Proof without words: Cubes and Squares(fermatslibrary.com)
fermatslibrary.com
Proof without words: Cubes and Squares
http://fermatslibrary.com/s/proof-without-words-cubes-and-squares
14 comments
Beautiful !
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?
Visual «proofs» are dangerous. Counter-example: Missing square puzzle: https://en.wikipedia.org/wiki/Missing_square_puzzle
Think deeper. You can categorize proofs in to two sets, one is safe in visual and one is not.
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.
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.
/\
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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
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
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.
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.
I never thought of that. Where did he say that?
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.
Jacobolus' comments are great.
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.
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.
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-id...
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'
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'
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
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
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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.
Damn cool, by the way.
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.
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:
[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)²/4The proof is simpler if you don't bother to simplify the total:
The inductive step in the OP theorem is:
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
> 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:
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
...
> 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.
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.
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)
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)
It's a "magazine", not a research journal. It's expoisitory recreational math for students.
I am not sure that this constitutes a proof. It seems induction would be required to show that the property holds.
This must be a reference to Mendelssohn's Songs without Words:
https://www.youtube.com/watch?v=y1uvYdW8MSk&t=1880
There are full books on these called "Proofs Without Words". On my wishlist. :) https://amazon.com/dp/0883857006
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)?
I still prefer algebraic proofs
Very recursive structure.
https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...