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177

The planet Neptune's discovery was an example of something similar to this. It was known that Newtons's Equations gave the wrong description of the motion of Uranus and Mercury. Urbain Le Verrier sat down and tried to see what would happen if we assumed that the equations were right and the universe was wrong. He set up a complicated system of equations that ...


115

Here's a rather different example which came up recently (see the Journal of Recreational Mathematics): A couple of mathematicians were studying juggling. They came up with a way to encode the 'ball catch' patterns as simple numeric sequences. Then they derived the sequences for all known juggling patterns, and inferred from them a set of rules governing ...


70

Write down Maxwell's equations in a vacuum: $$\nabla \cdot \vec{E}=0$$ $$\nabla \cdot \vec{B}=0$$ $$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \times \vec{B}=\mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$$ Note the vector identity $\nabla\times(\nabla \times \vec{X})=\nabla(\nabla\cdot\vec{X})-\nabla^2\vec{X}$. Apply this ...


67

Quasicrystals. Aperiodic tilings of the plane and space were discovered by mathematicians, starting from Robert Berger's work on Wang tiles in the 1960's. Physical materials exhibiting these properties were found in the 1980's by Dan Shechtman, who won the Nobel Prize for Chemistry in 2011 for this work.


45

Arago's spot is a classical (and classic) example of a beautiful mathematical theory anticipating a beautiful physical fact. Briefly, the story goes like this: Back in the 1800's, scientists were debating whether light was a particle or a wave. Following some convincing experiments by Young showing wave-like properties of light, Fresnel developed a ...


45

The memristor, the fourth passive electronic component (to accompany the resistor, capacitor, and inductor), was predicted by Leon Chua in 1971. An anomalous signal found by engineers in HP Labs in 2008 was, after much consternation, eventually attributed to the discovery of the memristor. The prediction follows from the relationship between voltage, charge,...


30

John von Neumann discovered some of the fundamentals of molecular biology back in the 1940s long before the field of molecular biology even existed. When von Neumann was developing his theory on universal constructors (UCs), machines that can build any possible physical structure including making copies of itself, he stumbled on a generic problem. The ...


30

The most notable ones that come to my mind at present are the Special Theory of Relativity and the General Theory of Relativity, both by Albert Einstein. Although Einstein published them in 1905 and 1915 respectively, the mathematical work of the theory had been done a long time before by Hendrik Lorentz, Henri Poincaré, and Hermann Minkowski. Einstein just ...


28

Dirac explains here how special relativity led to the Dirac equations for the electron which predicts its spin, magnetic moment and the existence of the positron.


20

Berry's Phase is a good example of mathematics uncovering new physics. In particular, a derivation in quantum mechanics assumed a one-dimensional domain in an integration. If the parameter space is higher-dimensional then the parameter domain can have nontrivial topology which ultimately leads to a nontrivial integral. This integral implies we can manipulate ...


20

I believe the story mentioned in the question is the story of the $\Omega^-$ particle, conjectured to exist by Gell-Mann and Ne'eman (see https://en.wikipedia.org/wiki/Eightfold_Way_(physics) and https://en.wikipedia.org/wiki/Yuval_Ne%27eman) as part of a representation-theoretic approach to quantum mechanics, and eventually discovered by a team at ...


19

The Fermi-Pasta-Ulam numerical experiments in the 1950's that led to theory of integrable systems. One day the computer simulation was accidentally left running longer than intended, showing that a nonlinear wave system almost returned to its original state instead of thermalizing. The theory that developed from this is enormous and applications include ...


13

I agree with some of the comments that the likely basis for this fuzzy memory is the Dirac "prediction" of positrons (the first known instance of anti-matter). However, slightly before Dirac's 1926 publication of a relativistic wave equation with negative energy states (which he didn't initially believe were "physical", so calling it a "prediction" was only ...


12

Bell's theorem on the foundations of quantum mechanics showed that not all philosophical questions are impervious to experiment, to the extreme surprise of pretty much every physicist on Earth. (It also showed that Einstein was soundly wrong, which some people might also find surprising.)


12

Kepler's attempted to match the orbits of the planets to a nested arrangement of platonic solids. Eventually, his data led him to the mathematics of Kepler's Laws. Kepler wasn't impressed by his three laws, but Newton found them in his papers. (from my post elsewhere)


9

Don't even forget I will Derive


8

Differential calculus was being developed by Leibniz and Newton in the 17-18th centuries at roughly the same time as Newton formulated his famous equations for mass, force and acceleration governing mechanical motion. I wonder what such laws would have looked like without having any differential and integral calculus available!


8

The Titius–Bode law (sometimes termed just Bode’s law) was an observation that the radius of the orbit of the $n\rm th$ planet in our solar system could be approximated by the formula $$r_n=(r_1+0.3\times 2^{(n-2)})\rm AU$$ for $n>1$, where “AU” represents an Astronomical Unit; i.e., the radius of Earth’s orbit (i.e., $r_3$). Setting $r_1$ to the ...


8

My favourite example of this is @SangchulLee's solution to @VladimirReshetnikov's question, which asks to verify the correctness of the identity $$\int_0^{\infty} \frac{dx}{\sqrt[4]{7 + \cosh x}}= \frac{\sqrt[4]{6}}{3\sqrt{\pi}} \Gamma\left(\frac14\right)^2 .$$ The other answers indicate the "toughness" of this integral, resorting to all sorts of special ...


7

Many results in the answers so far are examples from quantum physics. However there is also the example of quantum physics. As far as I remember, Planck wanted to explain the spectrum of black body emission by calculating a limit $\lim_{h\to 0}$ of a discretization. However, the results would only make sense if instead of letting $h\to 0$, he kept $h$ ...


7

There have been many attempts to prove Euclid's Parallel Postulate, which for around 2000 years was stumping many Mathematicians. Eventually it took Gauss to completely redevelop the notion of some of the equivalent properties in his development of hyperbolic geometry. Whilst it is true to say that Gauss' endeavours were not found by accident; it is true to ...


7

The Bieberbach conjecture is an example of a lonely result in the sense that, while it generated much interest and almost competition, its ultimate solution by de Branges pretty much closed the field. It turned out that the result does not have many applications, and is a kind of a very high-level olympiad problem.


6

For $a=\sqrt{\pi^2-\phi}$ and $k>0$, we have $$\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\large \int_0^\infty \ln \left( \frac{x^2+2kx\cos \sqrt{a^2+\phi}+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}=\phi}}$$ I hope you find this integral interesting. Extra: $$\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\large \int_0^\infty \frac{x^{...


5

"Banach-Tarski", on the music of Duck Sauce's "Barbara Streisand". I personally dream of the day someone does an Abstract-Nonsense cover of "Arrows" by Fences feat Macklemore & Ryan Lewis. Alas, that day is yet to come.


5

There is a lot of 'tough looking' integrals which can be solved by various tricks, but usually it requires more than a few lines of proof. This is a really soft question, because 'tough looking' integral is a very subjective term (note that I use it instead of just 'tough' because I agree with Robert's comment). I suggest the book Inside Interesting ...


5

Ramanujan's master theorem can be applied to a wide range of (sometimes extremely complicated) definite integrals, allowing them to be evaluated in less than a line of computations. The theorem states: If $f(x)$ has a series expansion of the form: $$f(x) = \sum_{k=0}^{\infty}(-1)^{k}\frac{\lambda(k)}{k!} x^k$$ then $$\int_0^{\infty}x^{s-1}f(x) dx = \...


4

Wikipedia has a few here. Here is Graham himself explaining what Graham's number expresses. (It's an upper bound for an geometric problem). It has been constructed to make a proof work so it's not a arbitrary construct. However I'm not sure if you would still consider this as "used in real life". It's large enough that you can't express it in a regular ...


4

Feynman's trick, Frullani's theorem, symmetry tricks, Glasser's master theorem, the Laplace transform, Fourier (Legendre/Chebyshev) series expansions and the Euler beta function gave me at least the $80\%$ of my reputation points, but the day still has to come, to be prouder than this massacre through the residue theorem. I have just realized I forgot to ...


3

Determinants can be used to see if a system of $n$ linear equations in $n$ variables has a unique solution. This is useful for homework problems and the like, when the relevant computations can be performed exactly. However, when solving real numerical problems, the determinant is rarely used, as it is a very poor indicator of how well you can solve a ...


3

Tom Lehrer's songs include: "New Math" and "Lobachevsky"



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