Bodies of Constant Width in Art, Design, and Engineering

Abstract

As we mentioned before, a normal wheel (rotating around a fixed axis) must be circular in shape to allow smooth forward motion without vertical bumps. However, a cylindrical roller does not require a circular cross section to allow smooth forward motion. In fact, any curve of constant width, such as the Reuleaux triangle, may be used as a cylindrical roller. Besides this nice property, the curves of constant width, in particular, the Reuleaux triangle, have been exploited by engineers, artists, and designers to obtain wonderful objects and number of ingenious mechanisms.

The one who understands geometry is able to understand everything in this world.

Galileo Galilei

Appendices

Notes

Constant Width in Engineering

As shown already in several parts of the book, curves of constant width occur in engineering, see, e.g., the references in [803] and [1200], pp. 48 and 246, and the intention of the book [190] goes into the same direction. One of the two authors wrote a nice introductory paper (for teachers) on constant width sets, also laying special emphasize on the mechanical point of view, cf. [1011]. A nice historical background for all this and more is presented in [854]. For example, besides as rotors one can see constant width curves also as “polygonal” profiles of so-called polygon connections in engines. (Here the notion “polygon” is also used for smooth constant width curves which only remind the observer how a Reuleaux polygon roughly looks like.) For this purpose we refer to DIN 32711 (German Institute for Standardization, DIN Standard 32711) and, in particular, the P3G profile presented there. Flexible continuous inner and outer contours for form- and force-closured connections are of constant width, and a nice “curve theoretic” discussion of this topic is given in [1222]; related are also the reports [376] and [375].

Drilling Holes and Rotors

As we mentioned before in Section 17.1, rotors can be regarded as a natural extension of constant width sets, and we also discussed the literature on rotors and how rotors are also used to drill holes with nonspherical (e.g., polygonal) cross sections. The idea to use Reuleaux triangles as rotors for this purpose goes back to Watts (1914). As already described in Section 18.5, this produces a hole whose cross section is almost a square, and in 1939 an anonymous contributor to Mechanical World modified this construction to obtain a perfect square, see § 10.4 (Plate 21) of the book [190], and also our Section 18.5. An extension is described in [266], see also [267] and [1160]. In these papers the problem of drilling a hole with a regular polygon (with an odd number of vertices) as cross section is solved; a polygonal trammel and families of rotors are used.

Another related mechanism that is mentioned in the Notes of Chapter 17 is the intermittent rotor [430]. That makes contact with a series of fixed elements but not always with all the elements. In the example shown in Figure 17.14, the rotor is restrained in its motion by contact with three of the four fixed points until all four of the fixed points are touched. This motion may then be continued with another set of three fixed points as constraints.

Natural Sciences

Reuleaux triangles occur also in the form of nanocrystals. The hydrolysis and precipitation of bismuth nitrate in an ethanol–water system yields disks having the shape of a Reuleaux triangle, see [246]. The (geometric) analysis of the homogeneous growth process provided insights into the mystery behind the formation of this constant width shape. This discovery (of Reuleaux triangles, possibly for the first time in a natural system) is fascinating!

Our next example is related to astrophysical observatories. Cross-correlation imaging interferometers designed in the shape of a curve of constant width offer better sensitivity and imaging characteristics than other designs; they sample the Fourier space of the image better than other shapes, see [619]. Namely, each pair of antennas measures one Fourier component with a spatial wavenumber proportional to the separation of the pair. Placing the individual antennas along a curve of constant width guarantees that the spatial resolution of the instrument will not depend on direction since the measured Fourier components will have the same maximum spatial wavenumber in all directions. And the optimum is attained with the shape of Reuleaux triangles; the constant width curve with highest symmetry (the circle) is the least satisfactory.

Sets of Constant Width in Potential Theory

Also in potential theory constant width sets occur, namely in connection with the farthest point distance function, and also this may be considered as an interesting application. Let $E \subset {\mathbb E}^2$ be a compact set containing more than one point, and let $d_E(x)$ be the distance from a point x to the farthest point of E. Then the continuous, subharmonic function $\log d_E$ is known to be the logarithmic potential of a unique probability measure with unbounded support. Laugesen and Pritsker [705] described some properties of this probability measure, reflecting from this also the topology and the geometry of E. They conjectured $\frac{1}{2}$ as general upper bound for this measure (and proved this, e.g., for polygons which can be inscribed to the circle), and they showed that this measure equals $\frac{1}{2}$ when E is a $C^2$-smooth set of constant width. They discussed also the case of regular triangles continuously expanding in a way to get the shape of Reuleaux triangles. In [396], the results of [705] are sharpened: In this paper the general upper bound $\frac{1}{2}$ for the described probability measure is confirmed, and it is also proved that equality holds iff the convex hull of E is of constant width and has the same measure as E. In [397], some more background is given. In [611], the analogous inequality for the n-dimensional situation (also conjectured by Laugesen and Pritsker) is proven for sets of constant width and for centrally symmetric sets provided $n > 2$. In addition, a proof of this inequality for the planar case is given, which yields an approach alternative to that from [396]. A nice explanation of these (and other) results on constant width sets is also given in [609].

Figure 18.32

Exercises

• 18.1. Prove the assertions (18.1) to (18.7).

• 18.2. Prove that if a body $\phi$ has constant width h, then $\phi _r$ has constant width $h+2r$.

• 18.3. Prove that the system of diametral lines of $\phi$ is exactly the same as the system of diametral lines of $\phi _r$. That is, an alternative way of producing the parallel body $\phi _r$ is to extend all diameters of $\phi$ outwardly by a distance of r from both ends.

• 18.4. If we use $t=45$ degrees in the construction of Figure 18.10, this shape is housed in a square hole of side length w. Prove that the locus of the point highlighted in Figure 18.19 must follow a square path of length $w(\sqrt{2}-1)$.

• 18.5*. Let us suspend another bucket at the axis of the water wheel (see Figure 18.32). Prove that there are two parallel curves of this triangle that touch all buckets at all times.