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# One Good Turn

360 is not the only choice. Mathematicians often use "radians" rather than degrees. There are about 6.283 radians in a circle. The exact number of radians in a circle is the same as the ratio of the circumference of a circle to its radius. This ratio is the same for all circles, regardless of the size.

Here's a different choice. We will make TURN 1 turn a full circle.

Click Go. On each turn, the turtle turns a fifth of the way around. So five turns make a pentagon. Similarly, FORWARD 80 TURN 1 / N makes a polygon with N sides. (Note: you must leave spaces around the "/".)

Now, try FORWARD 80 TURN 2 / 5. This makes another five-sided figure. But it's not a pentagon; it's a five-sided "star polygon." The turtle crosses its path before returning to its starting position.

The turtle makes two full turns while drawing the star polygon. In general, FORWARD 80 TURN T / N makes a figure with N points, and the turtle makes T full turns before returning to its starting position. (This assumes that T / N is a fraction that is fully reduced.)

Numbers that can be represented as T / N are called rational numbers. So if the input to TURN is any rational number, then the turtle will eventually get back to its starting position, having drawn a figure with N points. Some numbers (such as the number of radians in a circle) can not be represented as T / N. If the turtle turns "irrationally," it will never return its starting position.

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