Part 04: Unmasking the Tusi Couple

In the first three parts of this series, I have tried to establish what I mean by "contraction" and "protraction" of ideas. I have likened the process to data compression and logical representation, but in seeking to demonstrate my meaning, I have overlooked a simpler means of conveying my point: physical action.

Consider a spyglass. Hold one end up to your eye and you can see distant objects as though they were near. This amplification of your natural sense of vision is "protraction". It stretches, amplifies, and "makes large" what before appeared small.

However, the spyglass has another use. Were you to turn it around and look through it the other way, everything which was before very close and distinct would appear small, distant, and indistinct. This is "contraction". It compresses and "makes small" what before loomed large. Contraction also makes indistinct what before was clear.

Now, consider what actions the spyglass performs. Does it physically affect the actual material that is represented to the eye? No. The objects in the distant scene are not physically affected at all. However, their representation to the eye is "bent" or "warped". From one direction, the representation is contracted and the objects become less distinct. All of the detail that existed before is still there, though our eyes lack the ability to clearly process the contracted information. When turned around and used as intended, these indistinct representations are bent in such a way that our eye can better distinguish their parts and we are thus able to process a greater amount of information.

Protraction and contraction are simply inverse relationships, best considered as light passing through a lens from one and the other direction. As a spyglass bends light in contraction and protraction, so does the mind bend ideas. I will explain how and why at a later time. For now, I ask the reader only to consider "what if" this were the case, and regard my hypothesis as plausible if not true.

This notion of "contraction" comes to me from Nicholas of Cusa, and was a principal consideration in his study of the infinite arc segment, which I have reconstituted in my study of The Identity of Unity, Nullity and Infinity and further developed in How Unity Relates to Number. I illustrated the Tusi Couple, which is a "two-cusped hypocycloid obtained by rolling a circle of radius α inside of a circle of radius 2α. The curvature of the hypocycloid is commonly regarded in mathematical circles as a "line segment". That is, straight, but better thought of as flat. Armed with the Nicholas of Cusa's notion of contraction, I will now show that this is not true, and that the Tusi Couple is not a straight line at all!

What creates a line?

Is, as many mathematicians assert, a line nothing more than a "set of points" that fulfill a certain set of conditions in line with our expectations? I hold that this Euclidean conception is flawed because it places our definitions and axioms as the basis of the Idea. If the Idea of a circle is "that thing which meets the requirements of how I define a circle" then I might be begging the question and leading myself into a prison of my own subjectivity. What if I were to project an image of a circle (with, say, an overhead projector) onto a piece of rumpled white fabric? Are the points described by this new shape on the cloth "the sum of all points equidistant from a given point on a plane"? Certainly not.

But what if we take another definition of a circle--that physical action which encloses the greatest area with the least perimeter? The "isoperimetric" action, as this is called, actually translates to the new, more complex surface, so that the wavy and complicated line which passes through the peaks and valleys of the new curved landscape is in fact a circle, though you won't be able to prove it with a compass and straight edge. However, the curvature that defines the scope of possible actions on this new curved landscape precludes the possibility of the "set of all points equidistant from a given point" definition. The assumptions that underlie the Euclidean definition of "circle" just do not translate to the curved surface. However, the Idea expressed by physical action does.

So lines aren't "sets of points" that obey axiomatic assumptions. They are representations of physical action, or, better put, perceptive contractions of physical action. This realization casts the common understanding of the "Tusi Couple", which describes through physical action the relationship between number and unity, with a "flat line", into question. Allow me to illustrate.

First, the Tusi Couple from my entry on How Unity Relates to Number:

Consider this supposed "line segment" as the result of a physical action. Is this really an action that describes a constant and uniform motion, as would befit a constant and uniform surface? I mean, if this "line segment" were "flat", like a groove in a flat plane, and you rolled a marble along it, would the marble behave in the same way that the smaller circle's point of interaction on this supposedly "flat" line does? Would the marble, as the intersection of the smaller circle and the "line segment" it traces does, speed up and slow down near the end points, if the groove were on a "flat" surface? It would not. After all, uniform manifolds of action only permit uniform motions--it's implicit in their nature.

So, what is expressed by the variation in the rate of action witnessed in the Tusi Couple above? What is it we are seeing? A hint can be found in our earlier consideration of the projection of the circle onto the curved surface.

Recall in our earlier example that I suggested the image of the circle could be projected onto the curved surface using something like an overhead projector. If the surface of projection were "flat", we would perceive a circle that comformed with the Euclidean definition of what a circle ought to be. However, if the surface were anything other than flat, this was no longer so. Imagine now that instead of a projector, we used a laser to trace a circular path on a flat wall. The circle is now, more than ever, the product of a physical action upon the "flat" surface of the wall. Let us further suppose that were we to measure the rate of motion that the laser travelled along the wall, that we would find it uniform in every part of its pathway.

Now interject the curved surface of wrinkled fabric between the laser and the "flat" wall. The pathway of the laser on the surface of the cloth must travel up and down through the hills and valleys of this new surface, but it completes its cycle in the same amount of time. In effect, because the laser must now travel over a greater amount of survace area, punctuated for brief moments by "flat" areas, its action along the surface must "speed up" and "slow down". Though the motion of the actual laser pointer device has changed not at all, the effect witnessed--that of the laser traveling across the curved surface--is one of nonuniform speed. If viewed from the perspective of the laser pointer device, there is no distinction at all in the rate of speed. However, if viewed from the perspective of an observer on the curved surface--the dot which is the laser's intersection with the surface speeds up whenever it goes up or down hills, and slows down at the peaks and valleys.

Now, reconsider the physical action described by the Tusi Couple below. I have plotted equal-time markers on the previously considered "flat" line in order to show that, in fact, there is a nonuniform rate of action.

We can now, freed from our Euclidean axiomatic assumptions, and armed with Cusa's idea of "contraction", clearly see that the so-called "line segment" described by the Tusi Couple is not flat at all, but a contracted arc segment! Just when we thought that we had found the ever-elusive flat line as an infinitesimal relationship between unity and number, it again slips from our grasp. My next step will be to create visualizations which clearly protract this "flat line", which is not flat at all, and determine its true nature, if possible.