To the Infinite and Back Again, Part I, is an introduction to projective geometry. Projective geometry arose out of the science of perspective after the Renaissance. It is a modern geometry, mainly developed in the 19th century, and transcends classic Euclidean geometry by working with the concepts of point, line, and plane at infinity. 

The book is richly illustrated. As a fruit of the author’s many years of teaching adults, it is a workbook for self-study by the lay-person and a resource for high school and college math teachers. The book leads in a careful step-by-step fashion to the challenging idea of the infinite. We learn to think the mind-expanding concepts that open up new ways of understanding. They bring coherence and wholeness to geometric transformations that otherwise would not exist. 

The book wants to encourage the reader to actively engage in geometric drawings. In numerous exercises that the book provides we can foster clarity of thought and precision in imagination. We learn to think in transformations, and we can experience the beauty of ideas that grow, weave, and metamorphose.

TABLE OF CONTENTS

Form and Forming

The Harmonic Net and the Harmonic Four Points

The Infinitely Distant Point of a Line

The Theorem of Pappus

A Triangle Transformation

Sections of the Point Field

The Projective Versus the Euclidean Point Field

The Theorem of Desargues

The Line at Infinity

Desargues’ Theorem in Three-dimensional Space

Shadows, Projections, and Linear Perspective

Homologies

The Plane at Infinity

To the Infinite and Back Again, Part II, is an introduction to projective geometry and begins where Part I ended. In Part I, the concepts of point, line, and plane at infinity were introduced and tested within certain contexts. The goal was to show that they are meaningful and not arbitrarily conceived. This volume works with these concepts from the outset.

The principal of duality (or polarity), which was discovered in the 19th century, is central to projective geometry and is the main focus of this volume. Learning to think in polarities can facilitate a significant and beneficial expansion of modern thought and modern consciousness. 

The book provides numerous exercises that foster capacities of precise geometric imagination, thinking in transformations, and thinking in polarities. In a careful step-by-step fashion, the book shows how ideas form, grow, weave, and metamorphose. Richly illustrated, this workbook is intended for self-study by the layperson, and as a resource for teaching projective geometry in high school or college.

TABLE OF CONTENTS

Form and Forming

The Harmonic Net and the Harmonic Four Points

The Infinitely Distant Point of a Line

The Theorem of Pappus

A Triangle Transformation

Sections of the Point Field

The Projective Versus the Euclidean Point Field

The Theorem of Desargues

The Line at Infinity

Desargues’ Theorem in Three-dimensional Space

Shadows, Projections, and Linear Perspective

Homologies

The Plane at Infinity