User:MeYCChang/INTRODUCTION

INTRODUCTION A set of n points evenly distributed on a surface of a sphere is called an Even Distribution (E). There are five Es while n = 4, 6, 8, 12, and 20, and they are symbolized as 4E, 6E, 8E, 12E and 20E respectively. Every E has a lattice of polyhedron with all identical regular faces. Every point of an E is a vertex of its polyhedron. The polyhedron can be inscribed in a sphere. The figures of five nEs are shown on next page.

My goal of study is to establish extensions systematically from the five basic Es and to analyze them. The extensions I established are eleven First Generations of Even Distributions (E’), three Twisted E’ (TE’) and six String Even Distributions (SE), plus millions (actually infinite number) of Second Generations of Even Distributions (E”). All E’s, TE’s, SEs and E”s are called E-extensions. The calculations and analyzes are based on non-Euclidean geometry.

Each of the eleven E's is extended by cutting off the vertices, or the vertices and edges of a mother E. The cutting corners are the points which form the new E’. The eleven E’ are symbolized as nE’i, where “ n “ is the number of points contained in the system, and “ i ” is an index for some E’s containing same number of points but in different constitutions. The eleven E’s are 12E'1, 12E'2, 24E'1, 24E'2, 24E'3, 30E', 48E', 60E'1, 60E'2, 60E'3, and 120E’. Every E’ has a lattice of polyhedron with two or three sets of different regular faces. Every set of faces is a certain E or another E’ arranged distribution. Their figures are shown on P. 4.

There are three special figures of E's called Twisted E’s (TE’). They are not directly extended from Es. They are extended by twisting the 12E’1, 24E1’ and 60E’2 to form T12E’, T24E’ and T60E’ respectively. Each of them contains three sets of regular faces. And every set of faces is an E or E’ distribution. Their figures are shown on P. 5.

An String Even Distribution (SE) is formed by transforming all edges of a certain nE to be strings of unit squares or pairs of twin unit triangles,. There are five SEs, and are symbolized as SnE, while n = 4, 6, 8, 12, and 20. (Here the “n” is not the number of points contained in the system, it is the “n” from the mother nE.) The vertices of the mother nE are transformed to be k’-edged polygons, called the V-faces. The ” k’ ” is the number of edges intersected at the original vertex of the mother E, and the “ k ” is the number of strings connected on a V-face. Every unit square or paired unit twin triangles is called a pearl. Every string of an SE contains same number of same pearls. The figures in this class, especially while every string contains a large number of pearls, resemble to their mothers’ figurse. (For their figures, see Chapter IV).

There is a distinctive subclass in the class of SnE, that is S2E (n can be 2 for SnE). Each of the two V-faces of an S2E may connect with infinite number of strings (k’ can be       ). [This is very different from other SnEs, and this character makes S2E distinctive.] An S2E with every string contains only one square or one pair of triangles is called an “Equator Belt” (EB) figure, because the single square or single paired-triangle on every string of all strings are distributed along the circumference of the equator like a belt, especially while the number of strings is large. For the      - EB, every string can contain only one square pearl. For the      - EB, the “belt” can be thickened by every string containing multiple pairs of           pearls, which forms a new group called KM group.

All S2Es and KM figures are symmetric with respect to the equatorial plane. All other Es and E-extensions are symmetric with respect to their centers.

There are thousands (actually infinite numbers) of Second Generations of Even Distributions (E"s).  They are extended by adding layers of points in a symmetrical arrangement on one or two sets of faces of an E or E-Extension figure.  All the added points and the original mother points are floating on one surface.  The lattice of an E" does not have all faces regular. An E" formed from a certain E or E-extension is called a family member of that figure. All figures of Es and E-Extensions are having all their points floating on a spherical surface. So this kind of packing is also called the “Floating E Packing”. The practical way to study the structures and do calculations of Es and E-extensions is to assume that every point of a figure is a small ball of radius 1 unit in length packed in a spherical container.  This induces the study of Even Distributions and E-extensions into Spherical Ball Packing.

The essential rule of ball packing is to pack all balls in a container under a criterion that every ball should be in a state of having absolute 0 degree of freedom in motion (in all directions and distances). This regulates the arrangement and the structure of the pack. A pack with every ball under such a state is called a well-bound (WB) pack. All E-extensions should be established under WB criteria. All Es, E’s, TE’s and SEs are well established (are WB). While a new E” is constructed, it is necessary to have every ball in the system to pass five WB criteria. The one passed all is an established good pack.

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. Besides, there is an independent class of spherical ball packing called intruding packing, which is an asymmetric packing. All these will be my succeeding studies.

(** 1) The five Even Distributions: (** 2) The Eleven First Generations of Even Distributions


 * A special tribute to the 60E’1: The structure of 60E’1 is a well-known 60-carbon molecule structure in chemistry called  “Buckminsterfullerene” (also nicknamed “buckyball”).  Its structure and chemical characters amazed me for years.  And it initiated my study of Even Distribution and Spherical Ball Packing.

(** 3) The Three Twisted First Generations (TE’):

T12E’                             T24E’                                           T60E’


 * The sizes of all above drawings (the five Es, eleven E’s, three TE’s, SnEs and KM) are not in proportion.


 * For the radii R of all above figures are calculated from the following Chapters.

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