By Glen Whitney for the Museum of Mathematics

We’re still putting four-bar linkages through their paces. See the introductory column in this series for the MoMath Linkage Kit, an introduction, and general instructions.

Last time, we saw that a four-bar linkage can cause the floating bar to take on any two or three desired positions. What about four positions? As you may recall, each endpoint of the floating bar always lies on a circle defined by the fixed bar and one of the bars linked to it. And we know that there are four points through which it is impossible to draw a circle, such as the three corners of an equilateral triangle and its center. So we must be stuck, right? Not quite, thanks to Burmester Theory. What Burmester realized is that although the endpoints of the floating bar might not lie on a circle, there might be (and in fact are) points in fixed relationship to the floating bar which do lie on a single circle for all positions of the floating bar. Moreover, this theory gives a geometric construction to find the centers of the respective circles, which is all you need to find the linkage. I’ve applied that (somewhat involved) construction to four positions for a bar which lay out the strokes of a letter “M”, as seen in this image and Geogebra worksheet.

This produces our next recipe: