Mathematical Tutorial of the Peaucellier-Lipkin Linkage, S35

by David W. Henderson and Daina Taimina

 

How can we draw a straight line? One way would be to use a “straight edge”  something that we accept as straight. Notice that this is different from the way that we would draw a circle. When using a compass to draw a circle, we are not starting with a figure that we accept as circular; instead, we are using a fundamental property of circles that the points on a circle are a fixed distance from the center. Is there a tool (serving the role of a compass) that will draw a straight line? For history of straight line mechanisms, see tutorial How to Draw a Straight Line.

For an interesting discussion of this question see [Kempe, A.B. How to Draw a Straight Line. London: Macmillan, 1877]  online, which shows the linkage that is the basis of the model S35.

 

 

The fact that this mechanism draws a straight line is the subject of the problems below. See the accessible and informative book Geometry and the Imagination by Hilbert and Cohn-Vossen (pp. 27273) [Hilbert, David and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea Publishing Co., 1983] for another discussion of this topic. The discovery of this linkage about 1870 is variously attributed to the French army officer, Charles-Nicolas Peaucellier (1832-1913), and to Lippman Lipkin, who lived in Lithuania and studied in Saint Petersburg. (See also Phillip Davis’ delightful little book The Thread, [Davis, Phillip. The Thread: A Mathematical Yarn. Boston: Birkhäuser, 1983] Chapter IV, for some stories of the discovery of this linkage.)

 

We study the mathematics behind the Peaucellier-Lipkin linkage in three problems:

 

Problem 1. Circles in the Plane, which explores some geometry of circles.

Problem 2. Inversions in Circles, which explores properties of inversion in circles.

Problem 3. Applications of Inversions to the Peaucellier-Lipkin Linkage, which explores the applications of Problems 1 and 2 to an understanding of the linkage.