Projective Geometry: Creative Polarities in Space and Time

Projective Geometry: Creative Polarities in Space and Time

by Olive Whicher

Paperback(Reprint)

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Product Details

ISBN-13: 9781855843790
Publisher: Rudolf Steiner Press
Publication date: 07/28/2013
Edition description: Reprint
Pages: 292
Product dimensions: 7.44(w) x 9.76(h) x 0.76(d)

Table of Contents

Foreword 9

I Concerning Changes in Classroom Mathematics 13

II Geometry and the Development of Thinking 27

From Moving Form to the Fixity of Measure 27

Experience with Measured Shapes 29

The Question of the Infinite 38

The Birth of Modern Geometries; Two Pathways 46

"Projective Geometry is All Geometry" 49

III First Step in Projective Geometry; Movement 52

The Line-woven Net in Step-measure 54

The Infinitely Distant Point of a Line 58

The Infinitely Distant Line of a Plane 62

The Infinitely Distant Plane of Space 64

Real or Unreal? 70

Point, Line and Plane 73

Point, Line and Plane in Mutual Interplay 75

The Axioms of Community of Point, Line and Plane 77

Two Basic Theorems; Desargues and Pascal 78

Desargues, Two-triangle Theorem 79

Pascal's Theorem 84

IV Further Discoveries; Duality and Projection 89

The Principle of Duality 94

Perspective and Projective Relationships 95

The Projection of Three into Three 99

The Fundamental Theorem 102

The Theorem of Pappos 103

Projective Creation of Curves; the Rainbow 108

The Harmonic Forms 113

Harmonic Fours and the Anharmonic Ratio 115

The Invariance of the Harmonic Property 119

The Thirteen Configuration and the Diagonal Triangle 120

V Projective Laws of Curves 124

The Line-woven Net in Growth Measure 124

Projective Concentric Circles 127

The Theorem of Brianchon 127

Curves Through Five Points and Touching Five Lines; Pascal and Brianchon 130

The Theorem of Jacob Steiner 133

The Tangent and its Point of Contact 134

Identity of Pointwise and Linewise Circle-curve 136

VI Projective Transformations; Collineations 140

Projectivity and Involution within the Line 140

Projection with One Double Point 142

Projection with Two Double Points 142

Involution 144

Projection and Involution on a Circle-curve 146

Projection without Fixed Double Points; Potentizing Process 150

Breathing Involution 151

Cyclic Projectivity 153

Circling Involution 153

Indications concerning the Imaginary Double Elements 155

Plane Path-curves in Breathing and in Circling Involution; One-dimensional Transformations 158

Two-dimensional Projective Transformation of Curves; Homology and Elation 167

Curve Families and the Harmonic Net 177

Spiral Matrix 177

Projective Transformations in Space; Plastic Perspective 180

VII Polar Transformations on Circle-curves; Correlations 185

The Fundamental Polarity of Space 186

Pole and Polar with respect to Circle-curves 188

Self-polar Triangles; Polar Conjugate Pairs 193

Conjugate Diameters 196

Polar Reciprocation; Correlations 200

VIII Polar Forms in Space 218

Pole and Polar with respect to the Sphere 221

The Line-line Polarity of Space; Line Congruence 233

IX Geometry of tie Twentieth Century 238

Rudolf Steiner's Indications concerning Space and Counterspace 239

The Mathematical Concept of Space and Counterspace 243

Physical and Ethereal Spaces 247

Three Positive and Three Negative "Dimensions" 255

Gravitational and Anti-gravitational Forces 258

Sun and Earth 262

Notes and Bibliography 275

Index 287

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