yz=((y+z)^2-y^2-z^2)/2 (check this yourself!)

Since we can perform every operation on the right with a linkage, we can get multiplication by two complex numbers with a linkage.

This is very interesting, because now that we can both add and multiply many complex numbers, we have everything we need to construct any complex polynomial.

Now you may want to know why that is interesting. Especially because we have computers and they are far better at dealing with polynomials than a system made of steel rods and ball bearings. Mathematicians like polynomials because they have many fascinating properties, and whenever one can make a statement in terms of polynomials there are usually a lot of other results that follow. This is no exception. The following two theorems require very advanced mathematics, but they follow from standard theorems, once it has been shown that linkages can be used to compute polynomials. Since we have shown this, we get the following:

There is a linkage which traces out your signature (whatever that signature looks like!!)


and

For any from a family of nicely behaved spaces, such as a line, a circle, a plane, a sphere, etc (these are all called manifolds), of arbitrary dimension (e.g., higher dimensional versions of the previous examples) there is some linkage such that its configuration space is a number of copies of the given space (the manifold).


You don't need to fully understand these results to appreciate them. They are answers to very general questions one can ask about the nature of linkages, their configuration spaces, and complex polynomials.

Please take a look at the 2 problems given in the hard problems section. You can also return to the table of contents,
or back to the MEC home page