THE GOLD HELLIX


TABLE OF CONTENTS

Consec. numbers Subsection Summary Links Last update
1 MATHEMATICS WITHOUT NUMBERS This section shows that the "mathematics of Nature" does not use any numbers or formulas
Link 01 06/28/2010
2 REFLECTIONS ABOUT THE BASE OF NATURAL LOGARITHMS My thinking, that the "implied" the Nature under the uncomfortable number of "e"
Link 02 04/19/2011
3 OCKHAM'S RAZOR About how should apply the principle of "Ockham's Razor" to the geometric constructions
Link 03 06/28/2010




MATHEMATICS WITHOUT NUMBERS


Swiss mathematician Jacob Bernoulli (XVIIv.)
bequeathed to carve on his tombstone
logarithmic spiral
and near it write the phrase
in Latin:
«Eadem mutata resurgo» - "After the changes I am reborn old"

      «Mathematics without numbers» - is it possible? I do not know how this happened before me, but in this section, I dare you to demonstrate that it is possible. I undertake don't use no numbers and formulas till the end of this section, although the entire section will be devoted to mathematics. In this case, I will consider not simple mathematics, but a serious math.
      Shortly before the end of his life, Einstein wrote:

      "Unified Field Theory is now finished... Despite all the spent efforts, I can not verify it by any way. This situation will continue for many years, the more that physicists do not take the logical and philosophical arguments".

      Tell me, is it ok when people of the scientific world, spoiled by the formulas and numbers refuse to "take in information, which does not include these numbers and formulas". They had better be "deaf and blind", than to delve into any calculations that do not contain numbers. For my part I want to say that these people are just "so alienated from nature", that all their efforts to understand the laws of nature may be just in vain. Such "blind and deaf" scholar in their path just has risk to "fall into the pit", out of which he is not able to move, because in order to get out of the pit he need to rely on something more real than abstract formulas and numbers. If we went any further, we should probably admit that Nature even has no the "laws" in our understanding of them, but it has only the simplicity and naturalness, which is concentrated in a known principle of "Occam's Razor."
      We know the mathematics and physics, but does the Nature know them? The question, as they say "rhetorical". Why she must know "our math" and "our physics"? She can create all and can destroy everything without our knowledges. So why do we so persistently apply our numerical methods to all the objects of nature, if we certainly know that they are alien to it? In this world we all are simply "kids" who play in their sandbox and watch the world around them. Communicating with the world around us, we, by and large, can only "peek and copy," can "multiply", but we will not be able to impose on the nature processes our "rules of the game."
      Swiss scientist Jacob Bernoulli lived several centuries ago. Why is it so valued by descendants? Because, what the scientific legacy that he left us, just priceless. He left his mark in many areas of physics and mathematics. Today, we are only interested in one small part of its scientific heritage, namely, isogonally or logarithmic spiral, also known as "conformal spiral", "spiral of growth", "spiral of life" or "golden spiral".
      What caused such "honorary titles", by which the humanity awarded this spiral? See and judge for yourself.


      Above is a picture from Wikipedia (chapter "logarithmic spiral"), which immediately allows us to estimate all the power and universality of such a spiral.
      The question arises: Can we talk about one of the most powerful of the spirals in math and did not use any numbers and formulas? It turns out we can. The nature doesn't use any math, when she creates all the objects in the picture. And let's we try to do so.
      You can not believe me, but the real "mathematics of nature" is precisely located where "are no any numbers". Treatises of ancient scientists were much longer than modern ones. Then, to reduce the texts of the describing of various problems, people resort to formalize their descriptions, using a specially devised badges. But I am very sad to realize that using their badges, they then began to accept their own badges as the essence of natural phenomena, and did not even notice this substitution. Now down to business.

      Let's take one line segment and divide it into parts so that the length of the greater part to the smaller one was in the same ratio as the length of the whole line segment -to the larger part. Let's divide the largest of the obtained segments, so that the ratio of the length of this segment to length of the more large of its pieces was the same as the ratio of the more large of its pieces - to a smaller one. This process can be continued indefinitely. Let's use this sequence of declining line segments for the building a spiral. What do we get the result? And that's what.



      Look at the lower figure. I think you already guessed that the lowermost and largest horizontal segment - this is namely the original line segment mentioned above. This segment is divided into parts as described above. Right has remained a big part of the initial segment, and the left - lower. Then, based on largest of the segments was built a square, the left side of a square also was divided into parts by the known algorithm and so on...
      Logarithmic spirals pictured above contain movement: growth and reduction, deployment, and compression, progress and regress. These spirals are the essential expression of the phenomenon of natural growth, which is found everywhere in the universe. Such spirals show us such small objects as atomic particles and "our level" objects, such as sunflower, which has two spirals shown above (see graphic at the bottom of this article), and such a huge, as the galaxy. As pointed out by David Bergamini in his "Mathematics", written for the series «Time-Life Books' Science Library», comet's tail is bent in the direction out of the sun as a logarithmic spiral. Bacteria grow with an acceleration, which schedule has the form of a logarithmic spiral. Artificial crystal in considering it under an electron microscope also shows us the logarithmic spirals (see "Golden Spiral" on the site: http://www.techanalysisfx.com/precter56.html).
      These spirals differ from the others by the fact that the tangent line to a spiral at any point, lies under the same angle to the radius vector of the spiral in polar coordinates. That is why the spiral is "like herself." That is why this spiral (or more precisely: "loxodrome" - logarithmic spiral, which is plotted on a sphere) is used for precise plotting course of the ship when moving in the ocean. In this case it is very convenient that the angle between the course of the ship and the magnetic compass in the motion remains unchanged. That is why the Swiss mathematician Jakob Bernoulli, who studied this curve for a long time, commanded to draw a logarithmic spiral on his grave next to the inscription: "When I was changed, I restored my old form", while noting the most interesting property of this spiral.
      I have read several articles about the construction of the golden spiral and have penetrated into these constructions so seriously that developed my own way of its construction, which I intend to describe here.
      Let's imagine the rectangular coordinate system (I am too spoilt by civilization, more precisely, by Rene Descartes, man and therefore I decided to "afford myself such a free-thinking"). Let the x-axis will be directed to the right and y-axis - up. Let's defer from the origin point some segment to the right. We can consider it as the "unit interval" or "segment of unit length", but for my presentation it is not essential. I'll call it "the original segment".
      The left point of this segment will coincide with the beginning of our orthogonal reference system, and the right end of this segment (ie the end point of this segment) I will call "the first point of the helix".
      From this first point of the spiral to the right postpone another segment, which will be slightly longer than the original one. The length of the new segment should be such that it refers to the length of the initial segment such as the total length of the segments - to the length of more long of them.
       Considering the first point as the rotation center, let's draw a circular arc passing through the end of the second segment till the intersection of this arc with a positive branch of the Y-axis. The radius of this arc is equal to the length of the second (ie, larger) segment. Intersection point of the arc with the vertical Y-axis will be called the "second point of the helix". Let's connect this second point of the spiral with the oblique straight line till the first point. That's all. "The initial base" of our spiral completely ready. We have received a "original triangle of the golden spiral", let's call it the "golden triangle". The Golden Triangle is formed by two catheti, lying on the positive branches of the coordinate axes, with inclined hypotenuse, whose length equals the length of the larger of the original segments.
      Now you can draw a perpendicular to this hypotenuse from a second point till its intersection with the negative branch of the x-axis - we get the third point. Let's construct a perpendicular from the third point to the previous segment till its intersection with the negative branch of the Y-axis - we get the fourth point, etc. It remains to connect the first, second, third and fourth points by means of a smooth curved line - it will be a golden spiral.
      If someone is interested in not divergent but convergent spiral - take it, please. In the first point let's restore a perpendicular to the hypotenuse till its intersection with the negative branch of the Y-axis - you'll get a "minus one" point of the helix. If in the "minus first" point of the segment you'll construct a perpendicular to its intersection with the negative branch of the x-axis, get a "minus second" point and so on - ad infinitum. Take a look at a conch shell "Nautilus Pompilius" and try "to feel it..."



      The above-described method of constructing the golden spiral, I never met in the literature. However, namely this process of building it seems to me the easiest and the most natural. I think that the clam shown above in the construction of its shell uses roughly the same logic, what I have described. After all, he, as I'm overlying presentation, "do not uses the mathematics", just because he "doesn't know the mathematics"...
      The whole scheme of the build process and its result for the divergent spiral you can see in the picture below.



      In this figure, the initial segment, I marked by the letter «a» and the second one - the letter «b». The length of the smallest red segment is equal to the length of the second segment of «b», shown in light-green.
      Well, how did you like "mathematics without numbers and without formulas"? Yes, it is hard, it is unusual (I'm also received all my education in modern schools), but by means of such a presentation more clearly can be traced THE ESSENCE OF THE PHENOMENON. Please note: I clearly describe the specific, not vague laws, which are contained in countless natural phenomena. In this case, I intentionally used the "mathematics of nature", not "perverted minds mathematics".
      Einstein on this subject expressed his opinion much more categorical. I can not find the exact quote, but his phrase was something like: "Mathematics - this is the best way to obscure the merits of the case and hide them behind a string of formulas".
      The thought is primary, and the formula - it's just "crutches", by means of which is forced to walk, so to speak, "not quite a healthy person".
      I apologize to mathematicians for the above joke, and I want to confess that I at one time, paid tribute to "melee math" too, when reckoning a series of contour and volume integrals in the theory of elasticity. Plain paper was so small that I was forced to fill a series of Whatman sheets for my "hand-applied mathematics" calculations.
      However, the attainment of truth in the surrounding nature is not directly related to mathematics. The truth is apprehended only by pure process of thinking. On this subject, Einstein in his Oxford's lecture said:

      "In a known sense, I consider as the truth that pure thought can captures the real, as the ancients dreamed about it."

06/28/2010 Segrim

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REFLECTIONS ABOUT THE BASE OF NATURAL LOGARITHMS


      Have you ever wondered why the base of the natural logarithms, which is "so natural and pleasant to Nature" is number, which is "so not suitable for the person"? Probably not. I, for example, thought about it. I'm going to share with you with my thoughts about this.
      I will not repeat the description of the many observations and calculations on this subject (they already "overfilled the Internet"). If readers want to delve into this topic, I can recommend the most authoritative (in my opinion) book:

E. Sedov, "One formula and the whole world", published by "Znanie", Moscow, 1982.

      If something in the nature is "skyrocketing" or "rapidly decreasing", then after some time, this rapid process, which is usually described by a logarithmic curve, starts to slow down in its velocity and the timeline of the process begins to steadily approach to a horizontal line. Why everything is happening EXACTLY THIS WAY?. It's not accepted to look for answer to this question. People for the description of such processes simply use in their practice logarithmic curve with a base in the form of e-number and stop there. I went ahead and plunged into this matter so that the result of my reflections can be considered even as a "mystical revelation" in this field. Became clear: if in Nature there is the rapid growth of something, then as a result of this growth product begins to interfere with the very process of growth. Arises, as they say, negative feedback, the negative influence of obtained PRODUCT on the initial PROCESS. Because the feedback is NEGATIVE, then the product begins to slow down the process. But the process still continues, the product in this process is still produced. However, if product quantity becomes more and more, the BRAKE ROLE of this product in the process is still growing.

      One note: DECIMAL logarithms in Nature do not exist. By and large, we just "do not have the right" to use the decimal logarithms. If a person in some cases uses the decimal logarithms, he only makes a "rough estimate" of those natural processes, in which he failed to discover the "true logarithmic curve".

      Of course, if the "two opposing factors are fighting each against other", one of which is steadily gaining momentum, then in the end, this second factor, which is derived from the main process, can completely stop the process of self production.
      Believe you or not, but the process, which I described, contains the MAIN PRINCIPLE OF SELF-REGULATION IN NATURE.

      All that is described above, tells you the man, who for a long time (six weeks) worked in direct communication with the Absolute (or with the "information field Vernadsky" - whichever you prefer). Believe me: such "communication" during six weeks it is a lot. Communion with the Absolute requires a lot of tension and mental and physical strengths. Since the obtained by me information nobody needed (this information just nobody wanted to obtain), I was forced to disable this information channel as unnecessary.

      In the relationship, which I described, in the mutual influence of the process and the product of the same process there is THE ESSENCE OF NATURE. Nature, as I wrote, does not need our mathematics of NUMBERS. Numbers - are simply "the toys for a man". However, if a person is not just a "mathematician to calculate", but a serious intellectual scholar, he must not only be able to use numbers. Depth ANALYSIS of processes, the LOGICAL natural relationships searching - that's what differs this researcher from the simple "savvy man to calculate".

      If my thoughts are interesting to the reader, I will continue. The fact that such "so pleasant to Nature" numbers as "e", "pi", "ro" are expressed in our mathematics with such "unpalatable" numbers as it is now, says plainly, that our mathematics is very far from the Nature. If we find the courage and will redirect ourselves to the new math (in which the above described numbers will be "laconic as a hen's egg"), the rate of mankind development will be significantly faster. At the same learning time in all educational institutions will be greatly reduced and in our development we will can achieve such a peak, which was observed previously, apparently, only in the legendary Atlantis.

P.S. What I described above, to me, I think, was not documented in any book of the globe.

19/04/2011 Segrim

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OCKHAM'S RAZOR


      Franciscan monk and philosopher William Ockham in 14-th century formulated the important methodological principle of "Ockham's Razor." This principle remains valid until today and every serious reseacher of Nature is guided by the principle in his studies. This principle states:
      «Nobody should not attract new concepts without extreme need them».
      If you want to say this more transparent and more practical, it will look something like this: from the content of your research, from the rules, laws, hypotheses, which you are formulated, delete all, which you don't need (cut off excess with a "Ockham's razor"). The rest that after such removal will remain, and will be your rule's "main essence". Let's try to apply this principle to the proposed by me method of golden spiral constructing.
      Below I've posted three pictures that illustrate my construction of the golden spiral. At the first figure I led the essence of the constructing till the original "Golden Triangle", which lies at the heart of the process of constructing. To no one doubted that this triangle is enough, on the second picture I showed in details, how on the basis of this triangle helix is unwound as in the direction of its expansion, and in the direction of narrowing. Thus all the "essence" has been reduced to the original "golden triangle", shown on the first image.
      At the second figure I've highlighted in green segment "b", whose starting point is point 1, and the ending point is the point 2. In the third picture I have identified this segment as red, and with the same color I denoted its end points.
      In accordance with the principle of "Ockham's Razor", I would argue that it is point 1 and point 2, as well as "certain specified connection" between them completely predetermine whole the golden spiral process of constructing, what can be seen clearly at the second image.



      And now I clearly articulate the three conditions "living wage", by which is predetermined all further golden spiral process of constructing.
1.) There is the first point - the "point number one".
2.) There is the second point - the "point number two".
3.) The mutual arrangement of these points corresponds to the following condition: the segment length, defined by the initial points, is in the same relation to the horizontal projection of this segment, adjoined to his bottom point, is the same as the total length of the segment and its projection relates to the length of the initial segment.

      These conditions are necessary and sufficient to construct the next point of the golden spiral. In this case we have a choice: if we want to build a "point number three" for the expanding spiral, then we'll lead further building from the point number two, but if we want to get the next point for a shrinking spiral, we'll continue to build from the point number one and then get the "point of minus one". All I said above is easy to observe on the second of those shown above images. Naturally, in the further constructions we must consider as the well-known such concepts as "horizontal" and "vertical", without which the discussion on the relative orientation of points and line segments are meaningless.

      And now I'll come to this question with the other side. All this "other side" of the issue I expressed in these two schemes. By the first of these schemes is constructed an expanding spiral, and by the second one - a shrinking spiral.


      That is, there it is... Whether we like it or not, but a "ubiquitous trio", and the law of world harmony with it manifested here too.
      I think that now it will be clear to all, which spirals I had in mind when I formulated for publication in the newspaper "Anomaly" my law of world harmony and my explanations to it.
      At this stage I will allow myself this section of the presentation to consider as completed. Thank you for your attention.

06/28/2010 Segrim

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