Spirals

Other curves have an equation that has no degree (or rather, they cannot be expressed through a polynomial). Some of these are distinguished by some special properties, that make them particularly useful and interesting. One of these is Archimedes' spiral.

An ant starts from the beginning of a turntable's platter, and walks outward in a straight line. But if the platter starts turning at the moment when the ant starts walking, and if both the ant and the turntable maintain a constant speed, the ant will trace a spiral curve, called Archimedes' because it was firstly studied by the mathematician from Syracuse.

We can substitute the ant with a marker, which we move from the centre to the edge with the most constant speed possible. We will then see Archimedes' spiral form, with more spirals the slower we move the marker.

An interesting application of the spiral is found in sewing machines, in the part that spins the thread around the spool. The thread coming from the skein is kept taut, and it is rolled around the spool, which turns and moves up and down, in order to allow an uniform distribution of the thread. And this is where the spiral comes in. To make sure that the thread is spun uniformly on all the parts of the spool, the oscillation movement needs to keep a constant speed. If the oscillation movement were faster in the centre and slower towards the ends, when the spool must change directions (which is what would happen unless one takes specific measures), the thread would not be distributed uniformly, but would tend to pile up at the spool's extremities.

Therefore, one needs a mechanism that makes the spool oscillate with a constant speed. This is obtained by regulating the oscillation of the spool through a profile made of two coupled spiral arcs. Here we see this specific mechanism of the sewing machine, and a working enlarged reproduction of it. The vertical rod moves alternatively up and down, always with the same speed.