2. Some history
The idea of a chain of linked rods that can move with respect to each other, like the links on the robot arm of the International Space Station, suggests issues of rigidity and flexibility. Thus, it is not surprising that the mathematical investigation of linkages has from time to time overlapped with the mathematics of rigidity. However, people interested in rigidity are interested in rigidity of structures other than path linkages, for example, polyhedra. We will also be interested in linkages where the rods lie in a plane but the collection of rods can be more general than a chain. We will also allow circuits of rods.
One of the earliest people to raise the question of flexibility of structures was the awesome Leonhard Euler (1707-1783). Euler, whose collected works and letters have yet to be completely published (!), raised the question of the rigidity of closed polyhedra surfaces in 3-dimensional space. A conjecture which he made was not shown to be wrong until Robert Connelly (Cornell University) published a counter-example in 1978. (A model that Connelly used to help demonstrate what he did is on display at the Smithsonian's National Museum of American History in Washington, DC.) Another famous mathematician who pioneered work in rigidity was A.L. Cauchy (1789-1857). He proved an important theorem about the rigidity of convex polyhedra though Ernst Steinitz, many years later, had to correct a subtle error in his proof. Cauchy's seminal work, though interfacing with modern work on linkages, did not directly affect an understanding of mathematical issues involving linkages in his day.
A major figure in investigating linkages was A. B. Kempe. Kempe, who was neither an academic nor a mathematician by profession, but, rather, a lawyer, is most famous for producing a proof of the 4-color problem: the vertices of a planar graph can be colored with 4 or fewer colors so that vertices joined by an edge get different colors. In 1877 hepublished a small book: How to Draw a Straight Line; A Lecture on Linkages. The book is beautifully illustrated with illustrations of models prepared by Kempe's brother, H. R. Kempe, and grew out of a collaboration with J. J. Sylvester (1814-1897). Kempe in surveying work done prior to his own, mentions that of James Watt (1784) in conjunction with Watt's work on the steam engine. He also mentions the work of Sylvester, Richard Roberts (Manchester), P. L. Chebyshev (1821-1894), Harry Hart (1848-1920), M. V. Liguine, William Kingdon Clifford (1845-1879), Jules Antoine Lissajous (1822-1880), Samuel Roberts(1827-1913), and Arthur Cayley (1821-1895). (For some additional historical information and details of what was being done during this early period see the paper of Warren Smith in the references.) Chebyshev's work was especially broad in scope, and during a visit to London, he apparently inspired Sylvester, who had not been active mathematically for some time prior to his visit, to take up the subject of linkages.
One of the earliest people to do mathematical work directly on linkages was Charles-Nicolas Peaucellier (1832-1912). Peaucellier, whose background was in engineering, developed a way to convert circular motion into linear motion using a linkage. This work, unlike that of some of the more famous people mentioned above, is still known today. This is probably because of the relatively simple yet lovely connection Peaucellier made between circular and linear motion. Peaucellier published his work in 1867 but mentioned in an earlier letter that he had the idea. In 1873 he acknowledged that the Russian mathematician Lippman Lipkin (1846-1876) had made the same discovery independently and explained the ideas involved in much more detail. The Peaucellier-Lipkin linkage, however, is only one of many linkages that can convert circular to linear motion. Some of these linkages are simpler than the Peaucellier-Lipkin linkage in the sense that fewer bars (links) are used. A modern thread of work on linkage problems is to find the optimal linkage which achieves a particular goal, for example, converting circular motion to linear motion. Sometimes there are competing criteria for optimality.
Kempe formulated in his work a very sweeping and striking theorem, though he did not prove the result (despite some claims to this effect). This theorem stated that if one has a planar curve described by an algebraic equation, then one can design a linkage which which will generate this plane curve. In the 1970's William Thurston (U. California, Davis), in reinvestigating this circle of ideas, formulated this whimsical theorem related to Kempe's pioneering work:
One can design a linkage which will sign your name!
The idea here is that as long as the signature is the union of continuous curves, one can approximate the pieces by algebraic equations (Weirstrass's Theorem) and then use Kempe's ideas to realize the pieces with a linkage. Careful recent examination of the circle of ideas growing from Kempe's work by Denis Jordan, Michael Kapovich, Henry King, John Millson, Warren Smith, and Marcel Steiner has brought clarification, rigor, and numerous areas of extension to what was done in the 19th century.
Others who have kept the tradition of what was done in the 19th century alive have been people with mathematical inclinations in the engineering community. There is a vast literature which slowly built up the details of the complex phenomena that linkages lead to and which have formed the basis of the design of robots and robotic arms, as well as other kinds of kinematic machines. The more recent combinatorial approach to linkages will be treated in more detail in a future column. This work puts together ideas about rigidity, graph theory, and discrete geometry. One can generalize and consider a polygon (where the edges are hinged at the vertices), a tree, or other more general structures consisting of rods and plates (polygons spanned with a membrane).
The concept of a linkage is an area of mathematics that became dormant for a time, but has been recently revived by new applications.As science progresses it must discard information that is no longer valid because new insight shows that what was thought to be true is no longer true. By contrast, once a mathematical result is proven, it is valid forever. Although standards of proof have changed over the years, the fundamental reality is that theorems, unlike theories, are forever! Since it is rare in the process of proving one theorem that no new questions emerge, a vein of mathematics, unlike a vein of gold in a mine, rarely gives out. It is more likely that an area of mathematics will not be mined further for reasons of fashion than that all the questions mathematicians might wish to see resolved have been answered. In assessing the new problems emerging from the proof of a theorem, the mathematics community makes an assessment of the value and importance of trying to tackle the new problems. The decision as to whether or not to go forward is judged by the mathematical interest and depth of these new problems and their potential for application. If one knows that obtaining a new mathematical result enables one to build a better medical imaging system, this is an incentive to continue. If there are no known application consequences of the pursuit of a problem, the reasons to continue on in this area are more subtle. This means that from time to time an area of mathematics which has been pursued vigorously will go to sleep with little or virtually no activity, sometimes for long stretches of time.
Sneak preview: carpenter's ruler problems