PhronesisOne, to be
A wheel is a single circle. It has three freedoms — center across, center up, radius — which is exactly why three points pin one down. Three points make a circle; that is the whole of a wheel at rest.
Two, to roll
Set it rolling on a road and you need a second circle: the ground, which is just a circle of infinite radius. A point on the rim then traces a cycloid — two circles making a third curve. Rolling is always one circle on another; on a curved track the trace is an epicycloid or a hypocycloid, and a straight road is the limit case.
Five, to roll free
Let it roll free on a table — a coin that can steer and lean, not a wheel locked in a frame — and the count jumps. To say exactly where the coin is takes five numbers: where it touches the table (two), which way it is heading (one), how far it leans from upright (one), how far it has spun about its axle (one). The rolling disk is a five-dimensional system. And look which are which: heading, lean, and spin are circles — angles. The other two are just place. So the turning part of a free wheel is three circles; the whole of it takes five coordinates — though rolling without slipping ties their rates, so only three move independently.
The other five
Five points pin something too — but not a circle. Five points determine a conic: the general oval, of which a circle is the special, more symmetric case (needing only three). So the wheel is three, and the orbit is five. One honest caveat: the rolling disk's five (degrees of freedom) and the conic's five (points) are different fives — both real, and unrelated. It is tempting to braid them into a sign. Don't.
One to be, two to roll, five to roll free.
A footnote to The Logic of the Circle — dimensions, counted as turns.