Chapter 4: Curves of Constant Width

When given a job to move an enormously heavy load, one would think to move it with circular wheels. The axels on the wheels would buckle under pressure. The object should instead be placed on a platform with cylindrical rollers. The rollers would have to be picked up and placed in front of the object to keep it rolling. These rollers have what is called a constant width to keep the object from bouncing up and down.

Most people think that the only shape that has a constant width is a circle. This is a common mistake. There is actually infinity of curves that have a constant width. The simplest of these curves is the Reuleaux triangle. To construct a Reuleaux triangle, first draw and equilateral triangle with sides A, B, and C. Then with a compass draw and arc from A to B, B to C, and C to A. If two parallel lines forming a right angle limit a curve of constant width, such as the Reuleaux triangle, the bounding lines will form a square. If a Reuleaux triangle is placed within a square and rotated, this will be observed. In 1914 Harry James Watts took this into consideration and created a drill based on the Reuleaux triangle. This drill drilled square holes! The drill is simply a Reuleaux triangle made concave in three spots to allow for cutting. The Reuleaux triangle has the smallest area for the given width out of all the curves. The corners have angles of 120 degrees, which are the sharpest possible angles that a curve of constant width can obtain. Extending the sides of the equilateral triangle can round off these corners. This provides you with points D, E, F, G, H, and I.  Use a compass to first connect D to I, E to F, and G to H. Then use the compass to connect the letters that have already not been connected. This makes it a symmetrical curve of constant width. This process of can also be done with a regular pentagon.

It is possible to have unsymmetrical curves of constant width. One example is to start with a star of seven points. Draw line segments connecting each point to the point almost directly across from it. All of the line segments need to be mutually intersecting. Each of these line segments must be of equal length. Use a compass to connect the two opposite corners to form an arc. This will result in a curve of constant width.

A curve of constant with does not need to be made up of circular arcs. You can draw a convex curve from the top to the bottom of a square and touching its left side. This curve will determine a curve of constant width. This being said, the curve cannot include straight lines. The perimeters of all curves of constant width n have the same length. Since a circle is a curve of a constant width, the perimeter of any curve with a constant width of n must be pi n. This is the same as the circumference of a circle with a diameter of n.

Most people would assume that a sphere is the only solid on constant width. This is not true. There are many solids that when rotated inside a cube, the shape is touching all six sides of the cube. All solids with constant width are derived from the regular tetrahedron. It is a very common mistake to think that all solids of constant widths have the same surface area. This is definitely not the case. Contrary to this, all the shadows of solids with constant widths are curves of a constant width. In 1917 Sôichi Kakeya proposed the Kakeya needle problem. The problem is: What is the plane figure of least area in which a line segment of length 1 can be rotated 360 degrees? For a long time, many mathematicians believed that the answer to this problem was the deltoid. Ten years later Abram Samoilovitch Besicovitch, Russian mathematician, proved that this problem had no answer. He proved that there is no minimum area.

This chapter was very interesting to me. Prior to reading it I believed that circles were the only shapes with constant widths. To learn that there is infinity of shapes with constant width was really an eye opener. This chapter was sometimes hard to follow and I had to reread many sections more than once but in the end it was worth it.