The science of yard sticks

The game in yard sticks is to get the density of notches to match the derivative of the density of notches, and then the utilization of integers is optimal.  I can do this with hyperbolics. The chart is from my spreadsheet.




The X axis is N, the integer assigned to the angle. The Y axis is:
-cosh(angle)/137 * log(cosh(angle)/137); which is the -iLog(i), and i, the probability of any given angle is the cosh divided by the total number of sums in my Lebesque summation.  The angles are multiples of ln(Phi) and the 137 being the inverse of the Fine Structure.  As you can see, each angle, so constructed, contributes almost the same entropy as any other, the -iLog(i) all within a few points of each other, until the angle 12 is reached.  That angle corresponds to the dimensionality of a sphere I presume.  Now I am not restriced to the fine structure, I am restricted to making sure the delta angle and the maximum number of sums match; meraning I have to pick of sinh(max angle), that matches the delta angle.  The hyperbolic diffecential handles the rest:

 I can add motion to this formula by introducing the proper polynomial.  However, I await the experts in the field since I am still the amateur. But, I am pretty sure the equation above really is the standard equation for measure theory, adjustable by adding a polynomial.  It works because Tanh is cosh'/cosh, and integral of tanh is the log(cosh). That condition is what makes the differential equation above work.