Exercises with Polarity

Henrike Holdrege

From In Context #46 (Fall, 2021)

We’re pleased to share an excerpt from Henrike Holdrege's new publication To the Infinite and Back Again, Part II — A Workbook in Projective Geometry, the companion volume to Part I published in 2019. To order ($22), go to our online bookstore, or call us (518-672-0116), or email info@natureinstitute.org.

Building on the first volume, in Part II of this practice-based introduction to projective geometry, Henrike introduces and works extensively and intensively with the fundamental idea of polarity. Through a wealth of exercises, illustrated with Henrike’s drawings, the reader learns to see how every form has, implicitly, a polar opposite form that is related to it. Here we give you an intimation of the expansive tapestry of thought that those who work through the book can enter and begin to weave for themselves:

The blue disk is the “inner” of the growing point-circle. Shading the disk allows me to convey that all points within the circle have taken part in the growth process so far. The “inner” of the point-circle is “filled” with points. The “inside” of the corresponding tangent-circle is filled with lines and cannot be shown as easily. The figure shows only a few of the tangents of the tangent-circle itself. All lines that surround the circle make up its “inside.” All surrounding lines have taken part in the growth process so far.

This imagination exercise allows us to expand our concepts of “inside” and “outside.” We can develop the concept of an “inside” that is centered in the periphery. The peripheral perspective complements the point-centered one. It challenges us in our thinking. The reality of an “inside” centered in the periphery is difficult to express in words.

The characteristics of a curve determine in every detail the characteristics of the polar opposite curve. [The light blue curve inside the inner circle is polar opposite to the yellow curve.] In all of the exercises in this chapter, we realize how important the concepts of point at infinity and line at infinity are. Without them, a geometry of polar opposite curves would not exist. The concept “at infinity” is not a question of distance. It is not a question of something being very far away, something growing infinitely large, something being beyond measure. It is not a question of measurable quantity at all. Rather, it is a question of completeness or wholeness.

Through projective geometry, the wholeness of a parabola or of a hyperbola, for instance, become tangible. Even though we reach the limits of our ability of mental picturing, we can grasp these forms with full inner clarity.

A “whole,” as the saying goes, is more than the sum of the parts. But what does “more” actually mean? Wholeness is not available to us in the way the parts are. Wholeness is in and through the parts, but is not “another part.” Parts we can measure; wholeness we cannot measure. It is of a different nature, and we need to develop a new way of knowing if we wish to catch a glimpse of wholeness.