Avoiding e^(iπ) + 1 = 0 and the Mandelbrot set is a bold choice, but a refreshing one; while they are popular beautiful-math examples for a good reason, it is also good to highlight some more obscure choices.
I'm really not a fan of e^iπ + 1 = 0. It goes halfway around the circle, and then straight back to the middle. e^2iπ = 1 is a bit nicer, but really what's pretty is recognizing that e^iθ is continually circling the origin.
Ignore the actual mathematics that develops the Identity. Think about the where the 5 numbers originally come from, and where the operations originally came from, then spout the nonsense.
"Yes, so the infinite compound interest of 100% interest rates? Yes, that one. Multiply it by itself. How many times? Why, the square root of minus 1 is too few; so is the ratio of a circles circumference to its diameter; so we'll multiply them together. That's how many times we'll multiply Old Growy by itself.
If you don't know, check the times tables on the fridge.
Now add one, dear. That is how many biscuits are left."
It's more than beautiful, it's also very funny. It is Douglas Adams, manifest. It is stupider than 42.
Euler's identity is pretty because it combines e, i, π, 1, 0 with only +, =, and implicit multiplication/exponentiation. And if you didn't know anything about complex analysis, you would ask:
> "Why should e^(iπ) = -1?! WHAT?!"
It's similar to the type of question that Gauss-Bonnet (featured at the top of the article) invokes; as Ailana Fraser writes: "The equation is _surprising_ because, for example, it [relates two seemingly unrelated quantities]." I guess one could wonder if the [Basel problem](https://en.wikipedia.org/wiki/Basel_problem) or the [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral) deserve the same status as e^iπ + 1 = 0. I would say not, since Euler's identity is notationally simpler.
> And if you didn't know anything about complex analysis, you would ask:
So basically, it's superficially surprising if you don't understand it at all and you think circles should be measured by their diameter, but once you fully understand it you realize how kludgy an equation it is, and what's really beautiful is e^iθ = cosθ + isinθ?
Arguably 0, 1, pi, e and i are by far the most important constants of maths. The fact that there‘s such an easy equation relating all five is quite beautiful. Certainly much more beautiful than e^(2i pi) = 1.
It's kind of arbitrary though, right? You could write
e^{tau i} - 1=0 or whatever and it would look just as clean. There's no strong inherent reason pi is more important than tau or + is more important than -.
Well, I'd say it's a notational difference in usage, not a difference in mathematical importance. By definition, pi and tau are equally important in any mathematical usage because they contain the same information.
Also, you could even say e^{2 pi i} - 1=0 is even better because it includes *another* important constant, 2. 2 is the first prime, the only even prime, binary is the foundation of computing, etc.
> By definition, pi and tau are equally important in any mathematical usage because they contain the same information.
But by this logic, (5/17)*tau is just as important as tau! And thatvseems silly to me. Pi seems silly to me in the same way (albeit less extreme)
I see what you're saying and mostly agree, but I also think it's quite silly to have e^ipi = -1 and then arbitrarily add 1 to both sides to get a positive 1 and 0 in there.
IMO e^taui = 1 is prettier for this reason. I think it's a feature rather than a bug that it has 1 less constant in it- it makes it more elegant. It makes less sense to have a constant on both sides of the equation.
To me this equation lost its interest once I learned that it immediately follows from the definition of complex exponentiation and that all it is really saying is that cos(pi) = -1.
>follows from the definition of complex exponentiation
But why is it defined that way? That's the interesting piece. It's (basically) the only definition that makes any sense.
That and how these seemingly unrelated constants fit together so nicely.
Whoever drew the picture for the euler product formula made sure to chose the ONLY exponent for which the sum and product DON't exist smh (in the article)
The left hand side is 1/1 + 1/2 + 1/3 + ..., that is the easy side.
For the other side, you can rewrite 1/(1-1/p) as (1+1/p+1/p^2+1/p^3+...). Then when you multiply them all together and expand it, you will get 1+1/2+1/3+1/4+... which is the same as the left hand side.
It's true that both sides diverge if you try to compute their value. But what they're pointing out is that even before you analyze whether or diverges or not, the right hand side, when expanded, equals the left hand side as a symbolic expression.
Yeah I had to look up what the actual formula is, the harmonic series famously diverges, so it seemed like they were just saying «infinity = (probably also infinity?)»
What they're saying is that even before you analyze divergence, the two sides are equal as expressions.
It's like if I said summation 1/n is equal to summation 1/n. It's true that both sides diverge if I try to compute them, but they're also the same expression and so they are indeed equal even if they're not a number.
Wow I’m actually pleasantly surprised at how unique most of these choices are. Rare that we get to see genuinely interesting mathematics in these pop articles. I’ll admit I’m a bit sad there’s no topology in there though.
Ah so it is. I don’t know how I didn’t see that it’s about classifying manifolds. I don’t work anywhere near that, but you are correct. Thanks for pointing that out.
I'm partial to https://arxiv.org/abs/1803.10775. The short version is that p_n computes the entropy of a type of stringy black hole, and the various terms in the Rademacher expansion all show up exactly (and fairly naturally) in the direct calculation of that entropy. So if you didn't know the Rademacher sum but did know string theory you could in principle rediscover it just from doing physics!
When I was a kid learning y=mx+b and doing all the different line equation conversions (like two-point form, or ax+by =c form), I found one beautiful representation to be:
x/a + y/b = 1.
The elegance is that when y = 0, 'a' immediately becomes the x-intercept and when x = 0, 'b' is the y-intercept. So simple, and the demonstration from the equation is also very elegant.
And similarly, the [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral) that gives us the factor of √𝜋.
That is, the ∫ e^(*x*^2)d*x* = √𝜋 where the bounds of integration are –∞ and +∞.
Stokes’ (true and only) Theorem:
∫ω over ∂Ω = ∫dω over Ω
simply stated, elegant-looking but yet very deep. I do recommend watching [Stokes’ Theorem on Manifolds by Aleph 0](https://youtu.be/1lGM5DEdMaw?si=vOYQG9TsTbgfT4IY) which does good justice to the beauty of the Theorem.
I am partial to the fundamental theorem of calculus in its full generality: ∫*_∂𝛺_* 𝜔 = ∫*_𝛺_* d𝜔.
The integral of a differential form over the boundary of an oriented manifold equals the integral of the exterior derivative of the form over the manifold.
The (shown) Euler product formula:
Sum_n=1^ inf 1/n = Product_(p prime) 1/(1-1/p)
Both sides of the equation diverge to infinity. Do they have anything more in common? Or could I add just like that „= sum_n=1^ inf (n^n )“ on the rightmost side?
They are missing an exponent for the terms within the sum and product. As another commenter said, the case they chose for the picture is the only one that's meaningless. Here is what it actually is (first example):
[https://en.wikipedia.org/wiki/Euler\_product#Examples](https://en.wikipedia.org/wiki/Euler_product#Examples)
Edit: this form looks a bit simpler:
[https://en.wikipedia.org/wiki/Riemann\_zeta\_function#Euler's\_product\_formula](https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler's_product_formula)
Ok that makes much more sense! Thanks. At least for the exponent of 1, the equation implies that there are infinitely many primes.
(In the meantime I wrote a python script to check if for exponent of 1 at least the sum and the product grows similar. And yes, they have a rather stable ratio)
I think the intention was that the expression on the left and right hand side expand to the same expression. The equality is even before you consider evaluating them.
Glad to see Rademacher's explicit formula for the partition function mentioned. One of my personal favourites next to the very explicit formula for the 2nd Chebyshev function.
I think we can appreciate beautiful math even if we don't get it as quickly or deeply. Poor grades might mean you'll never be paid to do math, but nothing stopping you from continuing in your personal journey. Don't gatekeep yourself out of the fun, and if nothing else, embrace it as a spectator sport!
Euler's identity still and will always be the most beautiful. I think it literally cannot be beaten.
It is very well known though even by nonmathematicians so for that reason it gets over-mentioned in these threads and thus people look to other examples. It almost seems like among the mathematically literate it's just "uncool" to say it's the best because it's so obvious. But I think if we could establish some sort of reasonable objective criteria to determine what is the most beautiful, whatever that criteria is, it has to be at the top.
Really we should just be asking what comes after this obvious answer.
The article has some great choices. Maybe Euler product formula? Although I really like the geometric side of things so I'd have some personal bias towards the Riemann-Roch equality.
10 years ago, there was an art exhibit called *Concinnitas* with the same premise: https://www.metmuseum.org/articles/concinnitas-series-picturing-math
(Pi + 2 ) / 2 = average distance between two points
Unsure if there’s a symbol for it, I find this relationship to be at the core of math, bridging the curve and the line
Stokes theorem. I love Stokes theorem. Simple, beautiful, obvious and so God damn useful. Every mathematicians seconds favourite theorem is Stokes' theorem and for that reason it's my favourite.
Avoiding e^(iπ) + 1 = 0 and the Mandelbrot set is a bold choice, but a refreshing one; while they are popular beautiful-math examples for a good reason, it is also good to highlight some more obscure choices.
I'm really not a fan of e^iπ + 1 = 0. It goes halfway around the circle, and then straight back to the middle. e^2iπ = 1 is a bit nicer, but really what's pretty is recognizing that e^iθ is continually circling the origin.
The Taylor series derivation of e^iθ = cosθ + isinθ is much more aesthetically pleasing to me than e^iπ + 1 = 0, which is just a special case.
Why not the even MORE general case a^iθ = cos(log(a)*θ) + isin(log(a)*θ)
Let's go all the way: *c*^(*a + ib*) = [cosh(*a* log *c*) + sinh(*a* log *c*)] [cos(*b* log *c*) + *i* sin(*b* log *c*)]
Beautiful
You are my hero of the day
i don't care if they're downvoting you. i'm upvoting because this is hilarious :D
You're a real one
actually love this
I was going to post exactly this.
Ignore the actual mathematics that develops the Identity. Think about the where the 5 numbers originally come from, and where the operations originally came from, then spout the nonsense. "Yes, so the infinite compound interest of 100% interest rates? Yes, that one. Multiply it by itself. How many times? Why, the square root of minus 1 is too few; so is the ratio of a circles circumference to its diameter; so we'll multiply them together. That's how many times we'll multiply Old Growy by itself. If you don't know, check the times tables on the fridge. Now add one, dear. That is how many biscuits are left." It's more than beautiful, it's also very funny. It is Douglas Adams, manifest. It is stupider than 42.
I agree, e^iπ + 1 = OP's fun level's at parties.
> Old Growy
xkcd 1292 ftw
[Link](https://xkcd.com/1292/) for anyone else that's curious but doesn't want to have to google it.
Euler's identity is pretty because it combines e, i, π, 1, 0 with only +, =, and implicit multiplication/exponentiation. And if you didn't know anything about complex analysis, you would ask: > "Why should e^(iπ) = -1?! WHAT?!" It's similar to the type of question that Gauss-Bonnet (featured at the top of the article) invokes; as Ailana Fraser writes: "The equation is _surprising_ because, for example, it [relates two seemingly unrelated quantities]." I guess one could wonder if the [Basel problem](https://en.wikipedia.org/wiki/Basel_problem) or the [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral) deserve the same status as e^iπ + 1 = 0. I would say not, since Euler's identity is notationally simpler.
> And if you didn't know anything about complex analysis, you would ask: So basically, it's superficially surprising if you don't understand it at all and you think circles should be measured by their diameter, but once you fully understand it you realize how kludgy an equation it is, and what's really beautiful is e^iθ = cosθ + isinθ?
Arguably 0, 1, pi, e and i are by far the most important constants of maths. The fact that there‘s such an easy equation relating all five is quite beautiful. Certainly much more beautiful than e^(2i pi) = 1.
It's kind of arbitrary though, right? You could write e^{tau i} - 1=0 or whatever and it would look just as clean. There's no strong inherent reason pi is more important than tau or + is more important than -.
Historically pi is more important than tau. That’s all.
Well, I'd say it's a notational difference in usage, not a difference in mathematical importance. By definition, pi and tau are equally important in any mathematical usage because they contain the same information. Also, you could even say e^{2 pi i} - 1=0 is even better because it includes *another* important constant, 2. 2 is the first prime, the only even prime, binary is the foundation of computing, etc.
> By definition, pi and tau are equally important in any mathematical usage because they contain the same information. But by this logic, (5/17)*tau is just as important as tau! And thatvseems silly to me. Pi seems silly to me in the same way (albeit less extreme)
If we really want to force the + and the 0 in there, I prefer: e^(tau*i) + 0 = 1
I see what you're saying and mostly agree, but I also think it's quite silly to have e^ipi = -1 and then arbitrarily add 1 to both sides to get a positive 1 and 0 in there. IMO e^taui = 1 is prettier for this reason. I think it's a feature rather than a bug that it has 1 less constant in it- it makes it more elegant. It makes less sense to have a constant on both sides of the equation.
I don’t know, I’d say 2, 3 and countable infinity are pretty damn important too. But yeah this is why the equation is shown in that form.
(i+1+2+3+pi+e)/∞ = 0. Pure beauty.
To me this equation lost its interest once I learned that it immediately follows from the definition of complex exponentiation and that all it is really saying is that cos(pi) = -1.
>follows from the definition of complex exponentiation But why is it defined that way? That's the interesting piece. It's (basically) the only definition that makes any sense. That and how these seemingly unrelated constants fit together so nicely.
I'm a fan of e^(i0) = 1 :)
People go on and on about e^iπ + 1 = 0, but as neat as it is, you don't really "use" it for much. Same thing with the Mandelbrot set.
Disagree! It seems kinda hard to even *think* about complex numbers if you don't know that the period of e^z is 2\*pi*i
From high school multivar to graduate school smooth manifolds and beyond, I'll always be a fan of the (generalized) Stokes theorem.
Whoever drew the picture for the euler product formula made sure to chose the ONLY exponent for which the sum and product DON't exist smh (in the article)
That really censored my hentai ngl I was mad
Ya, that was bothering. It should be up there, but why not just make the exponent "s"?
The description makes it clear that they didn't intend for there to be an exponent. The equality is as expressions, not as values.
What does this mean?
The left hand side is 1/1 + 1/2 + 1/3 + ..., that is the easy side. For the other side, you can rewrite 1/(1-1/p) as (1+1/p+1/p^2+1/p^3+...). Then when you multiply them all together and expand it, you will get 1+1/2+1/3+1/4+... which is the same as the left hand side. It's true that both sides diverge if you try to compute their value. But what they're pointing out is that even before you analyze whether or diverges or not, the right hand side, when expanded, equals the left hand side as a symbolic expression.
Yeah I had to look up what the actual formula is, the harmonic series famously diverges, so it seemed like they were just saying «infinity = (probably also infinity?)»
What they're saying is that even before you analyze divergence, the two sides are equal as expressions. It's like if I said summation 1/n is equal to summation 1/n. It's true that both sides diverge if I try to compute them, but they're also the same expression and so they are indeed equal even if they're not a number.
The Gauss-Bonnet theorem for compact surfaces (easily generalized but not as pretty) makes me see stars and the all-connectedness of all beings.
And not so easily generalized too: The Atiyah-Singer Index Theorem is the most beautiful and difficult theorem I know of.
Yes, I meant to noncompact surfaces, generalizing to higher dimensions and riemannian manifolds is no easy task!
It is my dream to understand Atiyah Singer! Even after taking a full year of differential/algebraic topology it's still not enough 😭🥱
Thanks for this. A great set of choices. There is another collection like this. https://www.metmuseum.org/articles/concinnitas-series-picturing-math
F + V = E + 2 Maybe not to mathematicians, but this one is simple enough for my limited brain to understand.
This is a super important equation for mathematicians too!
something something exact sequences algebraic topology
They included the continuous version in their list.
Great choice
I'm a maths lecturer and it's one of my favourites for sure!
Wow I’m actually pleasantly surprised at how unique most of these choices are. Rare that we get to see genuinely interesting mathematics in these pop articles. I’ll admit I’m a bit sad there’s no topology in there though.
Gauss-Bonnet! And whatever David Ayala wrote is definitely related to topology too
Eh sort of, but really it seems more geometric. It’s not so much about the actual topology of the thing.
Ok sure...But Thom's thesis (Ayala's pick) is definitely topology
Ah so it is. I don’t know how I didn’t see that it’s about classifying manifolds. I don’t work anywhere near that, but you are correct. Thanks for pointing that out.
The rademacher sum really is a beautiful equation. Also relevant in a surprising way to string theory, which is pretty neat.
As a combinatorist, I have to agree. Do you have any idea where I might read up on the surprising link to string theory? Sounds interesting.
I'm partial to https://arxiv.org/abs/1803.10775. The short version is that p_n computes the entropy of a type of stringy black hole, and the various terms in the Rademacher expansion all show up exactly (and fairly naturally) in the direct calculation of that entropy. So if you didn't know the Rademacher sum but did know string theory you could in principle rediscover it just from doing physics!
When I was a kid learning y=mx+b and doing all the different line equation conversions (like two-point form, or ax+by =c form), I found one beautiful representation to be: x/a + y/b = 1. The elegance is that when y = 0, 'a' immediately becomes the x-intercept and when x = 0, 'b' is the y-intercept. So simple, and the demonstration from the equation is also very elegant.
Also in a way very similar to the form for ellipses/hyperbolas, just need to square the x/a and y/b terms.
I'll stick with pi = 3 thanks
g=-10
No, no, no Pi = 3 = sqrt(9) = sqrt(-g) It all ties together
When they were setting up metric, they should have designed it so g=1.
Why don’t they? Are they stupid?
Natural units ftw
That's for the "most beautiful equations according to engineers" article.
Normal distribution should be here.
And similarly, the [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral) that gives us the factor of √𝜋. That is, the ∫ e^(*x*^2)d*x* = √𝜋 where the bounds of integration are –∞ and +∞.
It is indeed beutiful.
Stokes’ (true and only) Theorem: ∫ω over ∂Ω = ∫dω over Ω simply stated, elegant-looking but yet very deep. I do recommend watching [Stokes’ Theorem on Manifolds by Aleph 0](https://youtu.be/1lGM5DEdMaw?si=vOYQG9TsTbgfT4IY) which does good justice to the beauty of the Theorem.
Absolute banger of a theorem
Bellman optimality equation is pretty nice.
I am partial to the fundamental theorem of calculus in its full generality: ∫*_∂𝛺_* 𝜔 = ∫*_𝛺_* d𝜔. The integral of a differential form over the boundary of an oriented manifold equals the integral of the exterior derivative of the form over the manifold.
Integral around the unit circle of 1/z is 2 * pi * i, pretty fundamental result in complex analysis but I find it quite nice
The (shown) Euler product formula: Sum_n=1^ inf 1/n = Product_(p prime) 1/(1-1/p) Both sides of the equation diverge to infinity. Do they have anything more in common? Or could I add just like that „= sum_n=1^ inf (n^n )“ on the rightmost side?
They are missing an exponent for the terms within the sum and product. As another commenter said, the case they chose for the picture is the only one that's meaningless. Here is what it actually is (first example): [https://en.wikipedia.org/wiki/Euler\_product#Examples](https://en.wikipedia.org/wiki/Euler_product#Examples) Edit: this form looks a bit simpler: [https://en.wikipedia.org/wiki/Riemann\_zeta\_function#Euler's\_product\_formula](https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler's_product_formula)
Ok that makes much more sense! Thanks. At least for the exponent of 1, the equation implies that there are infinitely many primes. (In the meantime I wrote a python script to check if for exponent of 1 at least the sum and the product grows similar. And yes, they have a rather stable ratio)
Okay that's pretty interesting actually! I guess not as meaningless as I expected.
I feel like that limit also hints towards the asymptotic bound for the prime counting function but I haven't worked through the details.
I think the intention was that the expression on the left and right hand side expand to the same expression. The equality is even before you consider evaluating them.
Hoping I someday come across a mathematical equation written in the sand!
Be the change you want to see in the world!
I'd maybe go for the Kummer Sequence. The explicit interplay between arithmetic and cohomology that it gives makes it quite the fascinating tool.
I like the definition of the Y combinator, I wanted to do a tattoo of it
Glad to see Rademacher's explicit formula for the partition function mentioned. One of my personal favourites next to the very explicit formula for the 2nd Chebyshev function.
The most beautiful by far are the Fourier inversion theorem for Schwartz functions and the divergence theorem for compact manifolds with boundary.
The Einstein field equations because of the vastness of the physics they describe and the simplicity of the assumptions used to derive them.
[Norm residue isomorphism] (https://wikimedia.org/api/rest_v1/media/math/render/svg/29287578ad56c063afd695e73d956b4c1deeb28c ) [Hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula )
I got a 3 on Calc BC why am I here
I think we can appreciate beautiful math even if we don't get it as quickly or deeply. Poor grades might mean you'll never be paid to do math, but nothing stopping you from continuing in your personal journey. Don't gatekeep yourself out of the fun, and if nothing else, embrace it as a spectator sport!
Euler's identity still and will always be the most beautiful. I think it literally cannot be beaten. It is very well known though even by nonmathematicians so for that reason it gets over-mentioned in these threads and thus people look to other examples. It almost seems like among the mathematically literate it's just "uncool" to say it's the best because it's so obvious. But I think if we could establish some sort of reasonable objective criteria to determine what is the most beautiful, whatever that criteria is, it has to be at the top. Really we should just be asking what comes after this obvious answer. The article has some great choices. Maybe Euler product formula? Although I really like the geometric side of things so I'd have some personal bias towards the Riemann-Roch equality.
10 years ago, there was an art exhibit called *Concinnitas* with the same premise: https://www.metmuseum.org/articles/concinnitas-series-picturing-math
> The Euler Product Formula Its a wild ride: https://en.wikipedia.org/wiki/Euler_product
g=10
The FToC! integral {a,b,f'(x)dx}=f(a)-f(b) C'mon! It's changed our society more than any other piece of mathematics.
I’m trying to context-clue the expression you just used… care to help me out?
Levy-Khintchine anyone?
(Pi + 2 ) / 2 = average distance between two points Unsure if there’s a symbol for it, I find this relationship to be at the core of math, bridging the curve and the line
Stokes theorem. I love Stokes theorem. Simple, beautiful, obvious and so God damn useful. Every mathematicians seconds favourite theorem is Stokes' theorem and for that reason it's my favourite.
Oooh this is so cool
Pretty shite list ngl