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Most first-graders don't know anything about tangents or diameters, and they certainly don't know the algebraic description of a circle. But if you ask them to form a circle, they know how to do it. They'll all hold hands and keep moving backwards until they are arranged in a circle.

Click Attract. Each turtle finds two "buddies" and tries to keep a fixed distance from each of them, as if holding hands. Now click Repel. The turtles try to move away from one another, while still keeping close to their buddies. The turtles, like the first graders, form a circle.


In this case, the turtles aren't drawing a circle; they are a circle. Two simple rules govern the interactions among the turtles. The circle is not mentioned explicitly in the rules; it emerges from them.

This circle is like a two-dimensional balloon. The particles in a balloon are pushed apart by the air inside the balloon. But because the balloon is made of an elastic material, the balloon particles try to stay as close to one another as possible. In other words, the balloon tries to minimize its surface area. In two dimensions, the resulting shape is a circle.

In three dimensions, the shape that minimizes surface area for a given volume is a sphere. That's why soap bubbles assume a spherical shape.

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Mitchel Resnick and Brian Silverman
Epistemology and Learning Group
MIT Media Laboratory

Last modified: November 17, 2003
by Rupert Russell