Colloquium

ABSTRACT: A set of n points evenly distributed on a spherical surface is called an Even Distribution (E). There are five E’s corresponding to n = 4, 6, 8, 12, and 20. My objective is to systematically establish extensions from the five basic E’s and to analyze them. The extensions I established are eleven First Generations of Even Distributions (E’), three Twisted E’s (TE’) and six String E’s (SE), plus an infinite number of Second Generations of E’s (E”). The calculations and analysis use 3-dimensional geometry. Practically, to study the structure and to perform calculations of E’s and E-extensions is to assume that every point of a figure is a small ball of radius 1 unit packed in a spherical container. This induces the study of even distributions and their extensions into spherical ball packing. The essential rule of ball packing is to pack all balls under a criterion that every ball should be in a state of having absolute 0 degree of freedom in motion. This regulates the arrangement and the structure of a pack. A pack with every ball under such a state is called a well-bound (WB) pack. There are five WB criteria that an E or E-extension should satisfy. All figures of E’s and E-extensions have all their points floating on a spherical surface. This kind of packing is called a “Floating E-Packing”. My goal is to establish and study the structure of E’s and their extensions, and is not to find a smallest container for a certain number of balls. Any WB pack system is worth analyzing. The spherical ball packing covers more than just the floating E-packing. There is a parallel packing to the floating E-packing, called solid E-packing. There is also an independent class of spherical ball packing called intruding packing, which is an asymmetric packing. These will be my succeeding studies.