Because we may now give the dimensions of any sphere as 5F n , we have no need for pi in developing spheres holistically. We know that pi is the length of a circle as expressed in the diameters of the circle, a relationship that holds always to the transcendentally irrational number 3.14159. 982.81 We recall also that both Newton and Leibnitz in evolving the calculus thought in terms of a circle as consisting of an infinite number of short chords.
Hexagons are, of course, cross sections through the vector equilibrium.
The hexagon's six radial vectors exactly equal the six chordal sections of its perimeter. 982.83 Assuming the vector equilibrium hexagon to be the relaxed, cosmic, neutral, zero energy-events state, we will have the flexible but not stretchable hexagonal perimeter spun rapidly so that all of its chords are centrifugally expelled into arcs and the whole perimeter becomes a circle with its radius necessarily contracted to allow for the bending of the chords. In the case of the square, the radius is taken from the center to the comer, not the edge. We take particular note that the radius of the square phase of the closest-packed circle is 1.060660, the synergetics constant. Therefore, it would take more equiperimeter triangular tiles or squares to pave a given large floor area than it would using equiperimetered hexagons.