Proof without words: Cubes and Squares

http://fermatslibrary.com/s/proof-without-words-cubes-and-squares

by luisb • 10 years ago

143 30 10 years ago

eykanal• 10 years ago

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.

Someone• 10 years ago

It is an actual paper, but it didn't appear in a high-ranking journal (https://en.m.wikipedia.org/wiki/Mathematics_Magazine)

https://www.researchgate.net/journal/0025-570X_Mathematics_M... gives it an impact factor of about 0.2.

The formula certainly wasn't novel in 1977, and chances are neither is the visual proof.

(Edit: http://hyrodium.tumblr.com/post/94237657514/inspired-by-this... has a nice animation of this proof)

gohrt2• 10 years ago

It's a "magazine", not a research journal. It's expoisitory recreational math for students.

Smaug123• 10 years ago

It's not novel (though the paper comes from the '70s and may have been novel then), but it's a beautiful proof. I was first shown the proof by a maths teacher at my secondary school, and he didn't tell me what it was proving; he let me work it out. The "aha!" moment has stayed with me.

jessaustin• 10 years ago

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 n²(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

gohrt2• 10 years ago

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.

mrcactu5• 10 years ago

the same proof is discussed on math.stackexchange

http://math.stackexchange.com/questions/61482/proving-the-id...

derknorton2• 10 years ago

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)?

iconjack• 10 years ago

I think the diagram found in wikipedia is the clearest.

https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...

aisofteng• 10 years ago

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?

anthonyreinette• 10 years ago

Beautiful !

_nalply2• 10 years ago

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

squidfood• 10 years ago

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

_nalply2• 10 years ago

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

est• 10 years ago

Think deeper. You can categorize proofs in to two sets, one is safe in visual and one is not.

agumonkey• 10 years ago

Very recursive structure.

acqq• 10 years ago

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

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

breakfastcereal• 10 years ago

I still prefer algebraic proofs

carapace2• 10 years ago

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.

jacobolus2• 10 years ago

And sometimes a circular area unit makes most sense: https://en.wikipedia.org/wiki/Circular_mil

The advantage of rectangles is that they generalize best to situations where the two dimensions are different types of quantities. For example, it wouldn’t make any sense to have a triangular joule or foot-pound.

delish• 10 years ago

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

jacobolus2• 10 years ago

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...

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.

carapace2• 10 years ago

I can't remember, and search failed (what's up with google these days? I search for "squaring" and all the results are for "square". :( ) but it was in one of his books, with a diagram.

Jacobolus' comments are great.

theoh• 10 years ago

Similarly, Pythagoras' theorem turns out not to be literally about "squares" but about squaring as what happens to area under linear scaling. Mathematicians seem to find this proof more satisfying than most of the many proofs available, as it points towards the reason for the identity... Something to do with right angles in Euclidean space.

I'm afraid this is not a prestigious source but my Google-fu is failing me. Polyps mentions this proof, as has Terry Tao more recently:

https://betterexplained.com/articles/surprising-uses-of-the-...

Edit: more credible discussion here: https://math.berkeley.edu/~giventh/papers/eu.pdf

taeric• 10 years ago

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

lyle_nel• 10 years ago

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

lodi• 10 years ago

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

justifier• 10 years ago

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'

Proof without words: Cubes and Squares

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