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

43 questions
81
votes
2 answers

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…
58
votes
2 answers

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…
2'5 9'2
  • 51,425
  • 6
  • 76
  • 143
53
votes
4 answers

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
  • 2,908
  • 15
  • 28
41
votes
6 answers

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…
18
votes
2 answers

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…
8
votes
0 answers

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has disappeared from Wikipedia. The Convolution Theorem…
7
votes
2 answers

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
  • 71
  • 1
7
votes
1 answer

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!
7
votes
2 answers

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
  • 702
  • 4
  • 14
6
votes
0 answers

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
  • 13,386
  • 9
  • 41
  • 79
6
votes
1 answer

Geometric proof of a trig identity on $\cos t \cos u\cos v$

Consider the following trigonometric identity, valid for any set of angles $u,v,t$: $$\cos t⋅\cos u⋅\cos v =\frac14\left[\cos(t + u + v)+\cos(t + u - v)+\cos(u+v-t)+\cos(v + t - u)\right]$$ This identity and its derivation have previously appeared…
5
votes
2 answers

Geometric proof in Concrete Mathematics

On page 32 of Concrete Mathematics by Graham, Knuth and Patashnik, they demonstrate that the sum of a geometric progression is $$ \sum_{k=0}^n a x^k = \frac{a-a x^{n+1}}{1-x}. $$ In the margin next to this, there's a figure with the note "If it's…
Nick Matteo
  • 8,583
  • 2
  • 22
  • 52
5
votes
1 answer

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 color…
cr001
  • 11,421
  • 10
  • 24
5
votes
2 answers

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
  • 910
  • 4
  • 11
5
votes
2 answers

Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$

$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$ I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$. But the two things look completely different and I can't think of an…
genepeer
  • 1,638
  • 10
  • 26
1
2 3