By Scott Kim|Saturday, December 01, 2001

Math Hits the Road

Photograph courtesy of Deanna Haunsperger/Macalester College
Brave bumpy byways and track traveling turtles in these two tempting teasers

Roads Less Traveled
Mathematician Stan Wagon (right) loves a smooth ride. Even though the wheels of his bicycle are square, the shape of the road allows the centers of the wheels to travel in a perfectly horizontal line. Below are some other wacky wheels and seemingly rocky roads adapted from Wolfram Research's CD-ROM The Mathematical Explorer by Stan Wagon. Pair each wheel with the road on which it would—in theory—travel smoothly, assuming the bicycle is going fast enough. (In practice, the sharp points on some wheels would get stuck in the roads' crevices.) Hint: The correct wheel will notch into a road's dip. Two of the roads match two different wheels.

Fractals on the Half Shell
Turtle geometry is a technique for drawing mathematical curves that was popularized in the educational programming language Logo, brainchild of Massachusetts Institute of Technology professor Seymour Papert. Imagine a robot turtle on the floor, facing right and holding a pen. The turtle responds to three commands: "F" tells it to go forward one unit, "+" tells it to turn left 60 degrees, and "-" directs it to turn right 60 degrees. The command string F+F--F+F, for instance, instructs the mechanical terrapin to draw the peaked shape shown above, starting at the black dot.

Using a substitution rule in the command string lets the turtle draw more complicated images. For example, F -> F+F--F+F means the turtle will substitute the string F+F--F+F wherever an F appears in the original command string. The string F--F--F produces an inverted triangle (top right). To get a six-pointed star (middle right), apply the substitution rule once: F--F--F becomes F+F--F+F--F+F--F+F--F+F--F+F. Apply the rule a second time and you get the prickly star shown at bottom right.

Match the images below with the commands that produced them. Each command is made up of a substitution rule, a command string, and a depth number that tells the turtle how many times to apply the substitution rule to the command string. For example, the command to draw the prickly star above is F -> F+F--F+F, F--F--F, 2.

1. F -> F+F--F+F, F--F--F, 3
2. F -> F+F--F+F, F++F++F, 2
3. F -> F+F--F+F, F-F-F-F-F-F, 2

4. F -> F+F--F+F, F---F, 2
5. F -> F+F--F+, F--F--F, 2
6. F -> F+F--F+F+, F--F--F, 2


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The Mathematical Explorer presents recent discoveries in mathematics and lets you try them out by plugging numbers into mathematical expressions. For more, go to, or call 800-965-3726.

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