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Second geometrical enigma – Children’s education

16. Chapter

Second geometrical enigma

Perimeter and surface of the circle (education for children)

The purpose and sense of all these geometric reviews was simply in good intentions, to bring sequences of simple solutions to new generations,that we, people, overviewed through ages, so we were left just „impossible unknown quantities“. Then we have played with geometry and with sense and unsense, not understanding that whole nature around us is geometrically harmonious, and every deviation is degraded perception of our human nature, „illness“ of healthy organism. We can’t say that individuals didn’t try hard during ages (of our written history), although they didn’t have social conditions for that (for example, Archimedes). There should be emphesized time of Greek philosophers. Later ages – medieval – were „dark“ of human existence, although there were a few names, but since then „theory“ was born or philpsophic thought, and consequences were many impossibilities till today, so the only way to get out of this is „to reach“ for ancient times. The safest way is to ask the one who gave purely geometric projects, along with 10 moral guidelines to man „Moses“ which are easy turned (in tradition) in acts. Finally, all that was confirmed by the one who was crucified. How? Lock yourself in the room anda sk. As much as today’s science „arogantly“ denies it or ridicules, but that is the best and simpliest way. Confirmation is even myself. Ask anyone in the world if anyone in the world has brought so many geometrical news in just two years. Me, simple man, 63 years old, worker, poet (in the last 10 years 1100 poems or over 300 000 verses). I wouldn’t be able to do it if I didn’t ask „in secrecy“ the one who said „call me Jahve“ on Horeb, 3300 years ago. How to make it act? Through this new media because I couldn’t do it in classical „written“ way. In this way it is functioning, so I can afford myself to repeat (on many requests from the world) looking for closest way to today’s geometric way, as int his chapter about perimeter of circle’s arc and surface of circle (universal) just with compass and ruler without measures, about number Pi (π), and along the way, to give respect to tragic person of „social situation“ of those times – Archimedes.

* * *1601

At the moment that we took some size in range of compass and stick it in one point, we have started one process (circular) and we’ve got limited space, which is circle that has its border – circular arc or circle and circle becomes circular facet. Cirle has its lenght or perimeter, and circular facet its surface.

* * *1602

If we stick compass with same range in one point of full arc (360º) or in circular arc of circle – we’ll see that that range of compass divides circle ins ix equal parts. That is natural constellation, law that we can’t avoid. That range of compass is called radius. If we connect opposite divisions or poles which pass through the center, we see that it has 6 radiuses or 3 double radiuse which we call diameters.

* * *1603

Now, if we connect neighbouring poles, we got 6 equilateral triangles so we say that we got hexagon. We see that those 6 equal lenghts are the same size as 6 radiuses and has to have smaller total lenghts than total lenght of circular arc. Now we have a problem that exists till today. What size is circular lenght and how to inscribe it with geometrical measurement. It doesn’t get egzact result. They said – impossible without measurement. And then it is found „eteranl compromise“ so called – numerical. We’ll think about it, for a second.


It is said that ancient Egyptians wre excellent experts in geometry, but something is strange there. It is strange that later, to the ancient greek philosophers who searched for knowledge int he area of geometric and mathematic sciences, they told that they got their knowledge, and strange story about existence of developed Greek civilization a long before their Greek civilization for which all traces are lost, a long time ago. And Egyptians got their knowledge from their gods. There are numerous stories, legends and myths about it, but it isn’t a theme of these reflections, but something else is.What a genious mind it had to be to inscribe a circle and saw its laws so precise, 3000 years ago, and by tradition, even before any letter anywhere, and the witnesses are ancient buildings and ruins of exactly geometrically processed stones with ideally regular geometrical facets.Even byblical story of genesis speaks about „ those who visited our land – angels“. And just one thing made us thinking. It is said that God’s day regarding to human’s day is a few thousands time longer. In evolutionary way, observing from a point of view of last act – man, evolutionary – byblical cyclus is compatible to our idea of evolution, but one small mystery in genesis is different, in dual sense. First, we think that universe has arised from Big bang, and then by developing systems of planets around queen stars and further. But there is something else about what is not thought. In the Bible it is said that in third cyclus (day) plants were made, trees with its seed, and in fourth cyclus – light and sources of light – sun (basic planet star) and stars. Mistake in manuscript or? From today’s point of view impossible. Or possible? Then? Weird questions. Unproveable.Even geometrical questions are weird. Number Pi. How did Archimedes know for number Pi and didn’t know to draw it. It is true that he knew it, otherwise he wouldn’t „bother“ to resolve it geometrically. And he would resolve it, for sure, if „conditions“ didn’t stop him. Because of that there was a question marka bove number Pi for ages, because it eas thought that number is number and not geometrical code – universal – simple. Where is „oversight“ or geometrical lapsus, known and used before every written(known) sign which we call number and letter. So, how?

* * *1604

Division of lenght on parts (picture). So, the goal is to calculate perimeter. Whole mess in geometrical sense has brought one number, especially nowdays. Mathematicians of medieval later times (and today) have played and that amount of mathematician’s effort have simply shortened. Number Pi. It came to number Pi, tool that has to have its purpose based on Archimed’s researches. And ancient stories told: „ We don’t know decimal number“, and then arised simple „ Whatever“. Stories said: „I know the whole one and its parts“. What i the whole? What number? It is one. And what is part? If you divide one with any number you get parts. That is what children learn at school today. How to divide lenght (1) on parts without measures. That is number Pi like.Archimedes said it is number 22 divided with7. He was right. 22 : 7 = 3,14285715. By seeing that decimal number, it was said – make it shorter for calculating. Mistake. If you short something it is like you cut off a part of it because, 1 : 7 = 0,14285715 or one seventh. That’s how number Pi is 22 sevenths or 3 whole and one seventh. The ancient number say so. So, how to draw number Pi – geometrically?

* * *1605

If we say that Archimedes made a base, someone else (and no one thought us or our children who was someone else) understood how to divide lenght on parts without measures. But, as geometry is an „orphan“ of mathematics and in those times was extremly important (Platon as witness after plaque in Athena – legend about duality of Zeuss’s altar) and today with extremly good things „acrobatics“ are made without natural harmony, especially in basics of geometrical science in elementary schools, so it is no wonder that most of the children don’t like geometry because it is not childly simple. How easy it would be to Archimedes if he knew division of the lenght on parts without measures. And some of them complicates it more by saying it is not 22 but 44, and it is in fronto f us. One more irritating thing, they say perimeter is lenght, here is formula 2rπ. Here is „oversight“ to not be understand.

* * *1606

Aren’t 2r (2 radiuses) equal to 1d (1 diameter)? Isn’t it simplier 1dπ (1 diameter times Pi) ? How happy would Archimedes be. He would run through Syracuse again,yealling: „Heureca!“. Every seventh grade pupil (13 – years old) can draw that lenght. It is extremly easy now to draw 3,14285715d or 3 whole and 1 seventh of diameter, and Archimedes would draw 22 diameters and divide it on 7 parts and would have perimeter’s lenght.
Transfer 3 diameters of circle (divisions of circle by radius on 6 parts) with compass on line, and fourth diameter divide on 7 parts without measuring anda dd to three diameters and we have perimeter’s lenght of any circle. Why is it numerically important that decimal part of number Pi? (1 : 7 = 0,14285715 or one seventh). So, three whole have 21 seventh plus 1 = 22 sevenths. It is only mathematics „true“. Extremly true, because to short means „to mutilate“. Now we can simply say, perimeter of any circle is three whole and 1 seventh of its diameter. He is drawn – perimeter – and is drawn as lenght. So,we can continue. How?

* * *1607

But, before we go on, draw perimeter’s lenght as square by the rules which we learned int his geometry, just with compass and ruler without measures (so, without mesuring), simply without shortening. By simmetrying of given lenght on 4 parts.
You’ve noticed that by drawing 1 seventh without measures angle of line isn’t important (p) but the last point of seventh’s (or any) division of line and the last point of given lenght, and then with paralels (2 triangles) divide lenght on 7 equal parts (or any). I tell this because many have forgot (older readers) basic elementary school geometry „reading matter“. Start with construction of perimeter’s cube, in a way to bring one fourth of total perimeter’s lenght that arises by summing 3 diameters plus 1 seventhof diameter on the line.

* * *1608

We’ll resume perimeter, geometrical way how we come to it, perimeter’s square, and comparison with its descriptive circle.

Resume of circle’s perimeter

If we draw a circle of some radius,we have limited some space – circle. Radius divides perimeter’s arc of circle on 6 equal parts, hexagon. Hexagon has 6 radiuses or 3 diameters, but, perimeter is bigger than 3 diameters or 6 radiuses. Archimedes concluded it is relation or proportion 22 divided with 7, relatively it is written today 2rπ. How to draw it geometrically when you got number that is concidered impossible to draw, 3,14285715. But it would be more real that it was used mark for diameter instead 2r, and instead decimal number which is liable to shortening, was used fractional number because it has’n got shortening but it is clearly said that perimeter is 3 whole and 1 seventh of diameter. Then we divide it by symmetring on 4 lenghts. By taking one fourth on the principle of construction of quadriangle, perimeter’s lenght we inscribed as square (quadriangle) of perimeter. In the same center, for comparison, we inscribed circle of first radius and we see that circle of perimeter’s quadriangle is bigger than perimeter’s circle of first radius’s circle. It would be about perimeter, its size and partially about number Pi. What are we looking for and based on what ? We are looking for surface of circle’s facet, to confirm it geometrically, without measuring – with compass and ruler without measures. But, we are meeting with something new here, unknown till now. Why? Because we are starting inside out from unbordered space. Because Pi itself is part of something, part of number 4. What does it mean? Let’s take a look.
To be drawn more clearly we’ll double circle’s diameter and descriptive circle’s diameter of perimeter’s quadriangle.

* * *1609

So, because of brightness of geometric picture we doubled all parameters of given geometric drawing of perimeter and its cube. Based on diameter as base we construct square 4 d (diameter) with center and simetrals of sides.

* * *1610

Now, in its center we inscribe circle of radius and circle of perimeter and squre of perimeter. As we see,logically, initial circle is inscribed circle of square. Then wy we start from square of perimeter 4d, and not from 3 whole and 1 seventh? We must clear it.

Abut number 4, Pi and Pit

It is easy to calculate but hard to draw, especially as universal code, just with ruler without measures and compass, without measuring surface of the circle, so it leaves what we, people, always leave, doubt. If we don’t convince ourselves as our old said „obvious“, because „obvious“ could be checked, measured, confirmed, or not confirmed. Someone will say.Great minds of mathematics concluded. It is not just like that. Nowdays, surface of circle is enigma because it needs to be drawn with al mentioned rules just with compass and ruler without measures. It must be known, just in this way is universal geometry or geometry of universum, geometry of harmony, natural, why it is called sacred (our human expression?), but whatever it is called, we can see geometry review which can be mastered by any elementary school child (14 – years old in our country). In that age children have already learned what is circle, diameter, radius, perimeter of regular forms, division of lenght on parts without measures, calculated perimeter, surfaces of forms (although not so exact, but children have learned). What is missing? It is missing visual review, and it is simple with existing knowledge of these 13 and 14 – years old. 4 or quadriangle. $ diameters of every circle. Descriptive quadriangle of every circle or surface  (diameter on square). What now? When number 4 divide with Pi or perimeter of 4 diameters with 3 whole and 1 seventh of diameter, we get one number (4 : 3,14285715 = 1,272727 or upside down 3,14285715 x 1,272727 = 4. What is decimal number 1,272727 ? It is 14 elevenths. That sign from ancient times is not well explained. It was and stayed letter. We’ll talk about it more later, and is not inscribed in computer programs and is not learned in schools anywhere in the world. And now you can draw „obvious“ surface of circle – as square surface because formula is simple :  (diamter on square – diameter times diameter) divided with 14 (parts 14 x 14 of diameter’s square, quadriangle of diameter) and divided with 11 = surface of circle – its squared – quadriangled review.So,let’s take a look.

* * *1611

This way, as children learn in school division of lenght on parts isn’t neccessary (but we are doing it for explanation). It would be ideal to divide sides of descriptive quadriangle of given circle on 28 parts. (relation 1,272727 or 14 elevenths is the same as 28 divided 22). Why isn’t it neccessary?

* * *1612

Because one part of relation we have. And that is quadriangle of circle’s perimeter.So,we have surface of circle’s facet as rectangle. Side ( x 11) x d

* * *1613

And from there everything is easy. Extended sides of quadriangle of circle’s perimeter on sides of descriptive quadriangleof circle or four diameters. Peak quadriangles divide with two.

* * *1614

In that way we got equal two surfaces of two quadriangles which are two surfaces of circle’s facet, and root gives us quadriangle’s side of surface of circle’s facet. And that’s it. Drawn what could be calculated just now and drahted (universal, for any circle if we know its measures of radius or not).

Closing word

What is the message of this 20 pages of geometry just with compass and ruler without measures. It revealed us the way of access in disclosure of unknown. It said that it is needed to value those who solved at least one element of some unknown fact and acknowledge their fame, and not reject it from the children’s books (school books). They have already „killed“ one, Archimedes. I asked school children if they are told a story about Archimedes’ way of researching phisical laws (laws of nature), How he runo ver town naked, and yealled: „Heureca!“ and how he died after the fall of Syracuse Which he defended with laser’s anecestor 2500 years ago.. Nothing from it. Just a container called „Heureca“, and exactly that „story telling“ was initiative a long ago to our children’s „mass“ to move us to make us „phantasists“. Grain which brings its crop because „grain“ is cause during time and space for numerous grains. It is not comendable for the writers of children’s books, although they like to take academic titles. In this chapter we should honour Einstein’s thinking about observing something from more points of view. Because more points of view talks about same thing, more details, and that is multidimensional. And we know that our knowledge lays on senses. And it is good as much as it seems paradoxal. „I don’t believe until I see.“ In geometry it means, if I can’t draw something then in vain is „convincig“ with number. Especially int his kind of universal, natural geometry in which hasn’t inbalance, asymmetry, whic doesn’t need measuring but just compass and ruler without measures (long ago „gauge“ had just measuring rope and measuring stick, but fantastic spirit of knowledge and skills – about many speaks the Bible). Here you found out for sign Pit (it is shortened, but it hasn’t mark of letter because it is preserved, undiscovered for greek ancient mathematicians or for some reasons, „hidden“ – as Pythagoreans hided their knowledge). But it doesn’t matter. It matters that you know solution of enigma now – surfaces of circle’s facet just with compass and ruler without measures.

HR – RIJEKA 07.03.2013

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