As everybody knows pi=L/(2*R)=L/D is the ratio of the circumference to the diameter for any circle. It is an irrational and transcendental number and since now we have discovered over 10^12 digits of it. But pi is the 'governor' of the geometry of our local universe and is apparent everywhere, even in the general relativity main equation. The question is very simple: Why is pi=3.14159? Which is the Law of the Laws?
irrationality of pi:
It is widely believed that the great Swiss-born mathematician Leonhard Euler (1707-83) introduced the symbol π into common use. In fact it was first used in print in its modern sense in 1706 a year before Euler's birth by a self-taught mathematics teacher William Jones (1675-1749) in his second book Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics based on his teaching notes.
Before the appearance of the symbol π, approximations such as 22/7 and 355/113 had also been used to express the ratio, which may have given the impression that it was a rational number. Though he did not prove it, Jones believed that π was an irrational number: an infinite, non-repeating sequence of digits that could never totally be expressed in numerical form. In Synopsis he wrote: '... the exact proportion between the diameter and the circumference can never be expressed in numbers...'. Consequently, a symbol was required to represent an ideal that can be approached but never reached. For this Jones recognised that only a pure platonic symbol would suffice.
The symbol π had been used in the previous century in a significantly different way by the rector and mathematician, William Oughtred (c. 1575-1 660), in his book Clavis Mathematicae (first published in 1631). Oughtred used π to represent the circumference of a given circle, so that his π varied according to the circle's diameter, rather than representing the constant we know today. The circumference of a circle was known in those days as the 'periphery', hence the Greek equivalent 'π' of our letter 'π'. Jones's use of π was an important philosophical step which Oughtred had failed to make even though he had introduced other mathematical symbols, such as :: for proportion and 'x' as the symbol for multiplication.
The irrationality of π was not proved until 1761 by Johann Lambert (172877), then in 1882 Ferdinand Lindemann (1852-1939) proved that π was a nonalgebraic irrational number, a transcendental number (one which is not a solution of an algebraic equation, of any degree, with rational coefficients). The discovery that there are two types of irrational numbers, however, does not detract from Jones's achievement in recognising that the ratio of the circumference to the diameter could not be expressed as a rational number.
http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi
Patrick, not exactly. Even if we find the rule for digits of pi in our Uneverse, probably for another Universe the relevant rule could be different!
π is independent of the geometry. Wether the universe is hyperbolic or euclidean, the series ∑∞n=0(−1)n2n+1 still converges to the same number.
pi is the name given to that number which results as the ratio of the circumference to the diameter of the circle in classical planar (Euclidean) geometry. In other geometries, such as spherical ones, this quotient is no longer equal to pi = 3.141592163..... In fact, in these other geometries, the sum of the inner angles in a triangle is no longer equal to pi (in radians). The deviation of the sum of angles, and the deviation in the already mentioned quotient, from the Euclidean framework, is expressed in terms of the curvature of the manifold (in the example, the curvature of the sphere).
Since in planar geometry the curvature is exactly 0, the ratio is exactly pi, and the sum of the inner angles of a triangle is also pi.
However, in general geometries, in which curvature may be not constant, these values may differ from point to point, and not be equal to 3.141592163 any more.
On the other hand, regarding the infinite decimal digits of pi, there is a (to be proven) property of pi called normality. Since pi is an irrational, trascendent and (let us assume) normal number (http://en.wikipedia.org/wiki/Normal_number), it encodes all possible numbers in its decimal part.
This means that if you think of any number sequence (expressed in any base) and with any length, it will occur INFINITE times within the decimal part of pi.
But this has an additional funny consequence: If you associate any unique character in a text a given number (such as the ASCII or the UTF-8 coding), and concatenate all numbers, this will be inside pi. So, the complete script of "Gone with the wind" or even this post (or their ASCII representations) are in pi.
Partial history of pi:
The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[23] In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[23]
In India around 600 BC, the Shulba Sutras (Sanskrit texts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[24] In 150 BC, or perhaps earlier, Indian sources treat π as \scriptstyle \sqrt{10} ≈ 3.1622.[25]
[23,24,25] refer to
Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation by Catriona and David Lischka.
That some authors have a pi in the field equation for the gravitational field has the same reason as why some authors have it in Maxwells equations. Einstein wrote his equation with a coupling constant kappa and no pi.
If you name pi and e the two numbers that describe the solution of z' = z (e growth constant on the real axis and pi periodicity on the imaginary axis) then these e and pi have the same value in any universe (since they are defined and fixed in the universe of logic). If you give the name pi to such physics-dependent quantities like the ratio between certain curve-lengths than it's your fault that it does not always come out as the same.
Domingo,
you wrote "pi = 3.141592163.....".
The last three digits of your sequence are not correct.
They should be 653... instead of 163... ;)
Dear Demetris,
I do not know if pi will unveil 'the law of the laws' what I know (for having computed it, I'm not so good in theory I'm a computatinal guy) that the complexity of the pi sequence of digits is lesser than the complexity of a microphone 'ope to the environment'...It is like the Shakespeare motto 'There are more things in nature than in your philosophy..' but you are right the problem is puzzling, and the fact pi is more deterministic 8as sequence of digits) than material noise, tells us that there is some 'law' on behind..I attached tha paper here below..
Pi occurs in all sorts of contexts not just as L/D ratio. If you define pi as:
4−4/3+4/5−4/7+… you can be sure that the answer you get will be the same in any other universe.
irrationality of pi:
It is widely believed that the great Swiss-born mathematician Leonhard Euler (1707-83) introduced the symbol π into common use. In fact it was first used in print in its modern sense in 1706 a year before Euler's birth by a self-taught mathematics teacher William Jones (1675-1749) in his second book Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics based on his teaching notes.
Before the appearance of the symbol π, approximations such as 22/7 and 355/113 had also been used to express the ratio, which may have given the impression that it was a rational number. Though he did not prove it, Jones believed that π was an irrational number: an infinite, non-repeating sequence of digits that could never totally be expressed in numerical form. In Synopsis he wrote: '... the exact proportion between the diameter and the circumference can never be expressed in numbers...'. Consequently, a symbol was required to represent an ideal that can be approached but never reached. For this Jones recognised that only a pure platonic symbol would suffice.
The symbol π had been used in the previous century in a significantly different way by the rector and mathematician, William Oughtred (c. 1575-1 660), in his book Clavis Mathematicae (first published in 1631). Oughtred used π to represent the circumference of a given circle, so that his π varied according to the circle's diameter, rather than representing the constant we know today. The circumference of a circle was known in those days as the 'periphery', hence the Greek equivalent 'π' of our letter 'π'. Jones's use of π was an important philosophical step which Oughtred had failed to make even though he had introduced other mathematical symbols, such as :: for proportion and 'x' as the symbol for multiplication.
The irrationality of π was not proved until 1761 by Johann Lambert (172877), then in 1882 Ferdinand Lindemann (1852-1939) proved that π was a nonalgebraic irrational number, a transcendental number (one which is not a solution of an algebraic equation, of any degree, with rational coefficients). The discovery that there are two types of irrational numbers, however, does not detract from Jones's achievement in recognising that the ratio of the circumference to the diameter could not be expressed as a rational number.
http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi
I attempt to think about pi as a solution of a superior Least Action Principle.
But, pi is a transcendental number, so it cannot be a solution of a common stationarity condition of every real valued 'objective function' that we can built 'from scratch'. I think that exists a superior to human civilization which has overcome our differential equations trap (z'=z) for describing a Universe. That civilization works with different 'apparatus' (still unknown to us) and in its framework such a question probably can be answered, because there is no need of algebraic equations to be used.
But, even so, what kind of Action could be an action or quantity or logic such that when we take the extremum of it we end up to the true value of pi?
Could it be something involving tensor geometry or a thermodynamic axiom?
[For a brief description of Least Action Principle look here:
http://en.wikipedia.org/wiki/Principle_of_least_action ]
@Janamejay & Issam: I think it is a good exercise to try to find proper geometrical sequences, relevant to the series you have written, such that their infinite sums converge to pi.
Dear Alessandro: So, the result about pi is that it is not so chaotic like logistic map and has a rather normal behaviour under RQA. What does this imply for the hidden law of pi-generator?
Dear Ulrich, I agree that our differential equations framework gives us the answer you wrote, but I' d like a 'superior apparatus', which does not exist.
@Domingo
your contribution seems to imply that the decimal digits of pi form a equidistributed random sequence. My question: do you know about proofs of this or of related properties?
@Ulrich
There is no actual proof of the normality of pi. It is strongly conjectured that such random equidistribution of digits occurs, but only based on empirical evidence...
The same is conjectured for every irrational algebraic number.
@Domingo,
thank you very much for your instructive note. I was not aware that my unsharp notion of an 'equidistributed random sequence' has a well-defined mathematical formulation as the concept of 'normality' about which Wikipedia tells us many interesting facts.
Many years ago (probably in 1965) I brought up the question of the statistical properties of the digits of pi during lunch in the 'Mensa' in a group of physics students. Next table there sat a solid state physics professor and angrily let us know that young physicists should not waste their time with such useless considerations. This verdict did not prevent me from looking in our library for information on the problem, without any success. Such things became much easier these days.
@Demetris
I don't understand: you would like an apparatus which does not exist? Sounds childish. Are there reasons?
@Ulrich, I ' d like the road to escape from the "wall" of non answer, that' s all.
Imagine that you are near the year 1613 and you are trying to answer difficult questions of that era, such as: "Which is the law of motion of a falling particle?" Could you be able to answer that question that year? No. After Newton and Leibniz, who founded Calculus, all those questions have been answered with accuracy. Now, in order to answer a whole set of questions, like: "Why is pi=3.14...?, "Why is τηε fine-structure constant a=1/137?", "Why is gravitational constant G=6.67×10−11 N·(m/kg)2 " and many other similar questions, we have to build a new work-frame.
What could this frame like? It is another interesting question.
A cosmological comment to the editors of RG: I write local universe without capitalized U because I think that there exist many or even infinite universes. If we study the set of all universes probably we can write its name with initial Capital letter.
Pi is the ratio of circumfence and diametar of any circle. It doesn't have to be equal to 3.14159... It depends on the way you measure these lengths. If you measure lengths with an euclidean metric (which is a standard way to measure in the plane) the ratio becomes 3.14159... but if you use some other metric, Pi ranges between 2 and 4. For example, in the Manhattan metric Pi = 4.
@Konrad, your comment is just expanding the question: Why is pi equal to its measured values according to the chosen norm?
Assume that the answer to your question is 42, what would you say to that ? The point is, even if we go off and try to search for an answer, we just might not recognize it as THE answer.
http://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker%27s_Guide_to_the_Galaxy#Answer_to_the_Ultimate_Question_of_Life.2C_the_Universe.2C_and_Everything_.2842.29
Robert, since everything what Miles Mathis 'proved' equals to nothing, I will just ignore your post.
Simple answer. Pi is a defined constant. It doesn't need to represent anything in the physical universe as it is a mathematical concept. Nothing magical. Keep in mind you forgot the ellipses on Pi in your question as it is irrational.
No Robert, I will take the wisdom of stating a mathematical claim as a fact only if I can (mathematically) prove it.
This particular thing that I claim is actually derived from the definition of arc length and by substituting the norm for some other equivalent norm. I can post the proof in detail if somebody is interested and then we can check if my proof is correct step by step, or disprove the whole thing. It all should be a basic calculation in analysis.
Now I see in detail that Miles Mathis doesn't do a pretty good job at really proving Pi = 4 except for some handwaving so I deleted his "proof" as a source.
P.S: The definition of arc length I have in mind is this:
http://mathworld.wolfram.com/ArcLength.html
Some geometrical properties, under the common Euclidean norm:
1)The surface of a unit disk is pi
2)The volume of a unit 3-sphere is 4/3*pi
3)The length of a unit circle is 2*pi
4)The surface of a unit 3-sphere is 4*pi
It seems that pi is something like a 'base' for many unitary objects.
For the unit ball in many dimensions look here:
http://en.wikipedia.org/wiki/Unit_sphere
Everything is a power of pi!
In this thread pi is mainly associated with geometric figures, the circle being the most prominent of those, but related quantities are listed in the previous note by Demetris. This reflects everybodies first steps into mathematics. With some experience added, the arithmetic origins of pi come into view: e.g. the apperance as sums of powers of reciprocal integers which are formalized as Riemanns zeta function:
pi²/6 = zeta(2), pi⁴/90 = zeta(4) or in Stirling's formua n! ~(n/e)^n sqrt(2 pi n). Trying to condense all these appearances of pi into a "Law of the Laws" according to Demetris' question looks far-fetched to me.
As I see the scientific community did not find the above question a 'good' one. Sorry for asking it. Of course I feel bad, since I had the illusion that RG could be a proper forum for asking ' transcendental questions', but who knows, maybe is my fault. Anyway, overcoming our scientific egotism will help us to reach the next level of scientific consciousness and then, probably, questions like the above could be treated more kindly than now.
Dear Demetris,
About Pi being a trascendental number, you may find useful: http://en.wikipedia.org/wiki/Trascendental_number
and the notes below this article.
The value of 'pi' is found by counting the number of words in the phrase "how i need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." i.e. 3.14159265358979
Regards,
Nitish
It is because of necessity. The ancient mathematicians discovered that the ratio of any circle's circumference to its diameter is always constant, and that was the number pi first defined. But the advancement of mathematics united its previous results and various branches, since the possibility to approximate value of pi by infinite number of polygons (as did by Archimedes up to 96-gons), for example, do not contradict the fact that pi is irrational as proved later by Lambert. Such developments, changed our mental pictures that pi must be imposed entirely to the geometric figures, since it was also known that pi can be associated with numbers, functions, and other mathematical entities, for instance in Machin's formula pi=4 arctan (1/5) - arctan 239. Arctangent function itself has a certain characteristic structures, therefore, it is by necessity that pi must be irrational and equal to 3,141592653589793... . It is also by conclusion follows the necessity that pi must be transcendental if it is brought to the concepts of algebra. Physicists try to understand and describe how universe works in unchanged forms of mathematics as the idealizations of real objects, that's why pi also appears in some fundamental equations of physics, but we know that in nature there is no any perfect circle, therefore, we can not say that special properties of pi is the most fundamental law of all physical laws.
@Fenda, pi is not the 'governor of all physical laws', but, I think, only of the geometry. Other great numbers are e=2.71828... (the governor of time or space evolution due to z'=z) and phi=1.618...(the governor of the beauty due to the golden section). My point is that all those numbers are the characterization set of our local universe and I think that another, local universe too, should governed by other values for the same quantities like pi,e,phi above.
Pi appears naturally in geometry, but only in dimension 2 or higher. In dim=1 it may exist naturally (I think) only as pi^0=1, thus going undetected/unnoticed. Funny, isn't it?
@Marek,
what is geometry in dim=1. Any interesting theorems? I know well that mechanics is already interesting in 1D, but geometry?
@Ulrich,
Yes, it is (1D geometry) degenerate, nevertheless it exists. Anyway, I see that I was too quick. The sum of angles even in a 1D triangle is equal to \pi again!
Dear Demetris,
I agree with you about the natural number e as the constant that govern evolution processes occur in nature, since real objects in space and time can be reduced into real number (e.g. radioactive isotopes or population) or imaginary number (e.g. wave or electrical impedance), but it is rather different case for the golden section: the word beauty is not well defined, and its realm is subjective human psychology. Mathematicians find the beauty of phi in its algebraic and arithmetical properties, for instance, the possibility to express Fibonacci numbers (which follows a special structure that eventually exists in nature, e.g. in plants) as the function of phi. But for the artists, the beauty appears visually as the comparison of the two measures, which they do not need to know exactly how to express in number and only perceive it more intuitively. It is also interesting to notice that many papers have been written about the golden section in Mozart's compositions. Therefore, to make a philosophical assertion that a set of special numbers play fundamental role in processes in both physical and psychological universe, we should assure first that there is a single unity in both realms. Mathematicians also, always seek unity, they have discovered remarkable formulas that relates two or more special numbers.
Regards.
@Fenda, I like this question, but if I keep serving it my RG reputation will be destroyed, since I still collect negative votes! :)
Oh Fenda,
you should have refreshed your memories befor writing this. Fibonacci numbers can be calculated from the golden section!!!! (Nothing known to me with pi).
Some links about phi or Phi, it depends on the writer:
http://en.wikipedia.org/wiki/Golden_ratio
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html
Probably I shall open a thread for the other two numbers (e, phi), but maybe later, not now.
Oh Ulrich,
I should have noticed that phi is not pi. Actually, I never encountered the golden section under the name phi. Sorry, Fenda for having suspucted you of a mistake which actually was mine.
Ulrich,
"Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the reciprocal of the golden ratio, 1/φ."
http://en.wikipedia.org/wiki/Golden_ratio
Dear Demetris Christopoulos Any change in "frame of reference" can make difference. Try it.
@Raja, do you mean about the value of pi? If yes, I speak for the inertial frame of reference for the circle under consideration.
Dear Demetris Christopoulos, its an idea which might can work with high probability. I think change in shape/geometry & angels of circle can tell some thing new. Consider a circle which looks like a circle but having some different internal shapes and properties from circle. I cant explain it well with out diagram but you may imagine a two bubbles but they looks like one circle but in internal shape it have more depth than a one circle. I think it will work because the frame of reference gets changed due to advance level of geometry, as Nature also used to store the Data/traits in circles.
Dear Dimitri, I know Pythagoras would have liked your question very much indeed. However, in the same way that I cannot find any way as to how to answer the question "Why is the charge of the electron what it is?", or "why is the value of G (the gravitational constant) what it is?", it becomes even more difficult to answer questions relating to certain mathematical concepts. I don't think that the question "why is sqrt(2)~1.4142?" is more "shallow" than your question... to put in another way, do you ask this question because you think there is some hidden truth in the decimal number expansion of π (in the sense of Pythagoras) ?
Dear Yianni, the starting point is a cosmological one: Our local universe is a part of a greater Set which contains all local universes and which we could name as the Universe with U. Thus for every local universe it has to be a value for its relevant pi, phi, e, sqrt(2) numbers etc. according to an unknown to us Law. That is the question: who is that Law? And, how, we the ants of a 3d cosmos can reach the way to answer it...
Dear Raja Khan, in order to avoid such problems, we are always concerning the proper inertial frame of reference, ie the frame relative to which our circle is always stationary.
@Vitaly, because 1-1=0, you could define x-x=epsilon, I am not speaking about Algebra, but rather about classes of universes.
Dear Dimitri, I don't think you will ever find any answer that even remotely becomes satisfactory. However, even though this is in the realm of meta-physics (but not meta-mathematics), I think that the values of π and e and sqrt(2) are purely mathematical concepts that have to hold no matter what universe we live in. So, even if there are "local universes", and multi-verses or whatever, I strongly believe that the values of these constants will be the same in any "alternate reality", "alternate universe", or "remote neighborhood of the universe". For example, if you travel on a spaceship and go very near a black hole (where the geometry will be quite different from ours), the value of π as the ratio of the circumference of an *ideal circle* on an *ideal plane* over its diameter will still be the same as here on Earth, since it's a pure construct of mathematics (and eventually, our minds).
I find more intriguing (and likely fruitful) questions regarding physical constants, such as the value of the charge of the electron, or the value of the speed of light, which may shed some light into other unexpected connections within our reality, but I'm not a physicist by training...
Dear Yianni, can you accept the next Axiom?:
"The Universe (:=the set of all local universes like ours) has to be more complex than any mathematical concept created by human beings"
“Why is pi equal to 3.14159?"
Dear Dimitris,
I understand that all geometric figures are related together apart the circle which is related to them by the constant π.
As this is an approximate value, by thus, geometry is turned into physics, where constants are used, in the form of approximation.
Thus we cannot expect any form of unification under this form of π.
I have worked for lengthy periods in my past, searching this question.
I went to Plato [Timaeus], and studied the original statement regarding the two orthogonal triangles as basic for the structure of solids.
These are the orthogonal isosceles and the orthogonal scalene.
I made my interpretation as to the form of the scalene one, which inscribed in a circle [after searching specially contrived conditions for relationships with it], get relationships with parallelogrammes and squares [ complying with Eucleidian theorems]
http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf
http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf .
Similarly this method proves [ in one page the insoluble of 3.141592654…..]
http://www.stefanides.gr/pdf/2012_Oct/PHOTO_11.pdf .
I attach the simplest form of these relationships in the photo below.
Many links may be found giving the solution [ 4/ sqrt{ [sqrt( 5 ) +1 ]/2 }, approx.: 3.14460551…..]
My Links, however, below give the compass and ruler solution, coupled in parallel with AutoCad graphics for checking further.
http://www.stefanides.gr/Html/piquad.htm
https://www.researchgate.net/publication/257469474_Qrt7a_1
P.S. a similar triangle [ but not the same ] is the Magirus triangle given to Kepler and known as Kepler triangle :
Regards,
Panagiotis Stefanides
Data Qrt7a 1
Dear Panagiotis,
Starting from last to first:
1)The equation x^4+16*x^2-256=0 has two real solutions (+-3.1446055110296931442782343433718...) and a pair of complex conjugate
(+-5.0880785980562758570096898469500i) but this is not pi, it is very close to pi...
2)Why is pi an approximate value? It is an irrational number, but why has to be an approximation? And if so, an approximation of what?
3)The status of constants in Physics is a little bit different than the relevant in Maths, by my opinion. In Physics it is matter of definition for the linearly independent set of describing variables, ie a linear science's option. In Maths it is a deeper and open question...
If we accept that everything has to be created from idealized polyhedra, then we have either a linear or a convex combination of them, but irrationality of pi is still present...
Best Regards and congratulations for your detailed work,
Demetris
TO DIMITRIS CHRISTOPOULOS
>…..1)The equation x^2+16*x^2-256=0 has two real solutions (+-3.1446055110296931442782343433718...) and a pair of complex conjugate
(+-5.0880785980562758570096898469500i) but this is not pi, it is very close to pi...>….….2)Why is pi an approximate value? It is an irrational number, but why has to be an approximation? And if so, an approximation of what?....…3)The status of constants in Physics is a little bit different than the relevant in Maths, by my opinion. In Physics it is matter of definition for the linearly independent set of describing variables, ie a linear science's option. In Maths it is a deeper and open question......…If we accept that everything has to be created from idealized polyhedra, then we have either a linear or a convex combination of them, but irrationality of pi is still present...
No need to go to quartic equations. If we think of geometry as referring to the properties of rigid bodies, we would have a hard time to prove that pi is not equal to 355/113.
Dear Panagiotis, I think you have to pay attention to complex numbers since they 'hide' valuable information. Take for example the series:
f(x)=1/(1+x^2)=1-x^2+x^4-x^6+x^8-x^10+...
If you treat it as a real function you have lost since its radius of convergence is 1, actually the open ball B(0,1) in the R^2. Only if you treat it as complex you will find the first singularity at +-i and deduce the right result.
So, try to bring into your models the complex functions.
For example see the conformal plot of f(z)=z^4+16*z^2-256*z for z=-π-πi...π+πi at the attachment.
Hi Dimitri, and sorry for the long delay in your question on whether I accept the axiom "The Universe (:=the set of all local universes like ours) has to be more complex than any mathematical concept created by human beings".
Well, I am with Einstein on this one: "the most incomprehensible thing about the universe is that it is actually comprehensible". So, I believe that we will also some day fully understand all the basic laws of nature (even if we cannot do much with them), in other words I believe that while the complexity of the real world is totally incomprehensible to us (and our computers), the basic laws that govern the universe will be fully discovered and described -in the language of mathematics of course- some day, but we will never be completely sure of course that we got them right. Whether this will have any serious effects in our every-day lives remains to be seen. Note that the above statement in no way "eliminates" or "reinforces" the need/wish/reality of God's existence...
Hello Yianni, you are welcome!
I think that Einstein had right just because everything can be projected with sufficiently high accuracy to a linear space, manifold or whatever depending on the field. More arguments about linearity you can find here:
https://www.researchgate.net/post/Are_you_satisfied_with_our_linear_science
As for the question of fully describe anything: Well even if we find the laws for our local universe, then we shall ask about why such a law and we shall want to go to the next level, that of 'universe design mechanics', i.e. which are the laws needed in order to build a stable universe (stability:=to live enough time in order to produce intelligent life inside it).
That's the reason why I have accepted the concept of increasing complexity: Try and explain a first level. Then wonder about next level. Try again and explain the following level. Then ... and so on.
So, my question actually has to do with the 'geometry of universe mechanics'.
>….Take for example the series:
f(x)=1/(1+x^2)=1-x^2+x^4-x^6+x^8-x^10+...
Please see the Gregory Galperin's method of determining the digits of pi by counting the number of elastic collisions between two balls rolling on a frictionless surface.
http://www.ijera.com/papers/Vol3_issue4/KL3418811903.pdf
Dear Panagiotis, I have something for you. Try to solve the simple equation:
1+1/8*x+1/2*x^2=0
It has the solutions:
x1=-1/8-i*(1/8)*sqrt(127)=-0.125-1.408678458698080*i
x2=-1/8+i*(1/8)*sqrt(127)=-0.125+1.408678458698080*i
The interesting point is that we have for the absolute value:
|x1|=|x2|=sqrt(2)
Is there any interesting geometrical representation for this result?
Dear Dimitri,
0.125 = 1/8 and Squared = 0.015625 = 1/64.
Also, [0.125 /10] + 1= 1.0125 =
=1/0.9876543209876543209876543209876543209876543209876543
[ from WolframAlpha]
So the relevant orthogonal triangle has hypotenuse sqrt(2) small horizontal side 0.125 , and the third side [ i.e 1.408678458698080 ] is related pythagorially by these two sides' values.
The interest is on this form of repeated series of numbers.
Panagiotis
Dear Dimitri,
Further to my above reply, and in connection with its contents, possibly of interest are the following:
[9/4]*[sqrt(2)] = [ 9/8]*[sqrt(8)] = 3.181980515..
[ 3.181980515. ]^2 = {[9/4]*[sqrt(2)]}^2 = 10.125
1/10.125 = 8/81= 0.098765432..
[ WolframAlpha : 0.0987654320987654320987654320987654320987654320987654…
1 - 1- 0.9876543209876543209876543209876543209876543209876544 =
= 0.0123456790123456790123456790123456790123456790123456... = 1/81
{[ 1/ 3.181980515..] = 0.314269681..}
Also, 10.125 = 81/80 and,
NOTES
By using sections of the four solids, we find the relationships, between them i.e. the Icosahedron with the Octahedron, the Tetrahedron and the Cube. In addition, if we add selective sections (one next to the other), of the three solids, Icosahedron, Octahedron and Tetrahedron, we find an angle [epsilon 41.8103149.Deg.=2*arctan{1/T^4}=arctan{1/sqrt (1.25)}] which we find also in a Section of the Dodecahedron.
In this manner we obtain a relationship, of the Dodecahedron with the other Four Platonic Solids. Dodecahedron was considered as the Fifth Solid, mentioned by PLATO in his Timaeus [a Fifth Consistency he states], which “God used it up to Paint the World”, and was given the name AETHER, by the philosophers.
41.8103149.Deg.=2*arctan{1/T^4}=arctan{1/sqrt (1.25)}
[41.8103149 deg = 0.729727656 rad]
Ref:
http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf [ page 17]
and,
http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_GREEK.pdf.pdf [Page 30]
http://www.wolframalpha.com/input/?i=arctan%7B1%2Fsqrt+%281.25%29%7D%3D
http://www.bing.com/videos/search?q=YOUTUBE+PSTEFANIDES&view=detail&mid=88D96B696307B955F1F988D96B696307B955F1F9&first=41#view=detail&mid=88D96B696307B955F1F988D96B696307B955F1F9
Panagiotis Stefanides
http://www.stefanides.gr
Dear Panagiotis,
Many thanks for posting the very interesting book. In addition to your remarkable introduction to the beautiful triangle from Plato's Timaeus, It is very nice to see the philosophy of geometry revealed in this book.
@Panagiotis, please take a look here:
http://mat.uab.es/~xarles/elliptic.html
It is probably close to your favorite topics.
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