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On Symmetric Structures within Symmetric Solid Objects

by Frank C. Fung ( 1st published in March, 2010. )

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Summary and Key Findings :

go to Table Of Content below

Key Finding One :

The 20-pieces Icosahedron is made-up-of 5 identical { Quad-Type Structures } ,
- with each { Quad-Type Structure } being in-itself-and-by-itself [ quad symmetric ] .
  
  ( see Section XXV / Part VIII for details ) .
And there are two (2) different-and-distinct ways to split the 20 pieces of the Icosahedron into 5 identical { Quad-Type Structures } .

Key Finding Two :

Duality for [ Dodechedron vs. Icoshedron ] may be linked via the [ OHATS ( 120-Split ) Sphere ] :

Key Finding Three :

This [ 5-colors configuration ] of the Icosahedron is [ quad-symmetric ] :

Key Finding Four :

This [ 4-colors configuration ] of the Icosahedron is [ 2-6-6-6-symmetric ] :

Table of Content

Table Of Content
	Part I --- Introduction and Preliminaries
Section I	Introduction
Section II	A Special Note on the Notations
	Part II --- Symmetric Solid Object Basics
Section III	Basic Properties of the Tetrathedron
Section IV	Basic Properties of the Cube
Section V	Basic Properties of the Octohedron
Section VI	Basic Properties of the Dodecahedron
Section VII	Basic Properties of the Icosahedron
	Part III --- The Tetrahedron, tbe Cube , and the Octahedron
Section VIII	The Bi-Symmetric Nature of the Tetrahedron
Section IX	The Bi-Symmetric Nature of the Cube
Section X	The Tri-Symmetric Nature of the Cube
Section XI	The Bi-Symmetric Nature of the Octahedron
Section XII	The Quad-Symmetric Nature of the Octahedron
Section XIII	Brief Summary on the Tetrahedron / Cube / Octahedron
	Part IV --- The Dodecahedron
Section XIV	Bi-Polar Structures within the Dodecahedron
Section XV	The Bi-Symmetric Nature of the Dodecahedron
Section XVI	The Tri-Symmetric Nature of the Dodecahedron
Section XVII	The Quad-Symmetric Nature of the Dodecahedron
Section XVIII	The Hexa-Symmetric Nature of the Dodecahedron
	Part V --- The Icosahedron
Section XIX	Bi-Polar Structures within the Icosahedron
Section XX	The Bi-Symmetric Nature of the Icosahedron
Section XXI	The Quad-Symmetric Nature of the Icosahedron
Section XXII	The Penta-Symmetric Nature of the Icosahedron
Section XXIII	The Deci-Symmetric Nature of the Icosahedron
Section XXIV	The [ 2-6-6-6-Symmetry ] co-existing within the Icosahedron
Section XXV	[ Quad-Type Structures ] within the Icosahedron
	Part VI --- Bringing-in the Spherical Solid Objects
Section XXVI	Defining Great-Circles , Great-Cricle-Planes and Great-Circle-Arcs
Section XXVII	The Tetrahedral Sphere ( 4-Split )
Section XXVIII	The Quasi-Cubic Sphere ( 6-Split )
Section XXIX	The Octahedral Sphere ( 8-Split )
Section XXX	The Dodecahedral Sphere ( 12-Split )
Section XXXI	The Icosahedral Sphere ( 20-Split )
Section XXXII	Ratio-Of-Areas and the Egyptian Fractions
	Part VII --- Duality Considerations
Section XXXIII	Duality - Tetrahedron vs. Tetrahedron
Section XXXIV	Duality Linkage via the [ TFS ( 24-Split) Sphere ]
Section XXXV	Duality - Cube vs. Octahedron
Section XXXVI	Duality Linkage via the [ FES ( 48-Split ) Sphere ]
Section XXXVII	Duality - Dodecahedron vs. Icosahedron
Section XXXVIII	Duality Linkage via the [ OHATS ( 120-Split ) Sphere ]
	Part VIII --- Structure of the [ Quad-Type Structures ] within the Icosahedron
Section XXXIX	Identifying 2 [ Quad-Type Structures ] for Our Analysis
Section XXXX	A First Look at the 2 [ Quad-Type Structures ]
Section XXXXI	Rotational Orientation of the [ Curved Triangular Pieces ]
	Part IX --- 5-Colors Combinatorics for the [ Quad-Type Structure Coloring Schemes ]
Section XXXXII	A Special Feature of the [ Quad-Type Coloring Schemes ]
Section XXXXIII	5-Colors Combinatorics for the [ ALPHA Series ]
Section XXXXIV	5-Colors Combinatorics for the [ OMEGA Series ]
	Part X --- Concluding Remarks
Section XXXXV	Concluding Remarks

On Symmetric Structures within Symmetric Solid Objects

by Frank C. Fung ( 1st published in March, 2010. )

Top Of Page

Summary and Key Findings :

go to Table Of Content below

Key Finding One :

Key Finding Two :

Key Finding Three :

Key Finding Four :

Table of Content

go to Top Of Page

go to the 1st section : Section I - Introduction

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Original dated 2010-3-31 / Updated 2010-5-08