البوابة
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 on the right illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments!
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)
which looks pretty close to 1, the area of the square.
Extend sides of the triangle the same distance beyond its vertices. This will create three 60o angles external to the triangle. In each of these angles draw an arc with the center at the nearest vertex. 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.
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 on the right illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments!
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)
which looks pretty close to 1, the area of the square.
Extend sides of the triangle the same distance beyond its vertices. This will create three 60o angles external to the triangle. In each of these angles draw an arc with the center at the nearest vertex. 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.
