Math Monday: Linkages – One Last Spin

by Glen Whitney

This is the final installment in our epic Math Monday series on the intricate world of mechanical linkages. See the Linkages series introduction for the MoMath Linkage Kit, an introduction, and general instructions.

Over the course of the series, we’ve seen practical linkages, like the scissor jack, pantograph, or Watt’s linkage; and we’ve seen the incredible versatility of even a four-bar linkage, which can be made to go through any four points desired. We’ve seen the limitations of the four-bar linkage, too — it can never trace out a perfectly straight line. But with more bars, and the mathematics of inversion in a circle, that limitation can be overcome, and numerous familiar, simple curves can be produced by multi-bar linkages. There’s much more you can explore concerning linkages as well: linkages used for heavy machinery, linkages found in the suspension of virtually every car on the road, paradoxical linkages that seem to rotate at different speeds clockwise or counterclockwise, three-dimensional linkages, and more. There’s no way to make a series like this one exhaustive.

So let’s wrap up the series with one last look at the diversity of the four-bar linkage. Here’s one with no practical purpose whatsoever — it’s just to produce a fanciful curve, reminiscent of a Spirograph.

Sangwin Linkage
Ingredients: One 40-bar (A); two 50-bars (B and D); one 60-bar with a 30-hole, and a pen.

Directions: Fix A horizontally. Link A to B to C-0. Link C-60 to D to A. Put a pen in C-30.

To use: Rotate B a full rotation, keeping the pen in the hole drawing on the paper. Note that the full curve has two loops. So you may need to go around again, urging the linkage to bend the other way.

Here’s a picture of the completed linkage:And finally, here’s the Sangwin linkage in action, drawing a rather lovely hypocycloid-like curve:dibieifhParting question: Is any segment of this curve a circular arc? Why or why not?

So that’s all for linkages in Math Mondays, at least for quite some time. I hope you’ve enjoyed this tour of linkages as much as I’ve enjoyed creating it!

Final P.S.: Would the person who stopped in at MoMath on 11 E 26th St. asking about linkages please send an email to with the word “linkage” in the subject? Thanks!

This article first appeared on Make: Online, February 4, 2013.

Return to Math Monday Archive.