Chapter 4: Curves of Constant Width

Chapter 4, “The Curves of Constant Width” starts off by talking about a heavy object being moved by a platform on cylindrical rollers. The heavy object rolls off and leaves the rollers behind. The reason why this so easily works is the rollers have a cross section that has come to be known as “constant width” by mathematicians. The circle is not the only curve that works, there is an infinity amount of curves that have the constant width. Franz Reuleaux demonstrated the constant with properties with the simplest noncircular shape, Reuleaux triangle (Seen Below), which was drawing curves off an equilateral triangle. This triangle has many mechanical uses, but none of which come from this property. There is also ways to draw unsymmetrical curves with constant width. One method starts with a star which is irregular. Using a compass one can then connect arcs. Another method is to draw as many straight lines and then arcs are drawn at intersection of two lines. A fact about all curves with constant width, is the perimeter of all shapes with constant width n have the same length. There is a famous problem related to this called the kakeya needle problem which asked what the least area is where a line segment can be rotated 360 degrees.

“Convex Bodies of Constant Width” by Chakerian and Groemer had much more about constant width in their article. There is one thing in this article that I am happy Gardner left out of his chapter and that is theorem 1 because this theorem is confusing. This theorem stated “A convex body K has constant width b if and only if K + (-K) is a spherical ball of radius δ” (Chakerian). This theorem is very confusing and would have made Gardner’s chapter more confusing in my opinion.  Theorem 3 in my opinion was not as confusing and Gardner had included it in his chapter, which I am very happy about. This theorem stated “Any plane convex body B of constant width 1 has area not less than (π — τ/3)/2, the area of a Reuleaux triangle of width 1” (Chakerian).  This theorem is a lot easier to understand and really helped describe the concept of constant width in Chapter 4. Gardner had not used this exactly, but a similar concept to this and I am glad he included this.

“Curves of Constant Width from Linear Viewpoint” by J. C Fisher had a lot more interesting information in the article than the chapter did as well, but I also saw many similarities. One thing I am very happy Gardner left out is the vector space of parametrizable curves. This concept, after reading, is still very confusing to me and would have probably confused me when I was reading chapter 4. The chapter itself was understandable and this would have thrown me off. The article also included the theorems that had been used in the other article and the chapter, which once again, I am glad Gardner included in his chapter.  Much of Fisher’s article for me was very difficult to understand and I am therefore glad that Gardner used this in the chapter because it is a reoccurring concept, which must mean it is very important.

In the “Kakeya Problem for Simply Connected and Star-Shaped Sets” by F. Cunningham Jr., talked more in depth about the Kakeya problem, which Gardner introduced late in his chapter. The very beginning of this article describes the problem and I wish Gardner would have described the problem in a similar matter because it made things a bit less confusing in my opinion. Cunningham Jr., by describing the “needle” being turned around until position is reversed, I thought this description was a bit better and wish Gardner would have included more from this article.  A topic I am happy was omitted from the chapter was once again all the theorems of this. All of these made the topic itself more confusing, but overall the article was well used in Chapter 4.