5 ms·

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

by 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.

by 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.