
Proof without words: Cubes and Squares - luisb
http://fermatslibrary.com/s/proof-without-words-cubes-and-squares
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iconjack
I think the diagram found in wikipedia is the clearest.

[https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...](https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Nicomachus_theorem_3D.svg/2000px-
Nicomachus_theorem_3D.svg.png)

~~~
aisofteng
The coloring helps very much, at least for me. The black and white image was
very unclear to me. Perhaps some grayscale coloring was lost in the linked
version?

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_nalply
Visual «proofs» are dangerous. Counter-example: Missing square puzzle:
[https://en.wikipedia.org/wiki/Missing_square_puzzle](https://en.wikipedia.org/wiki/Missing_square_puzzle)

~~~
squidfood
For real-numbered measurements, dangerous. For integer measurements that are
about seeing the counting re-arrangements, safer.

~~~
_nalply
Agreed. But this is an awfully fine point for proofs which are intended to be
«simple».

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carapace
Technically this is a demonstration, not a proof.

Does it _prove_ the pattern holds? My "intuition" says the pattern holds, but
I wouldn't bet my life on it just from this diagram.

Unrelated, I find it easier to see the pattern if you draw it with triangles.

    
    
             /\
            /\/\
           /\/\/\
          /\/\/\/\
         /\/\/\/\/\
    
    

Bucky Fuller pointed out that associating the second power with the area of a
square (as opposed to some other figure; and the third power with the volume
of a cube) is a choice.

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delish
I never thought of that. Where did he say that?

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jacobolus
Bucky Fuller was big on measuring shapes relative to triangular / tetrahedral
coordinates. “Synergetics coordinates”.

See e.g.
[http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966...](http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966.20)

The advantage of measuring area relative to a unit simplex is that it gets rid
of a factor of n! in the denominator which would pop up if you measured the
area of a simplex relative to a square grid. Likewise it simplifies the
formulae for area/volume of the _n_ –sphere.

An equilateral triangle grid in 2-dimensions, and its dual the hexagonal grid,
have much to recommend them. They’re often more efficient, e.g. it would be
noticeably better (less isotropic, more efficient) to represent images using
hexagonally shaped pixels than square pixels. Perhaps most importantly, they
broaden thinking beyond the square-grid culture we’re immersed in in every
aspect of our society (textiles, city planning, construction, carpentry,
cartography, visual art, written documents, analytic geometry, ...) and just
take for granted. It’s quite literally a way to “think outside the box”.

The downside is that it’s not a priori obvious exactly which way to generalize
the equilateral triangle grid beyond two dimensions, whereas for a square grid
there’s a natural choice. We can base our coordinates on just a single
simplex, but this doesn’t necessarily make measuring or relating arbitrary
shapes any easier.

The symmetry system of the tetrahedron has one type of 2-fold and two types of
3-fold axes of symmetry, and regular tetrahedra alone don’t tile space in a
regular way. The FCC and BCC lattices, based on the symmetry of the
cube/octahedron have 2-fold, 3-fold, and 4-fold axes of symmetry (this is the
symmetry system of a tiling of space using mixed octahedra and tetrahedra).
The dodecahedron/icosahedron have 2-fold, 3-fold, and 5-fold axes of symmetry,
don’t tile space, and don’t generalize beyond the 4th dimensional case.

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mrcactu5
the same proof is discussed on math.stackexchange

[http://math.stackexchange.com/questions/61482/proving-the-
id...](http://math.stackexchange.com/questions/61482/proving-the-identity-
sum-k-1n-k3-big-sum-k-1n-k-big2-without-i)

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justifier
But is this a proof?

It's seems like essentially the same as 'proving' Riemann by using a computer
to calculate the first 100,000 values of 0.. which is considered unsuitable as
proof

If using examples as evidence of proof I was under the impression you needed
to also prove the completeness and consistency of the pattern from the
examples

This reads to me like if instead disproving a finite hypothesis Euclids proof
of the infinitude of primes just consisted of listing the first 7 prime
numbers allowing the reader to make the assumption, 'well those are getting
bigger they must all get bigger'

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acqq
And of course ancient mathematicians knew it already at least around 2000
years ago:

[https://en.wikipedia.org/wiki/Squared_triangular_number](https://en.wikipedia.org/wiki/Squared_triangular_number)

"Pengelley (2002) finds references to the identity not only in the works of
Nicomachus in what is now Jordan in the first century CE"

Now it is Jordan, but at the time on Nicomachus there were almost 300 years of
Greek and 100 years of Roman rule of that area.

[https://en.wikipedia.org/wiki/Nicomachus](https://en.wikipedia.org/wiki/Nicomachus)

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eykanal
For those of us unfamiliar with this, is this an actual paper? Is something
novel being shown here or is the novelty entirely in the visuals?

Damn cool, by the way.

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jessaustin
Gauss noticed that the sum of the first _n_ numbers is _n_ ( _n_ +1)/2, while
in grade school. One can prove, via induction [0], that the sum of the first
_n_ cubes is _n_ ²( _n_ +1)²/4\. Clearly the latter expression is the square
of the former.

[0] The inductive step looks like this:

    
    
      (n-1)²((n-1) + 1)²/4 + n³
       (n⁴ - 2n³ + n²)/4 + 4n³/4
       (n⁴ + 2n³ + n²)/4
         n²(n + 1)²/4

~~~
gohrt
The proof is simpler if you don't bother to simplify the total:

The inductive step in the OP theorem is:

    
    
          ((1 + ... + n-1) +n)² - (1 + ... + n-1)²
        = n² + 2n(1 + ... + n-1)
        = n² + 2n(n-1)n/2
        = n²(1) + n²(n-1)
        = n²n
        = n³
    

Working out the algebra is about as much work, or maybe less, than
interpreting the "proof without words" (which BTW isn't a "proof", it only
illustrates n=1..7 and hints the generalization to the reader). The proof
without words is an illustration of the algebraic steps.

~~~
justinpombrio
> which BTW isn't a "proof", it only illustrates n=1..7 and hints the
> generalization to the reader

What you've written isn't a proof, either. All of those ellipses are terribly
informal. You should really write:

    
    
        (\Sigma_{k=1}^{n-1} k + n)² - (\Sigma_{k=1}^{n-1} k)²
        ...
    

Personally, I find the "proof without words" both easier to read and more
convincing than some algebra. I think I'd have a harder time spotting a
mistake in the algebra. Do other people really find it easier to find a
mistake in a dozen lines of algebra than in a diagram?

~~~
thaumasiotes
> I think I'd have a harder time spotting a mistake in the algebra. Do other
> people really find it easier to find a mistake in a dozen lines of algebra
> than in a diagram?

You're asking about two different issues as if they were one. It is
intuitively "easier" to examine the visual presentation. Nevertheless, it is
technically much easier to find a mistake in a dozen, or a hundred, lines of
algebra because there is a definite, known method of verifying the algebra.
With the visual presentation, you're just staring at it and hoping you notice
the mistake.

Analogously, it's much easier for people to produce an exhaustive list of
things that occur in a known fixed order (say, the 50 US states in
alphabetical order) than to produce an exhaustive list of unordered members of
some category (say, the 50 states if you never learned them in any particular
order). The algebraic proof is like the fixed-order recall; the structure it
imposes is, to some extent, error-correcting.

So, visual proofs, which are inherently _convincing_ , are a good tool for
persuading someone of their own truth _regardless of whether they 're actually
true or not_, and also a good tool for getting people to _remember_ a proof in
the future, but they are a terrible tool for formally establishing results.
They are best used for communicating results that you have other reasons for
believing in, or as an aid to memory or investigation.

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lyle_nel
I am not sure that this constitutes a proof. It seems induction would be
required to show that the property holds.

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lodi
This must be a reference to Mendelssohn's Songs without Words:
[https://www.youtube.com/watch?v=y1uvYdW8MSk&t=1880](https://www.youtube.com/watch?v=y1uvYdW8MSk&t=1880)

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taeric
There are full books on these called "Proofs Without Words". On my wishlist.
:) [https://amazon.com/dp/0883857006](https://amazon.com/dp/0883857006)

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derknorton2
This is a very nice visual proof! I wonder if there are similar proofs in
higher dimensions (or at least in 3 dimensions so that we have a hope of
visualizing them)?

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breakfastcereal
I still prefer algebraic proofs

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agumonkey
Very recursive structure.

