The theory of yard sticks

Here they are, the notches on a yardstick:

    F1 = { 0/1, 1/1 }
    F2 = { 0/1, 1/2, 1/1 }
    F3 = { 0/1, 1/3, 1/2, 2/3, 1/1 }
    F4 = { 0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1 }
    F5 = { 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1 }
    F6 = { 0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1 }



 We can make the unit circle with them.

The semi circles within the circle are the bounding value in a Lagrange theory. The size of the semi circle is really the 'frequency' in the corresponding hyperbolic functions.

But, here is the thing. If I have infinite dust, then, just like Fourier, I can make copies of the lowest bandwidth and superimpose them.  If my dust is finite, then, just like Shannon, I have to use my dust sparingly, regardless of bandwidth.  So, finite dust, maximum entropy, infinite dust, don't worry.