Swiss mathematician Jacob Bernoulli (XVIIv.)
bequeathed to carve on his tombstone
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
"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
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
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
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
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:
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
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
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
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
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."
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
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
P.S. What I described above, to me, I think, was not documented in any book of
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.