How to discover the secret:
- procure: compass for drawing curves, ruler, pencil or pen, and blank sheet of paper;
- with compass, draw circle, say, of radius 1 inch;;
- level with top and bottom of page, draw line segment on paper, just touching the "bottom point" of the circle;
- label this: "CROSS-SECTION OF WHEEL RESTING ON A LEVEL SURFACE".
You then have a drawing similar to that on upper left. The disk (circle filled in) on right resembles the surface of a wheel.
Mathematicians have long had a fanciful label for a line that just touches one point of a curve. They say "the line is kissing the curve". Ain't that sweet! Even more, we'll see that that KISS IS THE SECRET OF THE WHEEL -- why it's the most useful of MECHANICAL devices, appearing in many guises in our Technology.
Whyso? Well, a WHEEL IS A DISC: A STACK OF CIRCLES -- ONE WAY OF 2-D CIRCLE BECOMING 3-D. Imagine that 2-D SEGMENT BECOME A 3-D FLAT SURFACE (say, a floor). THEN EACH OF THOSE CIRCLES IN THE STACK TOUCHES THAT SURRFACE IN JUST ONE POINT!
That's the STATIC DESCRIPTION. But IMAGINE THAT WHEEL REVOLVING, ITS STACK OF CIRCLES ROLLING ON THAT SURFACE. AT EACH INSTANCE OF ROTATION, EACH OF THOSE CIRCLES IS KISSING JUST ONE SEGMENT OF THE SURFACE.
Mathematicians also have a term, "conservation", for describing what I said in that last sentence. For example, ON A FRICTIONLESS PLANE, ENERGY IS CONSERVED BY SUCH MOTION. Some ENERGY shifts from one part of the scene to another, but no ENERGY is lost.
Similarly, we can say: KISSING IS CONSERVED AS A WHEEL ROLLS ON A FLAT SURFACE. ("The Conservation Law of Kissing"!)
Another way of saying this is: THE KISSING RELATION BETWEEN WHEEL AND SURFACE IS INVARIANT UNDER THE TRANSFORMATION OF ROTATION.
OK. But so what? What's the big deal? Well, replace the wheel by a box to be dragged across the floor. You become aware of RESISTANCE almost unnoticed with the wheel. The heavier the box, the more the resistance. A physicist could make you aware of the friction invoked, robbing the system of ENERGY -- TRANSFORMING ENERGY into an UNRECOVERABLE form.
That's the big deal. THE KISSING ENABLES THE WHEEL TO MINIMIZE THE RESISTANCE INVOKED IN ROLLING ON A SURFACE. NO OTHER STRUCTURE CAN SO MINIMIZE!