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Projective Geometry

Lawrence Edwards

352 pp.  
6 1/2" x 9"
Illustrations: 200+ diagrams

Floris Books


Published:  January 2007


Projective geometry is a non-metrical form of geometry that emerged in the early nineteenth century. It originated from the principles of perspective art. Here, he presents a clear and artistic understanding of the intriguing qualities of projective geometry.

Projective geometry formalizes a central principle of perspective art—that parallel lines meet at infinity and, therefore, should be drawn that way. Essentially, projective geometry may be viewed as an extension of Euclidean geometry—one in which the “direction” of each line is subsumed within the line as an extra “point,” and in which a “horizon” of directions corresponding to coplanar lines is regarded as a “line.” Thus, two parallel lines will meet on a horizon because they possess the same direction.

In the spirit of projective geometry’s origins in synthetic geometry, some mathematicians have investigated projective geometry as a useful means to describe natural phenomena. The first research in this direction was stimulated by a suggestion from the philosopher and spiritual teacher Rudolf Steiner.

In the mid-twentieth century, Louis Locher-Ernst explored the tension between central forces and peripheral influences. Lawrence Edwards discovered significant applications of projective geometry (Klein path curves) to organic development. In the spirit of D’Arcy Thompson’s On Growth and Form, but with greater mathematical rigor, Edwards demonstrated that forms such as the buds of leaves and flowers, pine cones, eggs, and the human heart can be described simply through the use of certain path curves. Varying a single parameter—lambda—transforms the interaction of what projective geometry calls “growth measures” into surprisingly accurate representations of many organic forms, which are otherwise not easily described through mathematics. Moreover, negative values of the same parameter produce inversions representing vortexes of both water and air.

Lawrence Edwards researched and taught projective geometry for more than forty years. His book will reveal the secrets of space to those who work through its more than two hundred instructive diagrams and exercises. Projective Geometry is an essential resource for teachers of Waldorf education or for those who wish to strengthen their thinking and expand their view of mathematics.

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