### 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.

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.

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:Parting 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 mondays@momath.org with the word “linkage” in the subject? Thanks!