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Nov
10
comment Determining if Graphs are Isomorphic.
@wbrugato also, I'm pretty sure that, even if the number of edges is fixed, there's no way they could be acyclic (it would imply tree implying 7 edges and degree 1 for at least 2 vertices)
Nov
10
comment Determining if Graphs are Isomorphic.
@lhf Are two graphs isomorphic if they exhibit the same degree of nonexistence?
Nov
9
revised Assumptions needed for proof of the Pythagorean Theorem from examples
slight rewording
Nov
9
comment Is it possible to formulate category theory without set theory?
Russell's paradox happens in CT, too. And independently, you can take CT as a base formalization and embed set theory in it (similar to the usual reverse situation to bootstrap people into understanding of category theory by starting off stating CT in terms of sets of objects, morphism, etc.)
Nov
7
awarded  Benefactor
Nov
7
comment Assumptions needed for proof of the Pythagorean Theorem from examples
@SammyBlack Understood. It may very well be that to establish the form of the relation among the sides that you've already gotten the constants and proven PT. And if you knew the form were $d a^2 + e b^2 = f c^2$ only one instance would set the coeffs to 1. I realize my "any other assumptions" is a bit broad, but are there any restrictions on form that leave a non-trivial interpolation?
Nov
6
comment Tiling of squares in instances of Pythagoras Theorem
Anatoly: proof #3 at cut-the-knot.org/pythagoras OK, #4 too.
Nov
6
comment Solution of an integral with strange imprecision of gamma functions
Have you tried the usual strategies, like trying to solve (or have MMA solve) simpler versions first: $\epsilon -> 0$, $\alpha = 1$, $m=n$, etc?
Nov
6
asked Assumptions needed for proof of the Pythagorean Theorem from examples
Nov
6
revised Tiling of squares in instances of Pythagoras Theorem
explanation of usefulness of such a tiling
Nov
6
comment Tiling of squares in instances of Pythagoras Theorem
This is almost good enough for a very basic class in geometry. The only 'hard' part is that A + B = 90. But the smart ones in class should be able to see that has the basic idea of a proof of the full general PT using the algebraic identity $a^2 + b^2$ (the sum of the areas of the two leg squares) $ = (a-b)^2$ (the center full unit squares) $ + 4 (a b)/2 $ (the areas of the 4 triangles) $ = c^2$ (the area of the square on the hypotenuse)
Nov
6
accepted Tiling of squares in instances of Pythagoras Theorem
Nov
6
comment Tiling of squares in instances of Pythagoras Theorem
Anatoly, see my new image. This is the tiling I'm thinking of. In your tiling, the tiling of the square on the hypotenuse is 'inscrutable', you can't tell what the size and orientation of the tiles in it are.
Nov
6
revised Tiling of squares in instances of Pythagoras Theorem
added a12sqrt5 triangle
Nov
5
comment Tiling of squares in instances of Pythagoras Theorem
Anatoly, yes, eight possible orientations of the tile are allowed. Essentially so you can look at the picture and say "Yes, this is a correct tiling" and then just count tiles to see that they add up.
Nov
5
comment Tiling of squares in instances of Pythagoras Theorem
I think you can. You can tile the unit(ary) square as you've done in your picture. Also in your picture, the original 345 triangle is tiled. Rearrange 4 copies of the original triangle around the unit square to get the square on the hypotenuse, with all the mini-tiles axis-aligned. I don't have a good drawing program. What did you use? The smallest nontrivial one works with a = 1, b = 4, c = $\sqrt{5}$ all with 1/2, 1, $\sqrt{5}/2$ tiles.
Nov
5
comment Tiling of squares in instances of Pythagoras Theorem
Nice. But is there a way to rearrange the tiling of the 5x5 so that it is obvious that it works (without assuming PT)? Say by filling the 5x5 with filings of 4 3x4s and a single unit (also tiled with the mini triangles).
Nov
4
comment Tiling of squares in instances of Pythagoras Theorem
That tiles the triangle, but how does a 1/a,1/b triangle (or rectangle) tile an a by a square (and similarly b by b square). I think I see it but am not sure. Also, how do they tile the square on the hypotenuse? I don't see that at all. Can you draw a picture?
Nov
3
awarded  Promoter
Nov
2
awarded  Nice Answer