterministic system because the only thing operating within that system are molecules interacting (this is not the place to delve into the mysteries of quantum physics). The molecules have positions and velocities, and everything about the system can be derived from those positions and velocities. But the system is not predictable. There is no way to gather the information necessary to make a completely accurate prediction. A butterfly may flap its wings, sending your system–and your prediction–into futility.

So, when you teach Orbifold Pinball, you will be teaching concepts fundamental to modern science: determinism, measurement, linearity, non-linearity, prediction, and many others. Once you understand these concepts, you are ready to Play Orbifold Pinball!

You will notice three black dots on the screen in the middle of the "playing field." Think of these as peaks of mountains. The surface slopes downward from these points, and terrain slopes upwards and downwards at different places: all you can see are the three peaks (because a three dimensional surface is displayed in two dimensions.

You begin the game by picking an initial angle to "shoot" the ball at: 90 degree is straight up, 180 degrees is straight left. The program calculates and displays the path of the ball.

Perhaps the best way to teach this game is to simply ask every student to "shoot" the ball at any angle that he or she chooses. You will be astonished at the diversity of the results. And this diversity itself is the first lesson: while this is a deterministic system (the path of the ball being calculated using a mathematical algorithm), there is no correlation between angle and "complexity" of the path. Some paths will be simple lines, others will be highly complex figures. Take time to reinforce this principle by reviewing angles and the paths the initial conditions produce.

Next, focus in on two particular angles that are fairly close together: 26 and 28 degrees.

- Have each student shoot the ball at 26 degree.
- Next, have them shoot the ball at 28 degrees. There will be a great difference between the two answers.
- Ask the students if they can find the "boundary" between complexity and simplicity. If 26 degrees is defined as "simple" and 28 degrees in defined as "complex", where does the image change?
- The students will experiment with this until you introduce the idea of significant digits. Have them enter "27.5" degrees, then "27.6", etc. And then ask them to input a number half way between "27.5" and "27.6". (They should input "27.55").
- After the students continue this for a while, ask them to find
them boundary down to four decimal places. It is a good idea for
students to work in groups now, especially with pencil and paper
so that they can write down things like:
27.551 = simple 27.552 = complex

There are many other experiments that you can perform with Orbifold Pinball, such as asking student to "produce" the most complex design possible, finding two very different angles that produce very similar patters, and finding two angles that are extremely close together that produce very different patterns.

This lesson plan and explanation of Orbifold Pinball produced by Dr. John D. Eigenauer. Dr. Eigenauer has no affiliation with the creators of Orbifold Pinball, nor the Geometry Center at the University of Minnesota.

If you have any comments or additions to these URLs, please send
them to Dr. John D. Eigenauer at home
, or at
work.

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Carol Galica ( Carol.A.Galica@grc.nasa.gov) Beth Lewandowski (Beth.Lewandowski@grc.nasa.gov)