Shapes of constant width

Yes - there are shapes of constant width other than the circle. No - you can't drill square holes. But saying this was not just an attention catcher. As the applet below illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments!

18 July 2015, Created with GeoGebra

Now then let us define the subject of our discussion. First we need a notion of width. Let there be a bounded shape. Pick two parallel lines so that the shape lies between the two. Move each line towards the shape all the while keeping it parallel to its original direction. After both lines touched our figure, measure the distance between the two. This will be called the width of the shape in the direction of the two lines. A shape is of constant width if its (directional) width does not depend on the direction. This unique number is called the width of the figure. For the circle, the width and the diameter coincide.

The curvilinear triangle above is built the following way. Start with an equilateral triangle. Draw three arcs with radius equal to the side of the triangle and each centered at one of the vertices. The figure is known as the Reuleaux triangle. Convince yourself that the construction indeed results in a figure of constant width. Starting with this we can create more. Rotating Reuleaux's triangle covers most of the area of the enclosing square. For the width = 1, the following formula is cited in Eric's Treasure Trove of Mathematics (Oleg Cherevko from Kiev, Ukraine kindly pointed out a misprint in the original quote)

$\displaystyle A=2\sqrt{3}+\frac{\pi}{6}-3=0.98770039073605346013\ldots$

which looks pretty close to $1,$ the area of the square.

Now generalize the construction a little by extending sides of the triangle the same distance beyond its vertices. This will create three $60^{\circ}$ angles external to the triangle. In each of these angles draw an arc with the center at the nearest vertex.

18 July 2015, Created with GeoGebra

All three arcs should be drawn with the same radius. Connect these arcs with each other with circular arcs centered again at the vertices (but now the distant ones) of the triangle.

There are many other shapes of constant width. May you think of any? There are in fact curves of constant width that include no circular arcs however small.

Here are some problems concerning shapes of constant width.

  1. For the Reuleaux triangle, find its area. Does it exceed the area of the circle of the same width (diameter)?
  2. For the Reuleaux triangle, derive a formula linking the length of the perimeter to its width? Does it remind you of anything? (Check Barbier's Theoerm.)
  3. The angle between two intersecting curves is defined as the angle between their tangents at the point of intersection. Find internal angles of the Reuleaux triangle.
  4. For every point on the boundary of a figure of constant shape there exists another boundary point with the distance between the two equal to the width of the shape.
  5. The distance between any two points inside the shape of constant width never exceeds its width.
  6. Assume in the diagram above, the triangle's side is 50 while each side was extended 10 units in each direction. What is the width of the resulting shape?
  7. The length of the boundary of shapes of constant width depends only on their width.

An aside

The applet on this page could be in one of three modes. At the start, it's in the first mode. In the second mode the square rotates around the triangle. In the third mode I stop erasing the background so that one can see how big the area is traced by the Reuleaux triangle. Click inside the applet to change modes.

Reference

  1. M. Gardner, The Unexpected Hanging and Other Mathematical Diversions, The University of Chicago Press, 1991
  2. R. Honsberger, Ingenuity in Mathematics, MAA, New Math Library, 1970
  3. H. Rademacher and O. Toeplitz, The Enjoyment of Mathematics, Dover Publications, 1990.
  4. I. M. Yaglom and V. G. Boltyansky, Convex shapes, Nauka, Moscow, 1951. (in Russian)

On Internet

Note

A visitor to the CTK Exchange who only identified himself as Giorgis left this message (Jul 12, 2008):

You have an attention grabber by stating "One can even drill square holes" and later you state with emphasis "No - you can't drill square holes" so I rectify it for you.

All you need is a drill bit, an almost stock standard round drill bit!!

Make sure it has $90$ degrees cone at the bottom.

  1. Take a piece of shim, fold it in half.
  2. Clamp it in a vice edge wise.
  3. Drill a hole no deeper than the height of the cone.

Take the shim out of the vice, unfold ...

Square hole with sharp corners with a round drill bit no less !!

While this approach does illustrate good "thinking out of the box", it may be hard to hold the drill steady, for the drill will make a contact with a very narrow surface, an edge actually. I think it could be easier to use a pair of cutters to cut off an $45-90-45$ triangle. The result is the same.

Shapes of Constant Width

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