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Capturing Infinity Continued
Constructing the Scaffold
Escher wrote much about the designs for his regular divisions of the Euclidean plane, but nothing about the principles underlying the Circle Limit Series. He left only cryptic glimpses. From his workshop drawings, one can see that, in effect, he created a “scaffold” of lines in the “nothingness” exterior to the basic disk, from which he could draw the circles that compose the desired figure. However, one cannot determine with certainty how he found his way. Did he reconsider Coxeter’s letter? Did he discover (by trial and error) and formulate (in precise terms) the principles which underlie the design of Figures A, B, C, and D? Lacking the certain, I will offer the plausible.
Straightedge and Compass
Let me describe how I myself would reconstruct the critical Figure A, with straightedge and compass. Such an exercise might shed light on Escher’s procedures. To that end, I will suppress my knowledge of mathematics beyond elementary geometry. However, at a certain point, I will allow myself to be, like Escher, preternaturally clever.
To begin, let me denote by H the disk that serves as the foundation for the figure. Moreover, let me declare that the radius of H is simply one unit. I note that there are six diameters, separated in succession by angles of 30 degrees, that emphasize the rotational symmetry of the figure. I also note that, among the circular arcs that define the figure, there are six for which the radii are largest. By rough measurement, I conjecture that the radii of these arcs equal the radius of H and that the centers of the arcs lie √2 units from the center of H. I display my conjectures in the following diagram:
The bold brown disk is H. Clearly, the six red circles meet the boundary of H at right angles. By comparison with Figure A, I see that I am on the right track.
The diagram calls out for its own elaboration. I note the points of intersection of the six red circles. I draw the line segments joining, in succession, the centers of these circles and I mark the midpoints of the segments. Using these midpoints and the points of intersection just mentioned, I draw six new circles. Then, from the new circles, I do it all again. In the following figure, I display the results of my work: the first set of new circles in green; the second, in blue.
Now the diagram falls mute. I see that the blue circles offer no new points of intersection from which to repeat my mechanical maneuvers. Of course, the red circles and the blue circles offer new points of intersection, but it is not clear what to do with them. Perhaps Escher encountered this obstacle, called upon Coxeter for help, but then retired to his workshop to confront the problem on his own. In any case, I must now find the general principles that underlie the construction, by straightedge and compass, of the circles that meet the boundary of H at right angles. I shall refer to these circles as hypercircles.
The Polar Construction
To that end, I propose the following diagram:
Again, the bold brown disk is H. The perpendicular white lines set the orientation for the construction. I contend that, from the red point or the blue point, I can proceed to construct the entire diagram. In fact, from the red point, I can draw the white dogleg. From the blue point, I can draw the blue circle. In either case, I can proceed by obvious steps to complete the diagram. Now, with the confidence of experience, I declare that the red circle is a hypercircle. Obviously, it meets the horizontal white diameter at right angles.
I shall refer to the foregoing construction as the Polar Construction. In relation to it, I shall require certain terminology. I shall refer to the red point as the base point, to the blue point as the polar point, and to the white point as the point inverse to the base point. I shall refer to the red circle as the hypercircle, to the (vertical) red and blue lines as the base line and the polar line, respectively, and to the (horizontal) white line as the diameter.
By design, the polar constructions and the hypercircles stand in perfect correspondence, each determining the other. However, to apply a polar construction to construct a particular hypercircle passing through an arbitrary point, one must first locate the base point for the construction, that is, the point on the hypercircle that lies closest to the center of H. In practice, that may be difficult to do. I require greater flexibility.
By experimentation with the Polar Construction, I discover the elegant Principle of Polar Lines:
If several hypercircles pass through a common point then their centers must lie on a common line, in fact, the polar line for the common point.
and a specialized but useful corollary, the Principle of Base Lines:
If two hypercircles meet at right angles then the center of the one must lie on the base line of the other.
Principles of Polar/Base Lines
The following diagram illustrates both principles:
For the first principle, the common point is the red base point for a polar construction and the common line is the corresponding blue polar line. Moreover, the two green hypercircles pass through not only the base point but also the white point inverse to it. Finally, in accord with the facts of elementary geometry, the angle of intersection between the two hypercircles coincides with the angle between the two corresponding green radii.
For the second principle, the orange base line for the lower hypercircle passes through the center of the upper hypercircle.
At this point, I should note that, in his letter of December 29, 1958, Coxeter offered the Principle of Polar Lines to Escher.
With the foregoing principles in mind, I return to the former point of stagnation. I engage the diagram, as if in a game of chess. For any new point of intersection between hypercircles offered by the diagram, I draw the corresponding polar construction. I determine which among the other hypercircles passing through the point are required for progress. Applying the Principle of Polar Lines, I draw them. (Sometimes, the Principle of Base Lines provides a shortcut. Sometimes, good fortune plays a role. These elements lend a certain piquancy to the project.) That done, I look for new points of intersection offered by the diagram: those defined by the new hypercircles that I have drawn. And so I continue, relentlessly, until I encounter a failure of motor control, of visual acuity, or of willpower.
I present the following diagram, with a challenge: Justify the drawing of the orange and purple circles.
Detail of a challenge
The Circle Limit Series requires refined ground plans, defined by legions of hypercircles. In preparing the plans, Escher gave new meaning to the words “enthusiasm” and “tenacity.”
To draw such a figure as Figure A or Figure C, one must know where to begin. In primitive terms, one must be able to construct the triangles at the centers of the figures. For the case of Figure A, the construction is simple. As described, one begins with a hypercircle for which the radius is one unit and for which the center lies √2 units from the center of H. However, for the case of Figure C, the construction is more difficult. Of course, Escher must have found a way to do it, since he used the figure as the ground plan for CLII and CLIII.
In any case, I have posted a suitable construction on my website: people.reed.edu/~wieting/essays/HyperTriangles.pdf.
Perhaps it coincides with Escher’s construction.
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