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The Geometry Forum Newsletter

Fall 1995, page 4

A Geometry Forum Problem of the Week

http://mathforum.org/pow2/

The Path(s) of the Roll - October 2-6, 1995

I have a giant piece of paper and a number of shapes. I'm going to dip these shapes in paint and then "roll" them across the paper, making a note of the "path" that they leave behind. For example, if I take the cube, dip it in the paint, and "roll" it across the paper, I get a path that looks like

          __ __ __ __ __ __ __ __
         |  |  |  |  |  |  |  |  |
          -- -- -- -- -- -- -- --

which is just a bunch of squares in a row. No surprise. But I can also get a different path if I roll it along one of the edges as well. This path might look like

        *         *         *         *
      *   * *** *   * *** *   * *** *   *
        *         *         *         *

So here the cube sits on a face, then rotates up onto an edge, and then falls back on a face. This could be described as diamonds connected by line segments.

Here are four other shapes. For each one, describe every path that it might take when moving along the paper. It could be one, two, three, or even more! You might be able to draw some of them, but also try to describe them in words as best you can, sort of like I did with the cube.

The objects are

A rectangular parallelopiped (a cube stretched in one direction, like a box)
A sphere
A square pyramid
A cone
You choose a shape!

Good luck, and don't get paint all over yourself!

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Sarah Seastone
9 November 1995