Dividing circles into component parts
CHAPTER „A“ FOR KIDS (and grownups)
DIVIDING CIRCLES INTO COMPONENT PARTS
(in former times & now)
What we will now begin to teach is based on a congruent method of learning contemporary geometry, a basis which should introduce corrections or supplementations of issues that were omitted. It all started well, but then certain issues were said to be unnecessary, and were therefore omitted. Thereby natural equilibrium was disrupted – implying everything on this earth in which we live that has a purpose and a meaning – and if anything like that is omitted then the natural equilibrium is disrupted – and that further implies that a right result cannot be attained. It is quite a different story in the natural geometry that we were not taught in school and neither were our teachers and parents. Namely, a long time ago so many things were omitted, because of allegedly being unimportant and unnecessary that in present-day teaching they are not applied. Yet, it is indeed important (as will be shown in the supplements that follow) to present (in a simple manner easy for children to understand) results that are sometimes labeled “impossible” even in the encyclopedia. Nevertheless, we will start from the beginning by comparison with the way that we have been taught and the way that natural ancient geometry teaches, step-by-step, drawing after drawing and all this only with a compass and unmarked straightedge, without any markings with letters or numerals, but only with a couple of accompanying words that will take us into the “world of play” in which the eye sees and the mind comprehends. That is why we are beginning with the division of the circle into its simplest parts and then we will immediately perceive the difference, even among those who already know the basics: the circle, the radius, the line of symmetry.
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Therefore, with the compass, from the center we draw a circle of whatever radius or a given radius. We were taught that the radius divides the arc (the full angle) of the circle into six equal parts. It all started well because we were told to “draw a circle from one of its points and divide it with its radius until you come back to the initial point of division”. We thereby acquired 6 parts and a “flower” with six petals.
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In all probability, seeing a „flower“, somebody must have decided to cast it out because it was “disturbing”. That marked the beginning of “segmentation” of the circle’s arc. Therefore, we acquired the division into 2, 3 and 6 parts.
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It was now necessary to divide the circle into 4 parts. No problem. When we bisect 6 of the circle’s subtended poles with bisectors we get four parts.
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With bisectors it is now easy to get 8 parts.
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Ten, won’t go. A sequence of operations is necessary. Seven, “impossible” only with a compass and unmarked straightedge (so is the nonagon). Ten. Once we get 5. Eleven, requires “tools”. Twelve, is easy. 3 bisectors, 6 through the center. We acquire 12 parts. And if we always divide with bisectors we will keep on getting more and more parts. Then, what’s the difference, and does it exist at all? Well, let’s see …
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As we stated in our introduction, nothing is shortened or omitted in ancient geometry (except as an aim for the sake of clarity), since everything has its purpose, which is utilized only with a compass and an unmarked straightedge. Therefore, we have the circle and its division with its radius into six parts. Thus our reference that it all began well, and so the circle divided by circles of their radius – gives us 6 parts.
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The same circle divided into 4 parts.
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Same circle. Same way. 12 parts.
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Same circle. The first division from 4 circles of same radius. 8 parts.
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Only from a quarterly point – 24 parts. Thus, we have 2 (3×2) 4, 6, 8, 12, 24. We will stop here.
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And why did we stop? Because this will be sufficient for a beginning. The next bisector division is the route to solving an enigma with this drawing. In the process, we “skipped” a succession of data that are essential for solving “enigmas” and “impossibilities”. We didn’t do anything in particular except that we refrained from undertaking shortcuts. Here is what was not shortened: drawing bisectors in full circles of same radius as in the first example. What else? We didn’t exclude the flower-like pattern, and already in our next chapter we will learn why and what “secret” it holds. ASoon we will also comprehend the importance of “using lines of symmetry”.
Therefore, let’s recapitulate:
If we divide the circle with its radius, with full circles of the same radius we can promptly divide the circle into 2, 3, 4, 6 and 12 parts. And if from a quarterly point we divide the said circle with circles of same radius the circle can promptly be divided into 2, 3, 4, 6, 8, 12, 24 parts.
Thus, a compass for circular inscribing, and an unmarked straightedge for linear drawing.
Therefore, our ancient ancestors have for unknown reasons been hiding from us some simple cognitions for at least 2500 years (approximately since the time when a record was found in which Archimedes declared it impossible to construct a 20° angle only with a compass and unmarked straightedge, which means into 18 parts (2×9). Did Pythagoras know this? Did the Greek mathematicians know this?
So, the responsibility is not ours, nor our teachers, fathers, grandfathers, great grandfathers – but rather of someone who, for whatever reasons, decided to suppress that knowledge or any personal information about it. Yet knowledge is the goal of the human spirit. Therefore, an injustice was done to knowledge, because ignorance gives birth to irrationality (this is evident today through the existence of so many secret societies). Fortunately, the rational spirit does not deviate from its path of the explorer. Therefore, kids, so long until our next chapter.