The purpose of this brief essay is to bring closure to the geometry and trigonometry material on the jmanimas website. Why the geometry? It was always a study in history, in particular why did the Pythagoreans ask this question:

"Can we construct a square exactly equal in area to a given circle using only the compass and straightedge?"

I have always believed, since 1957, that there is a positive solution to this riddle and that the Pythagoreans knew what they were talking about. The resolution for me, which is at this time not accepted thy mathematicians, is that the riddle is presented with reference to "physical pi" and to real objects in the physical universe. Because the real, physical universe is governed by the lower limit of "the smallest possible particle," there are no Euclidean circles in the real world. The smallest possible particle (spp or "spip") that represents a stable piece of matter is probably the atom -- the smallest atom. Other alleged particles are either the fantasies of physicists or fleeting events that possess significance but are not the stable spip.

What this means is that the traditional interpretation of the riddle is the source of the agonizing and world renowned allegation: "You can't square the circle." The reason this is true is because the Euclidean circumference is defined as an infinite number of points -- each point having no dimensions. While mathematicians cling to this embarrassment tenaciously, there is also the equally accepted doctrine -- a far more reasonable doctrine -- that an infinity of zeros is still zero. We cannot accumulate "null" until it becomes "something." For this reason, a real circumference in the real world has to be composed of a finite number of spips, with each spip possessing dimension -- a length or a width -- however small or "short" it may be.

There is more than one way to examine this phenomenon, but I have found the following to be the most satisfactory:

Nature does in fact possess a "ruler." On Nature's ruler, the unit of measurement is 1 spip - one smallest possible particle. Every real circumference, meaning physically real and composed of matter, is necessarily a regular polygon. Every real circumference, meaning physically real and composed of matter, is composed of N spips and only N spips and N is an integer. The fact that we do not have half a spip, or any fraction of a spip, or ten billion and 0.759 spips, is simply because no fractions of spips exist. The decimal digital system appears in the human brain, and does serve some useful purpose, but it is anthropomorphic, as many human concepts are, to insist that Nature employs a decimal digital number system, or a binary number system, or any number system, except possibly proportion itself.

How does all this connect with the Pythagoreans? They were already around when Democritus argued that there is a smallest possible particle. The Pythagorean riddle about squaring the circle is a thought provoker. It's purpose is primarily to defend the concept of the smallest possible particle, and Democritus, and to stimulate the student to think about this issue. How could there be a material world if there were no smallest possible particle? Is it possible for matter to descend downward infinitely in size, until we arrive at nothing? This would mean that the physical universe is composed of nothing. There may be philosophers and gurus and even burnt-out "scientists" who want to tell us the universe is composed of nothing, or that we can travel to "another realm" or to the past or future, but I conclude that the universe is made of something and it is made of something because there is a smallest possible particle. Or, the better logic is the reverse: the universe that exists is "something" and is physical matter precisely because there is a smallest possible particle. The logic of the real universe is, then, that a smallest possible particle is necessary, absolutely necessary, in order for physical matter to exist and "prevail over nothing."

So, this is it. Yes we can construct a square exactly equal in area to a real, physical circle (not a Euclidean circle) because a real circle is a regular polygon. This is where the riddle leads, and to the logical philosophical concept that if there were no smallest possible particle, there would be nothing at all.

(But I still wonder… and sometimes try again.)

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