### PREFACE IX-XV CHAPTER 1 Introduction 1-3 CHAPTER 2 Some Geometry 4-11

Some Differences Between Plane and Spherical Geometry 4
General Definitions 5
Angles on a Sphere 7
Distances Between Points on the Surface of a Sphere 9
Definition of Polygons on a Sphere 9
Spherical Triangles 10

### CHAPTER 3 The Polar Stereographic Map 12-27

Construction of the Polar Stereographic Map 13
What is a Straight Line on the Polar Stereographic Map? 15
The Nature of Pattern Distortions Created by the Polar Stereographic Map 17
The Irregular Appearance of the Projection of the Centers of Circles 21
All Circles on the Earth's Surface Project as Circles on the Map 22
The Value of the Angle Between any two Arcs on the Surface of a Sphere is
Exactly Reproduced on the Polar Stereographic Map 26

#### CHAPTER 4 Organization in Space 28- 49

Space and Some of its Features 28
Space Filling Patterns on a Two-Dimensional Surface 34
Principles of Closest Packing of Spheres 41
Examples and Experiments in Closest Packing of Spheres 44
Angles of the Hexagon 48

#### CHAPTER 5 Organized Movement Patterns in Space 50-57

General 50
The Principle of Similitude 50
Kepler's Laws 51
Wolf's Law 52
Rings and Spokes of Saturn 53
Roche's Law 53
Laws of the Atom 54

#### CHAPTER 6 Waves 58-65

General 58
Equilibrium 59
Oscillations and Resonance 59
Harmonic Movement 60
Fourier Analysis 61
Wave Pulses 63
Interference 63

#### CHAPTER 7 Symmetry 66-81

Definition of Symmetry 66
Classes of Symmetry 67
Types of Symmetry Operations 68
Symmetry Along a Straight Line 71
Dilation Symmetry 71
Symmetry in a Network of Points 71
Equivalent Symmetry Points 76
Symmetry of Averages 76
How Many Points Will be Crossed by anyStraight Line in a Regular Network? 77
Symmetry and Equilibrium 79

#### CHAPTER 8 The Problem of Plateau 82-90

The Conic Sections 82
Surfaces of Revolution 84
Collapseof Cylindrical Structures 86
The Splash of a Drop 89
The Center of a Structure is Related to its Shape 89

#### CHAPTER 9 The Chladni Figures and Wave numbers 91-95

Two Modes of Vibration 92
Some Differences Between the Hemispheric Cap of Air and Chladni Plates 94

VII

#### CHAPTER 10 Quantum Units and Angular Numbers 96-106

Everything is Packaged in Quantum Units 96
Some Factors Determining the Quantum Sized Vortex on a
Surface Weather Map 97
"Diffraction Grating" for Finding the Quantum Vortex and also
for Exposing Simple Geometric Configurations 99
Construction of theWeatherTool 102
Other Possible Weather Tools 103

#### CHAPTER 11 General Explanation of the charts 107-114

Why December 7,1950 was Chosen for Analysis 107
Why June 6,1944 was Chosen 108
Some Rules Followed in the Drawing of the Charts 108
The Identification Chart 109
Some Unusual Aspects of the Charts 111
Some Simple Definitions of Simple Weather Terms 111
What is the Center of a Low or High and How is it to be Determined? 112
Two Types of Chart Analysis: Circumferential and Radial 114

#### CHAPTER 12 The Circumferential Charts 115-168

Some Rules Followed in Drawing the Circumferential Charts 115