In determining the area of sphere surface portions left uncovered by cylinders fitted to them diametrically for purposes of rounding a hard-angled transporter shell guided by the transport template, one planar convergence type is of special interest.
The convergence type in question is essentially the same as the bode (in its planar manifestation), where 2 opposing squares converge on each of the form’s 12 vertices, and where 2 opposing equilateral triangles occupy the spaces between the squares.
In the transporter shell convergence, the remaining sphere area (determined by spherical trigonometry) turns out to be PI r-squared / 3, or one twelveth of the sphere srvace’s complete area: 4 PI r-squared.
Similarly, if cylinders are fitted to spheres (of half edge length radii) situated on either end of lines joining the bode’s 12 vertices where they are centered, the area remaining for each vertex-centered sphere, multiplied by 12 spheres, equals the area of one full sphere!
That this should be so may be obvious to the average mathematician, but to a simple-minded design scientist such as myself who is easily entertained, I don’t see how this should necessarily be the case.
Does the equality hold for other forms? Yes, at least for the easily pictured cube with spheres centered on each of its 8 vertices, spheres that are then halved and quartered and halved again by cylinders converging identically from 3 directions. Other than the cube I’m guessing that the equality holds for all convex polyhedra with identical vertices, but beyond that, I’m not so sure. Whatever the attribute’s applicability, the bode holds a unique claim pertaining to that attribute: It is the only form innately possessing one such full sphere – in its very center.
It would seem reasonable to speculate that this relationship in some way manifests in physical law and phenomena – much as parabolas delineate focal points and projectile paths; or as ellipses define the orbits of celestial bodies.