You can use simple trigonometry (just observing that the congruent triangles remain congruent) to conclude that wherever z is put, w will be double it in length, and in the same direction. Therefore, the function which this linkage computes is the complex function w=2z, the doubling function.

Of course, we can't input any number, since our linkage is anchored at 0 and is of a fixed size. However, if someone wanted us to use a linkage to double a number, and the number given was arbitrarily large, then there is a linkage we could use to double the number. In other words, although the domain of the function is always bounded, it can be made arbitrarily large, when the linkage is constructed.

This particular linkage is called the pantograph. A 3 dimensional version was actually used by Italian artisans to scale down Michelangelo's famous sculpture David, (which rises some 17 in the air) to sell to tourists.

Our next task will be to use to pantograph to get a few more functions. Can you think of a way of getting w=-z? How about w=(1/2)z?