Dead Space Percentage in Ball Bearings

  Title:   Exact Closed-Form Solution for Dead Space in a Toroidal Packing of n Identical Spheres Author:   Thomas Blankenhorn, Corrupt Grants Pass Oregon Date:   November 2, 2025 Abstract: This work derives an exact closed-form expression for the percentage of dead space when n identical spheres (ball bearings) are packed inside a torus such that each sphere touches its two neighbors. The formula, P(n) = [1 - (2n sin(pi/n))/(3pi)] * 100%, is presented alongside a table of values for n=3 to 6. The result reveals that the dead space percentage decreases with n, approaching a limit of exactly 1/3 (33.333%) as n grows large. This limit is shown to be consistent with the classical packing problem of a single sphere inside a cylindrical sleeve. Derivation: Setup: Let n spheres of radius r be arranged in a circle. Their centers lie on a circle of radius R (the major radius of the torus). Each sphere touches its neighbors. Relating R and r: The straight-line distance betwee...