Questions tagged [proof-without-words]

For questions concerning the creation and understanding of pictorial proofs.

A proof without words is a proof of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.

For example, the following is a proof of the Pythagorean Theorem due to Martin Gardner.

enter image description here

See also: Wikipedia

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Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles" , Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality of square and curved areas is based on another…
VividD
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Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a $n\times(n+1)$-rectangle. Similarly,…
2'5 9'2
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What is the explanation for this visual proof of the sum of squares?

Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
Nitin
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Is there a sufficiently reachable plausibility argument that $\pi$ is irrational?

I was teaching someone earlier today (precisely, a twelve-year-old) and we came upon a problem on circles. Little did I know in what direction it would lead. I was able to give a quick plausibility argument that all circles are similar and that they…
Allawonder
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Is it possible to explain geometrically why $\arctan (1/2) +\arctan (1/3) = 45$ degrees?

$\arctan(1/2)$ seems to be some strange, irrational angle, and the same goes for $\arctan(1/3)$ , but those two angles seem to sum up to $45$ degrees. This seems like a mystery to me even though I can derive the result algebraically as follows, by…
A A
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A Proof with no words that $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2$

Question What are the words to describe the method in the image below? (from Nelsen's Proofs without Words II ) Attempt I was thinking and could define the sequence $u_1=2; u_{n+1}=f\circ g^{−1}(u_n)$ where $f(x)=\sqrt x$ and $g(x)=x−2$ , as…
Pierre
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Geometrical proof that $\|a+b+c\|^2+\|a-b\|^2+\|b-c\|^2+\|c-a\|^2=9$ for unit vectors $a,b,c$

The accepted answer to an older Q&A amounts to the identity $$\|a+b+c\|^2+\|a-b\|^2+\|b-c\|^2+\|c-a\|^2=9$$ for unit vectors $a,b,c$ . As noted there, the identity follows immediately if we use vector algebra to distribute these squares into inner…
Semiclassical
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Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
kjo
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Explain this calculus proof without words

This demonstrates that $\int_0^1 t^{p/q} + t^{q/p} dt = 1$. Could you please explain how the proof without words shows that?
user103493
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Generalized Pythagorean Dissection Tiling Result

I would sincerely appreciate any critical feedback or evaluation on the following result. Here's the context. Many of you are probably familiar with dissection proofs of the Pythagorean Theorem, such as Bhaskara's. I tried to generalize this idea.…
kingyoon
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Visual or intuitive proof of $\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=\frac{\pi}{4}$

It is well known that the alternate sum of the reciprocals of the odd numbers adds up to $\frac{\pi}{4}$ . That is $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=\frac{\pi}{4}.$$ A proof can be obtained by…
A. Bellmunt
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Can someone explain this “visual proof” of $(1 + 2 + 3 +…+ n)^2 = 1^3 + 2^3 + 3^3 +…+ n^3$?

I see a lot of people saying this visual proof is beautiful and I really want to be able to understand it. If anyone could help me out, I'd really appreciate it! Thanks in advance!
FreshWoodJohnson
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Geometric proof that the product of the $x$-intercepts equals the $y$-intercept for a monic quadratic

I know you can prove that the product of the roots of the monic quadratic $x^2+a_1x+a_0$ equals the $y$ -intercept $a_0$ by comparing its coefficients to the coefficients of $(x-m)(x-c)$ where $m$ and $c$ are the roots. So $a_0 = mc$ . This is how…
mihirb
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How to prove a tiling of a hexagon must form a 3D cubic stack?

How to prove that a tiling of a big hexagon consisting of triangles, using only $2$ -triangle tiles (three possible orientations), must resemble a continuous, convex (for each small cube), manifold, $3$ D cubic stack (i.e. a 3D staircase) if we…
cr001
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Is there a graphical proof that $9n+1$ is triangular if $n$ is?

It is straightforward to prove that if $n$ is a triangular number then $9n+1$ must be. Is there a systematic decomposition of a triangle made of $9n+1$ dots into nine triangles of $n$ dots plus one left over to illustrate this fact visually?
dbmag9
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