6 ms·

Proof without words: Cubes and Squares

143 points by luisb 10y ago

eykanal 10y 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 10y 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://en.m.wikipedia.org/wiki/Mathematics_Magazine) https://www.researchgate.net/journal/0025-570X_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-twocubes-post-and-asked-to-make http://hyrodium.tumblr.com/post/94237657514/inspired-by-this... has a nice animation of this proof)
  - gohrt 10y ago
    It's a "magazine", not a research journal. It's expoisitory recreational math for students.
- Smaug123 10y 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 10y 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
  - gohrt 10y 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.
    - justinpombrio 10y ago
      > 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 10y ago
        > 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.
deleted 10y ago
[deleted]
mrcactu5 10y ago
the same proof is discussed on math.stackexchange http://math.stackexchange.com/questions/61482/proving-the-identity-sum-k-1n-k3-big-sum-k-1n-k-big2-without-i http://math.stackexchange.com/questions/61482/proving-the-id...
derknorton2 10y 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 10y ago
I think the diagram found in wikipedia is the clearest. https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Nicomachus_theorem_3D.svg/2000px-Nicomachus_theorem_3D.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...
- aisofteng 10y 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 10y ago
  Beautiful !
_nalply 10y ago
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 10y ago
  For real-numbered measurements, dangerous. For integer measurements that are about seeing the counting re-arrangements, safer.
  - _nalply 10y ago
    Agreed. But this is an awfully fine point for proofs which are intended to be «simple».
- est 10y ago
  Think deeper. You can categorize proofs in to two sets, one is safe in visual and one is not.
agumonkey 10y ago
Very recursive structure.
acqq 10y ago
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
breakfastcereal 10y ago
I still prefer algebraic proofs
carapace 10y 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.
- jacobolus 10y ago
  And sometimes a circular area unit makes most sense: https://en.wikipedia.org/wiki/Circular_mil 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 10y ago
  I never thought of that. Where did he say that?
  - jacobolus 10y 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#966.20 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.
taeric 10y ago
There are full books on these called "Proofs Without Words". On my wishlist. :) https://amazon.com/dp/0883857006 https://amazon.com/dp/0883857006
lyle_nel 10y ago
I am not sure that this constitutes a proof. It seems induction would be required to show that the property holds.
lodi 10y ago
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
justifier 10y 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'

143 points by luisb 10y 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 10y 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://en.m.wikipedia.org/wiki/Mathematics_Magazine) https://www.researchgate.net/journal/0025-570X_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-twocubes-post-and-asked-to-make http://hyrodium.tumblr.com/post/94237657514/inspired-by-this... has a nice animation of this proof)
- gohrt 10y ago
  It's a "magazine", not a research journal. It's expoisitory recreational math for students.
Smaug123 10y 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 10y 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
- gohrt 10y 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.
  - justinpombrio 10y ago
    > 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 10y ago
      > 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.

deleted 10y ago

[deleted]

mrcactu5 10y ago

the same proof is discussed on math.stackexchange http://math.stackexchange.com/questions/61482/proving-the-identity-sum-k-1n-k3-big-sum-k-1n-k-big2-without-i http://math.stackexchange.com/questions/61482/proving-the-id...

derknorton2 10y 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 10y ago

I think the diagram found in wikipedia is the clearest. https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Nicomachus_theorem_3D.svg/2000px-Nicomachus_theorem_3D.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...

aisofteng 10y 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 10y ago
Beautiful !

_nalply 10y ago

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 10y ago
For real-numbered measurements, dangerous. For integer measurements that are about seeing the counting re-arrangements, safer.
- _nalply 10y ago
  Agreed. But this is an awfully fine point for proofs which are intended to be «simple».
est 10y ago
Think deeper. You can categorize proofs in to two sets, one is safe in visual and one is not.

agumonkey 10y ago

Very recursive structure.

acqq 10y ago

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

breakfastcereal 10y ago

I still prefer algebraic proofs

carapace 10y 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.

jacobolus 10y ago
And sometimes a circular area unit makes most sense: https://en.wikipedia.org/wiki/Circular_mil 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 10y ago
I never thought of that. Where did he say that?
- jacobolus 10y 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#966.20 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.

taeric 10y ago

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

lyle_nel 10y ago

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

lodi 10y ago

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

justifier 10y 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'