Proof without words: Cubes and Squares
fermatslibrary.com
fermatslibrary.com
https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Ni...
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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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.
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.
Jacobolus' comments are great.
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
http://math.stackexchange.com/questions/61482/proving-the-id...
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'
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.
Damn cool, by the way.
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)
[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 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
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
...
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.