The most common closed and curved plane figure that has a constant width as it rotates is the circle. Surprisingly, however, there are other figures that have this property. You can construct a variety of these shapes with a compass and straightedge. The rollers you can build with this Snack behave in seemingly paradoxical ways.
- Access to a computer printer
- Glue, tape, or stapler
- File folders, poster board, or similar stiff material
- Metric ruler
- Drawing compass
- A piece of foam board, poster board, or Masonite, about 6 × 18 inches (15 × 45 cm)
In this Snack, we offer several options. You can use a pre-drawn template to build a simple non-round roller (Version 1), perform the geometric constructions for the rollers yourself (Version 2), or construct simple non-round rollers of your own design (Version 3).
Version 1: Predrawn template for a simple non-round roller:
- Print several copies of this template (if possible the printed size should match, more or less, the measurements indicated on the template).
- Glue each printed sheet in its entirety to stiffer material (file folder or poster board).
- Cut out the patterns. The large pattern is the axle and the two smaller ones are the ends.
- Fold the axle on the horizontal lines to make a triangular prism.
- Fold Tab A over area A, and glue or tape them together.
- Fold the axle flanges x and y outward.
- Staple or glue the axle flanges to the corresponding lettered areas on the ends.
- Make two or three of these rollers.
Version 2: Geometric constructions for a simple non-round roller:
- Follow steps 1–4 in the diagram below (click to enlarge) to construct an equilateral triangle with circular arcs connecting its vertices. We suggest you start with a triangle about 2 1/3 inches (6 cm) on a side. After you are familiar with the process, you can make larger versions. This particular constant-width, non-round roller is called a Reuleaux triangle.
- On a piece of file folder, draw four identical Reuleaux triangles and cut them out. These will serve as the wheels for a set of two rollers.
- To make an axle, draw a rectangle measuring 3 inches × 7 inches (8 cm × 18 cm) on another piece of file folder. Divide this into three smaller rectangles, each measuring 3 inches × 2 1/3 inches (8 cm × 6 cm). On one end, add a 1/2 × 3-inch (1 × 8-cm) tab.
- Finally, use the compass to make three arcs of radius 2 1/3 inches (6 cm) along each side of the three rectangles. Your drawing should be identical to the drawing in this template.
- Draw and cut out two or three rollers. Assemble as noted in Version 1.
Version 3: Geometric construction of a general case of non-round rollers:
Non-round, constant-width rollers of many different shapes can be made as noted in the following set of steps and shown in the diagram below (click to enlarge). You’ll need to build two or three of these rollers.
- Draw a triangle of any size on a piece of cardboard. It does not have to be any particular type of triangle—any variety will do. Extend each side of the triangle beyond the triangle’s vertices.
- Find the longest side of the triangle and open the compass so that its gap is a little longer than that side. (In the picture shown, the longest side is BC.) Set the compass at point B and make arc EF between BA and BC.
- Set the compass at point A, matching the pencil to point E. Make arc DE.
- Set the compass at point C, matching the pencil to point F. Make arc FG.
- Continue this process until the closed curve is complete. (Note: If, after the first arc is made, one of the subsequent arcs ends up going into the triangle, then you will not be able to continue around the outside. If this happens, you should consider how the triangle might be redrawn to avoid this, and try again. The drawing after Step 6 in the diagram shows all the circles that contribute to the roller.)
- Construct the axle by drawing a diagram similar to the one shown (Step 7 in the diagram above) onto a piece of cardboard.
- Attach the axle to the roller ends. It is important that the roller ends be aligned with each other. The easiest way to accomplish this is to match the side of the axle with the corresponding sides of the triangle.
Build at least two identical rollers. Put them on a flat surface. Place the piece of foam board, Masonite, or poster board on top of the rollers and roll it gently from side to side. The rollers should roll smoothly and the board should stay level.
There is no standard definition of the center of a non-round roller. Define your own center, and see if it stays a constant height above the surface on which the rollers are rotating. Notice that there is no point on the roller that stays a constant height as the roller rolls. Any point you choose will bob up and down.
In Versions 1 and 2, the roller always pivots on a vertex, even at the top and bottom, and the distance to the opposite contact point is always the same.
In Version 3, the width of the non-round roller at any point is defined by a straight line that runs through one of the vertices of the triangle and through the triangle itself. Each such straight line is the sum of two radii—the radius of one large arc and the radius of one small arc. For example, the straight line PO in the diagram below (click to enlarge) is the sum of the radii of arc HG (a large arc) and arc DE (a small arc).
Suppose you have a roller like the one shown in the diagram above. Let’s say the roller is resting on point O. As the roller rolls on arc HG, with its resting point approaching H, the board on top of the roller will be rolling on arc DE. The board will remain level because the roller’s width (which is the sum of the radius of arc DE and the radius of arc HG) will be constant.
Suppose the roller rolls until it rests on point H. Its width is still the sum of two radii—the radius of arc HI and the radius of arc EF. Because arc HI and arc HG have point H in common, and because arc DE and arc EF have point E in common, the width of the roller must still be constant. As the roller continues rolling and reaches point I, the same argument applies, and the width of the roller is always the same.
When you tried to choose a center for your non-round roller, you might logically have chosen the triangle’s centroid. Another logical choice would be the point halfway between the top and bottom of the roller for a particular orientation. Neither of these points, however, stays at a constant height as the roller rolls. Instead, they describe an up-and-down motion. As you observed, no point on the roller stays a constant height as the roller rolls. For this reason, the rollers would make lousy car wheels. (Where would you put the axle?)
A drill bit made in the shape of a Reuleaux triangle can be used to drill a square hole!