Picture a simple steel bar. Now picture a second bar, and attach it end-to-end to the first one with a simple hinge.

What you've got is the simplest possible polygonal linkage. Its properties are easy to deduce: its single joint can rotate through 360° -- we're assuming here that the bars have infinitesimal width and they only "block" one another when they're actually parallel.

But what if you kept adding more and more bars and hinges? And what if some of the hinges were `rusted' solid so that they couldn't move? What about that object's properties? In particular, can it be straightened -- that is to say, is there a way to extend it all the way out so that all its flexible joints are all at 180°? Answering this question will be the focus of this project.

The answer, of course, is that its properties are rather complicated. In fact there are no theories which would allow you to calculate the possible motions of such an object in the general case. We can, however, derive some relatively concrete results in certain special cases.

Why, you may ask, would we possibly want to? Polygonal linkages are mostly important in mechanical engineering, so describing their properties and the kinds of motion they are capable of can be useful in working with many types of devices. Four-bar linkages, described below, are used in many mechanical systems, ranging from children's toys to automotive engines and drivetrains.

# Contents

Here's what we've put together for your geometrical enjoyment...

Cauchy's Lemma proves that any convex linkage can be straightened.

Four-bar linkages are an interesting special case which have found wide-ranging applications in mechanical engineering..

Most other cases are open problems!

Try some for yourself with our Java Applet, which demonstrates how polygonal linkages work, or don't. (Java 1.0; anyone can play!)

Links to related sites on the web.