When building a fence.

Yes but I build stuff so…

Checking if something is squared. For example, if you measure 3’ from one corner and 4’ from the other corner and you measure from the 2 lengths if it’s square it will equal 5’

The 3 4 5 method is an old school technique in construction

This is what I do, lol. I tell my friends I'm gonna walk the hypotenuse so I can walk less.

Pythagoras = Others

Bro acting like Pythagoras discovered the triangle inequality

If building one side of the square road takes 190 bricks, how many bricks takes to build the desired path?

I don't know... musing out loud, but a cone is solid of revolution of a right triangle. Any relationships in a cone that would imply the pythagorean theorem?

I think there is unlikely to be a satisfying answer to this question, although I cannot rule out the possibility.

Every proof I have seen of Pythagoras' theorem is essentially in one of three categories.

1. Purely geometric proofs.
2. Proofs that exploit properties of the sine and cosine functions.
3. Sleight of hand arguments about metrics induced by inner products.

I will focus on only the first, because it seems most relevant to your question. Among geometric proofs there are essentially two avenues of approach.

The first avenue argues from the scaling properties of area (a contraction by a reduces area by a^(2)**). Area is an essentially two dimensional phenomenon so in three dimensions this would have to focus on surface areas.

The second avenue chases two pairs of similar figures, to show some segment is composed of copies of itself scaled by a twice and b twice, respectively. An argument of this type in three dimensions would involve looking at lines and angles, so care would need to be taken that it does not reduce to a two dimensional argument.

You may be interested in this collection of 122 geometric proofs of the Pythagorean theorem, non of which extend beyond the plane.

Lastly, you mentioned dissections of 3d shapes so this closely related problem might be relevant. Is there a 3d polyhedron that can be dissected into two pieces similar to itself? The first geometric proof of Pythagoras would apply here also, and yield a^(3)**+b*^(3)=c^(3)**.*

I'm not aware of any proof like that. But I also think you might be wasting your time looking for one.

Sometimes a statement about objects in one number of dimensions can be proven by adding dimensions. There are two situations I can think of where that's interesting:

1. if the statement becomes considerably easier/more intuitive to prove if you add more dimensions (this suggests that the statement might have been about higher dimensions all along)
2. if the statement is well known but it appears as a surprise corollary within a completely different context

The usual proofs of the Pythagorean theorem that work by cutting up squares and moving the pieces around are very simple. That means Option 1 is out of the picture.

Option 2 can't really be planned for. Either you do work in some unrelated theory and suddenly the Pythagorean theorem pops up, or not.

I think you should pursue it. This sounds interesting.

Does the Riemann sphere count? Since on the sphere straight lines become circles, the rank 2, three term Grassmann-Plücker relation proves Ptolemy which proves Pythagoras. 

A question, isnt f(θ) always a²sec²(θ) ? As in f(θ) cannot be an arbitrary function, it is a fixed function.

The ladder problem is of the classic "Solve Pythagoras' Theorem so that a, b, and c are all integer values".

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/ul there’s a difference between using something because it works and proving exactly why it works. pythagoras did the latter. it’s called “pythagorean theorem” because he is who proved the theorem, not because he is the one who thought of it first.

/ul wait, he didn't invent it?

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HOWTO?

To prove the theory, you must first calculate the square root of one of the sides, it's done using the Vieta's formula:

Step1:
For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid whenanis not a zero-divisor andP(x)factors asan(x−r1)(x−r2)…(x−rn). For example, in the ring of the integers modulo 8, the quadratic polynomialP(x)=x2−1has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say,r1=1andr2=3, becauseP(x)≠(x−1)(x−3). However,P(x)does factor as(x−1)(x−7)and also as(x−3)(x−5), and Vieta's formulas hold if we set eitherr1=1andr2=7orr1=3andr2=5.

Step2:
Using the above solution, you apply the resulting 2 squares to Schroedinger equation for the hydrogen atom. U^(t/N)N=U^(t)for anyN>0. ThenU^(t)depends upon the parametertin.

Step3:
Using the solution you can prove this post being a lie.

Step4:
???

Step5:
PROFIT!

I'm in the process of building a primary root acoustic diffuser (sometimes called a PRD or skyline for aid in googling). I've yet to really grasp why the design is the way it is, but it's built over number theory concepts. Reading over the patent, the cited math papers were sometimes within a year or two of the release of the patent itself in the 90s - so it was done with some cutting edge math for the time.

If anyone has a key intuition for how it works, please share!

Depends what you mean by number theory...

Modular forms are foundational in conformal field theory which is a theoretical physicists playground for a number of topics...

See the book “From Number Theory to Physics”.

Ok so i got my PhD in connections between string theory and number theory. I think my favorite example of this connection is the fact that black hole entropies are given by the Fourier coefficients of modular forms. This is particularly clear in the context of so-called small black holes, where literally the number of states in a black hole of mass $n$ is given by the $n$-th Fourier coefficient of 1/Delta(tau). where Delta is the weight-12 cusp form. In fact, the Fourier coefficients of this modular form admit an exact series expansion known as a Rademacher series, which takes the form of a sum over elements of (a quotient of) SL_2(Z). The exact same series appears in the evaluation of the gravitational path integral which is supposed to compute this entropy, so we have a highly nontrivial matching between the number theory result and actual string theory.

I can go on for hours about other connections between string theory and number theory (or, ya know, write a thesis about it :p), but I think this is the most striking demonstration.

Just as an example, here, a hamiltonian is constructed whose eigenfunctions given an appropriate boundary value have associated eigenvalues corresponding to the zeroes of the Riemann Zeta function.

https://arxiv.org/abs/1608.03679

Check out: A Panoramic View of Riemannian Geometry

Differential geometry IS pure mathematics! It’s connected to lots of other fields of pure math, including topology, dynamical systems, harmonic analysis, algebraic geometry, category theory, PDE, etc.

Differential Geometry is pure maths, isn't it ?

Ricci flow is the key ingredient behind Perelman’s solution to the Poincaré conjecture.

Sure, it has many applications in topology, especially low dimensions and this has repercussions for higher dimensional topology, often via obstruction theory. See things like the Poincare conjecture and computations of diffeomorphism groups via hyperbolic geometry.

There's Invitation to Non-Linear Algebra which touches upon various applications, though it's main focus is introducing the math with a view towards them.

Since I have them already, here's a link to some slides from an informal presentation I gave recently on applications of abstract algebra to software engineering.

https://docs.google.com/presentation/d/1WnKadU0fVers_VUihQO_UGjeu2e4kHLHxYjz4H6aPtc/edit?usp=sharing

You'll notice a strong flavor of universal algebra especially in the beginning, setting this apart from the focus on specific structures you see in most algebra in the math world. The applications in general have a character that is perhaps unfamiliar to many mathematicians. Although there are a few places where deep theorems might matter, what matters a lot more is having a strong intuition for how to model the structure of things with algebraic properties and structures, and recognizing that when things like closure, associativity, commutativity, and idempotence arise, they are good signs that you have modeled ideas in a good way.

This isn't to say that you won't also end up appying software engineering to problems where there's algebra involved in understanding the problem domain. If you're working in cryptography, for instance, you'll think a lot about number theory elliptical curves, etc. If you're working in machine learning, you'll use a lot of linear algebra. And so on.
But this presentation tries to stay focused on the role algebra plays in the essence of software engineering as an activity, not in the accident of which domain you are applying the skill.

I mean what isn't algebra applied to? It's pretty versatile. Galois theory goes into encryption algorithms, eigen vectors go into computer graphics. Linear transforms goes into statistics. Group representation goes into quantum mechanics.

My students certainly agree. Everything is linear to them. Sine, cosine, logs, arctangents, exponents, roots,...

Here's a video describing how group theory was used to solve a complex problem in computer science

https://youtu.be/KufsL2VgELo

Haha same. I just wanted to share this to try and kind of show how AA relates to other fields of science

This discussion on MathOverflow describes 'real-world' applications of various areas of mathematics, including algebra.