Is it Euler’s identity eiπ + 1 = 0
where e is Euler's number (2.718), the base of natural logarithms, i is the imaginary unit, which satisfies i2 = -1, and π (3.14) is the ratio of the circumference of a circle to its diameter. Probably the numbers 0, 1, e, π and i are the most used numbers in mathematics. These 5 numbers play important and recurring roles across mathematics, and are the five constants appearing in the above Euler's identity.
Or could it be one of such seminal theorems as
(a) The number of primes is infinite;
(b) There is no rational number whose square is 2;
(c) π is transcendental;
(d) A continuous mapping of the closed unit disk into itself has a fixed point.
David Wells, the writer of a number of popular mathematical works, suggested for a mathematical statement to be beautiful it must be simple, brief, important, and, obvious when it is stated but perhaps easy to overlook otherwise, surprising. Lots of mathematician think Euler’s theorem (eiπ + 1 = 0) scores high on all four counts. What is your favorite theorem? What others you may think as beautiful as Euler's theorem?
Dear @Mahmoud, we can not avoid Euler, but we have to admmit that he has beautiful theorems. Top Ten beautiful theorems follow:
"A 1988 poll of readers of the Mathematical Intelligencer ranked some of the most well-known theorems in mathematics thus:
1.Euler’s identity, eiπ=−1superscripteiπ1e^{{i\pi}}=-1
2.Euler’s formula for a polyhedron, V+F=E+2VFE2V+F=E+2
3.There are infinitely many prime numbers. See Euclid’s proof that there are infinitely many primes.
4.There are only 5 regular polyhedra
5.The sum of the reciprocals of the squares of the positive integers is π26superscriptπ26\frac{\pi^{2}}{6}. See theBasel problem.
6.A continuous mapping of a closed unit disk into itself has a fixed point
7.The square root of 2 is irrational
8.ππ\pi is a transcendental number
9.Every plane map can be colored with just 4 colors
10.Every prime number of the form 4n+14n14n+1 is the sum of two square integers in onlyone way..."
http://planetmath.org/thetop10mostbeautifultheorems
Dear @Mahmoud, we can not avoid Euler, but we have to admmit that he has beautiful theorems. Top Ten beautiful theorems follow:
"A 1988 poll of readers of the Mathematical Intelligencer ranked some of the most well-known theorems in mathematics thus:
1.Euler’s identity, eiπ=−1superscripteiπ1e^{{i\pi}}=-1
2.Euler’s formula for a polyhedron, V+F=E+2VFE2V+F=E+2
3.There are infinitely many prime numbers. See Euclid’s proof that there are infinitely many primes.
4.There are only 5 regular polyhedra
5.The sum of the reciprocals of the squares of the positive integers is π26superscriptπ26\frac{\pi^{2}}{6}. See theBasel problem.
6.A continuous mapping of a closed unit disk into itself has a fixed point
7.The square root of 2 is irrational
8.ππ\pi is a transcendental number
9.Every plane map can be colored with just 4 colors
10.Every prime number of the form 4n+14n14n+1 is the sum of two square integers in onlyone way..."
http://planetmath.org/thetop10mostbeautifultheorems
I would like to accept your suggestion, since the equation involves all irregular numbers in mathematics: e, p, i!
"Euler's Magic Formula. eiπ + 1 = O.
This has been called a "magic" formula because the five most important numbers in mathematics (0, 1, e, i, π) are tied together in a very simple equation. If one thinks about how diversely these five numbers appear - o as the additive identity, 1 as the multiplicative identity, e as the natural exponential base, π as the ratio of the circumference to the diameter of any circle, and i as the imaginary unit, this result is truly amazing."
In Benjamin Fine, Gerhard Rosenberger, The fundamental Theorem of Algebra. Springer 1997, p. 16.
Dear Prof. Dr. Mahmoud Omid and others,
I have my greetings to you and your staffs from my country, Albania!
Dear colleague, Prof. Dr. Mahmoud Omid, I have a suggestion about your question, and concretely you asked eveyone: What is the most beautiful theorem in mathematics? My proposal is to change the word beautifull with other word. In mathematics no beautiful, ...
Dear Prof. Dr. Mahmoud Omid and others, for me!
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem can be written as anequation relating the lengths of the sides a, b and c, often called the "Pythagorean equation".
a2 + b2 = c2,
where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.
Although it is often argued that knowledge of the theorem predates him the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC) as it is he who, by tradition, is credited with its first recordedproof. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.
Dear Colleagues,
Good Day,
I DO love Mathematics, so I see all the theorems beautiful, Here are a list of some of them:
Gödel’s Incompleteness Theorems
Fourier's Theorem (Function Theory, Joseph Fourier)
Euclid's Theorem of the Infinitude of Primes (Number Theory, Euclid)
The Minimax Theorem (Game Theory, John von Neumann)
Schubert's Prime Knot Factorization Theorem (Knot Theory, Horst Schubert)
Cantor's Theorem (Set Theory/Transfinite Analysis, Georg Cantor)
Cauchy's Residue Theorem (Complex Analysis, Augustin-Louis Cauchy)
The 4-Color Theorem
Fermat’s Last Theorem
.......
Maybe one should write Euler's formula, as Mahmoud did, in the form ei𝝅+1=0, since then ei𝝅 represents -1 in that adding one to it equals zero, while introducing at the same time "the strangest number in the universe" the Zero.
Anyway my vote for the most beautiful formula so far discovered is Euler's formula, since it relates the transcendentals e, 𝝅, the imaginary unit i, as well as the number one, a representation of -1 and the "controversial" Zero.
I'll have to agree with the passionate comments posted as I am not a Mathematician.
Best regards,
Debra
Dear All,
Thank you for your beautiful answers, new suggestions, and detailed explanation. Here I quote Keith Devlin writing of eiπ + 1 = 0: "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence."
Euler Himself is the Beauty of Mathematics! so whatever he produced is the beauty
I once had a professor exclaim, "Euler's Identity proves the existence of God!" While questionable, the claim does comport with Mahmoud's assigning beauty to an equation. I find all equations "beautiful" that are purposeful to the problem at hand. An equation's beauty increases when the equation has a physical interpretation.
Euler's identity includes two fixes to mathematics. The most recognized is "i" because it fixed the problem of the square root of a negative number. It was difficult for many to accept "i" so it was called imaginary. The other and more important fix was to invent zero as a place holder. These fixes make Euler's Identity more interesting, because it could not exist without them.
Dear all,
today, I begged Arvjen, Arnisa and Vera to accept my answer provided to our colleague about your question, namely: "What is the most beautiful theorem in mathematics?"
My answer was: "Dear Prof. Dr. Mahmoud Omid and others,
I have my greetings to you and your staffs from my country, Albania!
Dear colleague, Prof. Dr. Mahmoud Omid, I have a suggestion about your question, and concretely you eveyone Asked: What is the Most Beautiful theorem in mathematics? My proposal is to change the word beautifull with other word. In mathematics no beautiful ... "
Answering that were more deserving of vemdje estimates, such as that of Prof. George, Prof. Mohamed etc.
Answering that Were more deserving of attention estimates, such as that of Prof. Gregorio, and so
Answering, opinion or anything else about my opinion given in the form of an answer was not directly opposing arguments? Why? ...
Please, undersatand me, now I ask you, was right my answer?
... AND MY ANSWER IS NO ... I was very wrong with my answer given to you! Thanks a lot for everything!
.... BECAUSE ...
...
... the correct arguments are given during the time that the Mathematics is beautiful by:
Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158
It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true.
Bertrand Russell (1872-1970), The Study of Mathematics
Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Aristotle (384 B.C.-322 B.C.), Poetics
Beauty depends on size as well as symmetry.
J.H.Poincare (1854-1912), (cited in H.E.Huntley, The Divine Proportion, Dover, 1970)
The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.
J.Bronowski, Science and Human Values, Pelican, 1964.
Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process.
J.W.N.Sullivan (1886-1937), Aspects of Science, 1925.
Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of Mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.
G. H. Hardy (1877 - 1947), A Mathematician's Apology, Cambridge University Press, 1994.
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
Lawrence University catalog, Cited in Essays in Humanistic Mathematics, Alvin White, ed, MAA, 1993
Born of man's primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art.
I.Newton, Letter to H.Oldenburg, the Secretary of the Royal Society, October 24, 1676, in A Source Book in Mathematics, D. J. Struik, ed, Princeton University Press, 1990
I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz's method for obtaining convergent series is certainly very elegant...
Jane Muir, Of Men & Numbers, Dover, 1996.
Gauss: You have no idea how much poetry there is in the calculation of a table of logarithms!
F.Dyson, in Nature, March 10, 1956
Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half-joking, "My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." (Herman Weyl (1885-1955))
O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956
To Goethe again we owe the profound saying: "the mathematician is only complete in so far as he feels within himself the beauty of the true."
O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956
"A mathematician," said old Weierstrass, "who is not at the same time a bit of a poet will never be a full mathematician."
Jakob Bernoulli, Tractatus de Seriebus Infinitis, 1689 (quoted in From Five Fingers to Infinity, F.J.Swetz (ed), Open Court, 1996)
So the soul of immensity dwells in minutia.
And in narrowest limits no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!
S.Lang, The Beauty of Doing Mathematics, Springer-Verlag, 1985
Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?"
Scott Fitzgerald, This Side of Paradise
The infinite boredom of conic sections ... their calm and tantalizing respectability ...
etc...
Dear friends and colleagues it was the surprise of the ordinal in RG ...
If, however, concludes that each of you have worried, I apologize, excuse me!
All of you who are involved in the debate, I wish all the best!
One of the most beautiful, brief, simple and important to me, is the Pythagorean theorems, not by elemental and ancient has outlived its usefulness, beauty and simplicity in mathematics (algebra and trigonometry)
The Pythagorean theorem refers to a right triangle:
The sum of the squares of the legs equals the square of the hypotenuse
C2 = a2 + b2
Dear Prof. Dr. Mahmoud Omid and others, for me I repeat:
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem can be written as anequation relating the lengths of the sides a, b and c, often called the "Pythagorean equation".
a2 + b2 = c2,
where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides ...
Dear Prof. Dr. Bashkim Mal Lushaj and All, Thank you for your contributions so far. This is one way to show why the mysterious formula by Euler is 'beautiful' as one limerick puts it:
e raised to the pi times i,
And plus 1 leaves you nought but a sigh.
This fact amazed Euler
That genius toiler,
And still gives us pause, bye the bye.
This is a very beautiful identity with complex numbers. Feynman has chosen it as the most beautiful identity and many important people, but I prefer another no so obvious
log(-1)=i
and simpler.
Dear Prof. Mahmoud, thank you and others too!
Yes, I know something about it, but I repeat again, I am for the most beautiful is
"Pythagorean equation".
a2 + b2 = c2,
where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides ...
See you in next question ...
Taylor's theorem is also one of the most beautiful theorems in pure as well as applied mathematics.
In the fall 1988 issue of the Mathematical Intelligence [Please See the attached PDF], a scholarly quarterly journal of mathematics sponsored by the prestigious publisher of mathematics books and journals, Springer-Verlag, there was the call for a vote on the most beautiful theorem in mathematics. Readers of the Intelligence, consisting almost entirely of academic and industrial mathematicians, were asked to rank twenty-four given theorems on a scale of 0 to 10, with 10 being the most beautiful and 0 the least. The results, from a total of 68 responses, were announced in the summer 1990 issue. Receiving the top average score of 7.7 was eiπ +1 =0. The scores for the other theorems above, by comparison, were 7.5 for (a), 6.7 for (b), 6.5 for (c), and 6.8 for (d), See question details for (a)-(d) explanation. Accordingly, eiπ +1 =0 is the most beautiful equation in mathematics!
http://press.princeton.edu/chapters/i9438.pdf
but the article says this is NOT an equation ...For me, It implies that transcendence of e is in the orientation of pi , no? And I don't quite get the transcendence of pi itself... loved this limerick ( with my own daring variation) :
ego raised to the piety times i,
And plus 1 begs you naught but a sign.
This fact amazed Euler
That genius toiler,
And still gives us mortals aches without a cause, bye the bye.
so if one concurs that this is the most beautiful theorem in math, it is also the uttermost self-evidently useless kind, for all matter of facts worldly concerned.
Dear @Costas has pointed out that Euler's identity involves in the same e,π,i. Furthermore, it involves 0 and 1, the most important mathemathical constants, as well as three important mathematical operations addition, multiplication and exponentiation!
Dear Ljubomir Thank you for your clarification, but I already had this in mind when I asked the question: " Probably the numbers 0, 1, e, π and i are the most used numbers in mathematics. These five numbers play important and recurring roles across mathematics, and are the five constants appearing in the above Euler's identity."
Dear @Susan Laird, Thank you for your comments. You said "but the article says this is NOT an equation .." You are right I did not say in the question I said it is it "theorem", a very beautiful theorem, indeed. Some mathematicians call eiπ +1 = 0 ‘‘an expression of exquisite beauty’" In fact, eiπ +1 = 0 isn’t an equation and it isn’t an identity. It is a formula or a theorem. This is explained in the mentioned article, too.
Dear all,
I think that all of debaters including in this very nice debate are winners, because each of theorem in mathematics is the most beautiful ... let's believe and hope in that I write ... I wish all the best having wise using of teormes of mathmatics during our works in our life ...
Dear Prof. Dr. Bashkim Mal Lushaj, thank you for your nice comments. Here I give some link about Pythagorean theorem, which is a celebrity. Please see the attached article to see how this 2500-year-old idea can help us today to understand computer science, physics, even the value of Web 2.0 social networks! It is time to make Pythagorean Theorem more beautiful. It works in geometry and it works with numbers. I also include a graphical proof of the theorem.
http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/
Dear our colleague, Prof. Mahmoud Omid, I thank you very much for your surprise given to me with "Surprising Uses of the Pythagorean Theorem"!
“Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.” - Deepak Chopra
I added the above quote for everyone participating in this discussion, with special thanks to Prof. Bashkim Mal Lushaj, Arnisa Lushaj, Vera Malsia, Arvjen Bashkim Lushaj, and Arnisa Lushaj for their encouraging words.
Dear our colleague, Prof. Mahmoud Omid, I agree with you: “Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.” ― Deepak Chopra
World's beauty can be depicted by the poet and the mathematician
Pure mathematics is, in its way, the poetry of logical ideas. -Albert Einstein
“Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.” - Francis Bacon
However there is a difference between a poet and a mathematician:
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” -G.K. Chesterton
I think, the most beautiful theorem in Mathematics is Newton's Binomial Theorem.
https://en.wikipedia.org/wiki/Binomial_theorem
http://www.intmath.com/series-binomial-theorem/4-binomial-theorem.php
1.Theorem (Fundamental Theorem of Algebra): Every polynomial p:C→C of degree n has n roots (counting multiplicity) in C;
2. Euler’s identity eiπ + 1 = 0
3. The Pythagorean Theoem (Geometry, Pythagoras).
4. Euclid's Theorem of the Infinitude of Primes (Number Theory, Euclid)
5. The Brouwer Fixed Point Theorem (Topology, Luitzen Brouwer)
6.Cauchy's Residue Theorem (Complex Analysis, Augustin-Louis Cauchy)
Let me show another beauty of Euler's identity, which has not been discussed here yet. The term "Euler's identity" is also used elsewhere to refer to the related general formula eix = cosx + isinx. The principal motivation for introducing the number e, particularly in calculus and AC circuit analysis in electricity, is to perform differential and integral calculus with exponential functions easily.
Let y=eix, Then d(eix)/dx = ieix = iy and ∫ eixdx =-ieix =-iy,
That is to say we can eliminate differential/integral operators altogether, i.e, they are gone/vanished and we end up with manipulating simple algebraic equations. This is another beauty! This property is very useful when we talk about alternating current and move from time-domain to frequency-domain.
One of the most useful identities for working with square matrices is that the determinant commutes the exponential or with the derivative. Working with eigenvalues associated to operators this can be fundamental in Quantum Mechanics or in Statistics.
[det,exp]=0
[derivative,det]=0
being det the determinant operator and exp the exponential on the square matrices. The bracket is the commutator.
Hi dear doctor.
I think :eiπ+1=cos(a)+isin(a)+1=cos(π)+isin(π)+1=0
Thanks.
Dear Yaser, Thank you for your answer. It is more elegant if we multiply both sides of eiπ+1=0 by e--iπ/2:
From eiπ+1=0, we get e--iπ/2(eiπ+1) = e--iπ/2(0)=0
But the LHS becomes eiπ/2+e-iπ/2 = 2cos(π/2) = 0 = RHS Q.E.D.
P.S.: This also shows why cos(π/2) = 0. Yet another beauty!
Dear Mahmoud, Pythagoras theorem is the most beautiful in mathematics. People, who aren't very good in mathematics can easily understand a2+b2=c2.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty. In particular, the Euler’s identity has been likened to that of the soliloquy in Hamlet.
People who appreciate the beauty of mathematics activate the same part of their brain when they look at aesthetically pleasing formula as others do when appreciating art or music, suggesting that there is a neurobiological basis to beauty.-
See more at:
https://www.ucl.ac.uk/news/news-articles/0214/13022014-Mathematical-beauty-activates-same-brain-region-as-great-art-Zeki
Dear Ljubomir, Following your top 10 beautiful theorems selection, here I present these 11 Most Beautiful Mathematical Equations. Mathematical equations aren't just useful, many are quite beautiful. And many scientists admit they are often fond of particular formulas not just for their function, but for their form, and the simple, poetic truths they contain. While certain famous equations, such as Albert Einstein's E = mc2, hog most of the public glory, many less familiar formulas have their champions among scientists.
http://www.livescience.com/26680-greatest-mathematical-equations.html
Maths and nature - harmony, symmetry, perfection, beauty: arithmetical progression, geometrical progression, Fibonacci number, golden section (aurea mediocritas).
http://e-science.ru/node/4521
http://wiki.pskovedu.ru/index.php/
http://matemonline.com/2010/09/nas-okruzhajyt-4isla-fibona44i/
http://mydocx.ru/4-5060.html
any equation implies a negation. Most identification relies on an explicit expression. What lacks from an equation and identification to revelation and redemption is the irrational plus transcendence of pure math, such as eiπ +1 = 0 or my incomplete spherical go theorem.
https://www.researchgate.net/post/Do_you_see_Googles_DeepMinds_AlphaGo_defeat_of_professional_9-dan_Go_player_as_good_as_IT_gets
In a new study by Semir Zeki and colleagues [1], they recruited 16 mathematicians at the postgraduate or postdoctoral level as well as 12 non-mathematicians. All participants viewed a series of mathematical equations in the fMRI scanner and were asked to rate the beauty of the equations as well as their understanding of each equation. After they were out of the scanner, they filled out a questionnaire in which they reported their level of understanding of each equation as well as their emotional experience viewing the equations. Leonhard Euler’s identity we are talking here was most consistently rated as beautiful. Other winners of the equation beauty contest included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler’s formula for complex analysis, and the Cauchy-Riemann equations.
[1] Semir Zeki, John Paul Romaya, Dionigi M. T. Benincasa and Michael F. Atiyah. (2014). The experience of mathematical beauty and its neural correlates. Front. Hum. Neurosci., 13 February 2014
http://dx.doi.org/10.3389/fnhum.2014.00068
Dear Irina Pechonkina, Thank you for "Maths and nature" and colorful pictures. You are absolutely right. Mathematics is full of beauty and it comes in all shapes and numbers. Can you guess why nine veils instead of eight? Any dedicated seeker of truth eventually stumbles upon the incredible symmetry and structure of mathematics, which is especially true in fractal geometry involving the integers 1-9. For a most basic example, just take a look at these nine equations:
(1 x 8) + 1= 9
(12 x 8) + 2 = 98
(123 x 8) + 3 = 987
(1234 x 8) + 4 = 9876
(12345 x 8) + 5 = 98765
(123456 x 8) + 6 = 987654
(1234567 x 8) + 7 = 9876543
(12345678 x 8) + 8 = 98765432
(123456789 x 8) + 9 = 987654321
Very amazing, indeed!
http://www.oneworldofnations.com/2015/02/understanding-nine-veils.html
Dear Mahmoud,
Thank you for the formula of Tree of Life, DNA-to our Jasna Davidovic, the infinity
Phi or Golden Ratio (1.618) Myth: “Beauty is in the phi of the beholder.”
What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section.
A template for human beauty is found in phi and the pentagon
Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed cross-cultural surveys on beauty and found that all groups had the same perceptions of facial beauty. He also analyzed the human face from ancient times to the modern day. Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as its foundation, which embody phi in all their dimensions. For more information and other examples, see his site at Marquardt Beauty Analysis.
Please click on the following pentagon to see the animation!
http://www.beautyanalysis.com/
Dear Mahmoud, on the other hand, it's the perverted Probability theory (financial pyramid) which is associated with a fantastic octupus nowadays...
http://fonxl.ru/wallpaper/monstry-suda-kraken-sushhestvo-temnoj-fentezi-okean-more-korabli-lodki-strashno-zhutko-poxozhij-na-prividenie-osminog-vody.html
Stokes' theorem in the exterior algebra of differential forms
∫MdF=∫∂MF
where the integral, on the manifold M, of the derivative d of the F differential form, is the same that the integral of the F differial form on the boundary of the M manifold. This is basis of the concept of cohomogy and homology in mathematics.
Maxwell's equations and light
Mathematics showed that Maxwell's equations (see wiki link attached) had wave-like solutions. Careful mathematical analysis by Maxwell showed that these equations predict EM radiation at this speed c = 2.998X108 m/s. Maxwell proceeded to calculate the speed of those waves. Using some not-so-simple calculus, Maxwell's equations can be used to show that the electric and magnetic fields obey wave equations. The speed c of an electromagnetic (EM) wave is determined by the constants of electricity and magnetism that you know so well:
c = 1/(ε0µ0)1/2 = 2.998 X 108m/s.
where ε0 and µ0 are physical constants known as the permittivity of the vacuum and the permeability of the vacuum. So light is an EM wave: this was realized by Maxwell circa 1864, as soon as the equation c = 2.998X108 m/s was discovered, since the speed of light had been accurately measured by then, and its agreement with c was not likely to be a coincidence.
Today, Maxwell's equations are so well-known that they decorate T-shirts:
https://en.wikiversity.org/wiki/Maxwell%27s_equations
To me the fourth equation is the most beautiful one. In this equation the genius James Clerk Maxwell inserted the displacement current term which gives the wave -like solution to the set of equations.
Maxwell's most notable achievement was to formulate the classical theory of EM radiation, bringing together for the first time electricity, magnetism, and light as manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realized by Isaac Newton.
“One scientific epoch ended and another began with J.C. Maxwell.”
Don’t believe me? Well, I wasn’t the first person to say it – Albert Einstein said it first. When Einstein was asked if he had stood on the shoulders of Newton, he replied: “No, I stand on Maxwell’s shoulders.” And Richard Feynman, another of the 20th century’s greatest physicists said: “…the great transformations of ideas come very infrequently… we might think of Newton’s discovery of the laws of mechanics and gravitation, Maxwell’s theory of electricity and magnetism, Einstein’s theory of relativity, and… the theory of quantum mechanics.”
http://www.famousscientists.org/james-clerk-maxwell/
Mathematics is the language with which God wrote the universe. — Galileo
Dear Colleagues,
Good Day,
For the things of this world cannot be made known without a knowledge of mathematics.
Roger Bacon
“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.” ― Richard Courant
The heat (or diffusion) equation: Yet another beauty, simplicity or elegance of equations or theorems:
Δf = f'
where the Laplace operator, Δ or ∇2, the divergence of the gradient, is taken in the spatial variables. When this equation is solved for the function f with a given condition on its boundary, it will describe how heat would actually flow over time on a specified surface. Is it not astounding that we can describe such a powerful physical law with just 5 symbols?
http://www.askamathematician.com/2012/06/q-are-beautiful-elegant-or-simple-equations-more-likely-to-be-true/
Dear Mahmuod,
Thank you for invitation. I feel obliged to give an answer to your question.
I guess the favorite theorem for a mathematician is “the theorem” that he has in his head and is trying to prove it. The most exciting part of the process of creating a theorem is related with the moment when one finds the proof. Then came the boring period of checking the parts of the proof and writing dawn the details. When you finish the manuscript and send it for publication everything is finished.
Dear Kazaros, Than you for your comment. Let me write a beautiful quote from Michael Faraday. It is on record that when a young aspirant asked Faraday the secret of his success as a scientific investigator, he replied, 'The secret is comprised in three words — Work, Finish, Publish.' I think, the same principle applies here in proving mathematical theorems.
"The beauty of mathematics is experienced when the physical reality objects are formulated using totally abstract mathematical models." (see link).
So, one can see the beauty of Maxwell equations, Heat equation, Dirac function, Fourier transform, Einstein field equation,...
https://en.wikipedia.org/wiki/Mathematical_beauty
"Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told that the young man had gone off to become a poet. Hilbert is reported to have remarked, 'I never thought he had enough imagination to be a mathematician'." -George Polya (1887-1985)
When in the 18th century Euler discovered those formulas which today still delight the mathematical phantasy, he seriously stated that his pencil was more clever than himself. This impression that mathematical structures can include a kind of self-determination concerns me at this time. ... Mathematics and Philosophy attack the world's problems in different ways. Only by their complementary action do they give the right direction. -E. Kaehler
Dear Colleagues,
Good Day,
"The Hundred Greatest Theorems
The millenium seemed to spur a lot of people to compile "Top 100" or "Best 100" lists of many things, including movies (by the American Film Institute) and books (by the Modern Library). Mathematicians were not immune, and at a mathematics conference in July, 1999, Paul and Jack Abad presented their list of "The Hundred Greatest Theorems." Their ranking is based on the following criteria: "the place the theorem holds in the literature, the quality of the proof, and the unexpectedness of the result."
The list is of course as arbitrary as the movie and book list, but the theorems here are all certainly worthy results. I hope to over time include links to the proofs of them all; for now, you'll have to content yourself with the list itself and the biographies of the principals.
Please, press on the link to see the 100 greatest theorems:
http://pirate.shu.edu/~kahlnath/Top100.html
Dear all
A rather interesting formula is given by the following symmetric n×n matrix (k,l=1,2, .. n)
Qkl=(δkl–1/n)exp( i𝝅(k+l–2)/n)
Let us say that n is Avogadros number ≈ 10–24 then every existing computer today would consider this a diagonal matrix. However, as can be easily shown this is nothing but a symmetric form of a so-called Jordan block, i.e. an n×n matrix of zeros except with ones in the superdiagonal.
Question has anybody seen other examples of “symmetric” Jordan blocks?
BOLTZMANN’S ENTROPY FORMULA
S=k log W
The equation describes the tight relationship between entropy (S), and the myriad ways particles in a system can be arranged (k log W). The last part is tricky.k is Boltzmann's constant and W is the number of microscopic elements of a system (e.g. the momentum and position of individual atoms of gas) in a macroscopic system in a state of balance (e.g., gas sealed in a bottle).
Nature loves chaos when it pushes systems toward equilibrium, and geeks call this universal property entropy. Austrian physicist Ludwig Boltzmann laid entropy’s statistical foundations; his work was so important that the great physicist Max Planck suggested that his version of Boltzmann’s formula* be engraved on Boltzmann’s tombstone in Vienna (above).
http://www.wired.com/2011/11/equations-for-geeks/
The number of states D(E) associated to Hamiltonian with E eigenvalues(energies) for a tight-band model in many-body physics can summarized by the simple formula
D(E)= (1/Pi) lim ImG(E+ie)
Where the limite lim is taking on the ImG imaginary part of the Green function when the imaginary infinitesimal ie tends to positive zero.
THE RAZOR'S EDGE OF OUTBREAK: R-NOUGHT
R0 > 1
Brought to mainstream attention by the thriller Contagion, R0, pronounced R-nought, is a very simple figure: It refers to the average number of people an individual infected with a pathogen will go on to infect. If it's less than one, the disease will burn itself out; if greater than one, it will spread. In a world where a flu virus from Mexico can infect millions of people around the world in a matter of months, this equation is as symbolic as it is straightforward.
Image: Subway riders in Mexico City during the 2009 swine flu outbreak. (Eneas de Troya/Flickr)
http://www.wired.com/2011/11/equations-for-geeks/
In my opinion, Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle, is one of the most beautiful theorem in Mathematics.
https://en.wikipedia.org/wiki/Pythagorean_theorem
http://www.cut-the-knot.org/pythagoras/index.shtml
http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html
100 Most useful Theorems and Ideas in Mathematics!
Fine blog and related links are included.
http://artent.net/2012/11/27/100-most-useful-theorems-and-ideas-in-mathematics/
Fermat's Last Theorem
"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
As a result of Fermat's note, the proposition that the Diophantine equation
xn+ yn= zn
where x, y, z, and n are integers, n > 2 no nonzero solutions for has come to be known as Fermat's Last Theorem. It was called a "theorem" on the strength of Fermat's statement, despite the fact that no other mathematician was able to prove it for 300 years.
In 1993, a bombshell was dropped. In that year, the general theorem was partially proven by Andrew Wiles (Cipra 1993, Stewart 1993) by proving the semistable case of the Taniyama-Shimura conjecture. Unfortunately, several holes were discovered in the proof shortly thereafter when Wiles' approach via the Taniyama-Shimura conjecture became hung up on properties of the Selmer group using a tool called an Euler system. However, the difficulty was circumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995) and published in Taylor and Wiles (1995) and Wiles (1995). Wiles' proof succeeds by (1) replacing elliptic curves with Galois representations, (2) reducing the problem to a class number formula, (3) proving that formula, and (4) tying up loose ends that arise because the formalisms fail in the simplest degenerate cases (Cipra 1995).
Andrew Wiles Wins 2016 Abel Prize for Fermat's Last Theorem". He has won one of the top prizes in maths for solving a problem that dogged number theory for three-and-a-half centuries. Wiles won 6 million Norwegian Kroner as part of the prize,
http://mathworld.wolfram.com/FermatsLastTheorem.html
http://www.dailymail.co.uk/sciencetech/article-3493189/British-mathematician-Sir-Andrew-Wiles-gets-Abel-math-prize.html
"The scientist does not study nature because it is useful; s/he studies it because s/he delights in it, and s/he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.”- Henri Poincaré
De Moivre's Theorem:
(A truly beautiful theorem)
For every real number θ and every positive integer n, we have
(cos θ + i sin θ)n = cos nθ + i sin nθ
A formula useful for finding powers and roots of complex numbers. The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that θ is real, it is possible to derive useful expressions for cos(nθ) and sin(nθ) in terms of cos θ and sin θ.
More on contemprory theorems: Edward Witten for theory of everything
Edward Witten, whose work on the mathematical underpinnings of string theory has made it the theory of everything to beat. He is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics.
In 2004, Time magazine stated that Witten is widely thought to be the world's smartest living theoretical physicist. By the mid 1990s, physicists working on string theory had developed five different consistent versions of the theory. These versions are known as type I, type IIA,type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8). The thinking was that out of these five candidate theories, only one was the actual correcttheory of everything, and that theory was the one whose low energy limit matched the physics observed in our world today. Speaking at the string theory conference at University of Southern California in 1995, Witten made the surprising suggestion that these five string theories were in fact not distinct theories, but different limits of a single theory which he called M-theory. Witten's proposal was based on the observation that the five string theories can be mapped to one another by certain rules called dualities and are identified by these dualities. Witten's announcement led to a flurry of work now known as the second superstring revolution.
https://en.wikipedia.org/wiki/Edward_Witten
Neuroscience of Mathematical Beauty: Beautiful V. Ugly equations
In a study ( http://dx.doi.org/10.3389/fnhum.2014.00068), researchers found some connection between how much the mathematicians understood an equation and how beautiful they found it. But not always. Certain equations were ranked ugly regardless of understanding. The researchers, led by Semir Zeki of University College London, also asked nonmathematicians to view the equations while their brain was scanned with a functional MRI. Again, certain equations caused the medial orbitofrontal cortex to light up—an area of the brain that’s also involved in integrating emotion, sensory experience and decision-making.
Most beautiful equation: The formula most consistently rated as beautiful (average rating of 0.8667), both before and during the scans, was Euler's identity 1+eiπ= 0
Most beautiful equation:: The one most consistently rated as ugly (average rating of −0.7333) was Srinivasa Ramanujan's infinite series for 1/π,
Some identities depends a lot of how they are represented for understanding its containing. For instance, the De Moivre's turns out trivial if you think that
(eia)n=eian
which is the same that the trigonometric identity of Moivre
(cos θ + i sin θ)n = cos nθ + i sin nθ
Khayyam's or Pascal’s Triangle
In algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)n.
This remarkable pattern of coefficients (see attached) was studied in the 11th century by Persian poet and astronomer Omar Khayyam. Here in Iran it is known as Khayyam's triangle, but it is also named elsewhere as Pascal’s triangle for the 17th-century French mathematician Blaise Pascal, but it is far older (Plz refer to attached link for originality).
(x + y)0 = 1, (x + y)1 = x + y, (x + y)2 = x2 + 2xy + y2, ...
The triangle displays many interesting patterns. For example, drawing parallel “shallow diagonals” and adding the numbers on each line together produces the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21,…,), which were first noted by the medieval Italian mathematician Leonardo Pisano (“Fibonacci”) in his Liber abaci. Another interesting property of the triangle is that if all the positions containing odd numbers are shaded black and all the positions containing even numbers are shaded white, a fractal known as the Sierpinski gadget, after 20th-century Polish mathematician Wacław Sierpiński, will be formed.
http://www.britannica.com/topic/Pascals-triangle
Decoding the Math Behind Beauty
More than five centuries ago, Leonardo Da Vinci tried to approximate the ideal in human beauty with his Vitruvian Man. Da Vinci's simple sketch makes a radically-sweeping implication: that there are architectural principles behind human beauty that can be investigated scientifically and defined mathematically. “The height of the man is in golden proportion from the top of his head to his navel and from his navel to the bottom of his feet. The Vitruvian Man illustrates all of the divine proportions within the human being” He called it the Golden Mean. He used this proportion in all of his paintings, including Mona Lisa. So what exactly is this ratio or number? The Golden Ratio or Phi
a/b = (a+b)/a
Leonardo Da Vinci has long been associated with the golden ratio, reinforced in popular culture by Dan Brown’s 2003 best-seller, The Da Vinci Code. The plot contains pivotal clues involving the golden ratio and a series of numbers known as the Fibonacci series.
http://paintedbrain.org/la/newsstory/decoding-the-math-behind-beauty/
Dear Colleagues,
Good Day,
Here are some of my favorites (either because of the elegance of its proof or because of its wide spread application):
- Badges
- Science topic
- Similar topics
- Engineering
- Modeling