archiver 8

Tuesday, February 07, 2006

reposting 3 posts.

tag: recovery math

My unlisted three posts on February 4:

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link: EWD975 Dijkstra's Pythagorean Theorem

tag: math. auxiliary. construct. formulate.

good example of constructing an auxiliary figure.
good example of symmetric formulation of a given theorem.

pnt: He takes into account that the formulation be made symmetric.

pnt: In constructing the auxiliary figure, Dijkstra considers that the difference alpha+beta-gamma should be constructed and that it should be done in a way that alpha and beta are symmetric. hence that germ of the figure. Lesson is that the auxiliary figure is not pulled out of a magicians hat, but follows naturally from the theorem to be proved.

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an inductive proof: example
link: Wallace-Bolyai-Gerwien Theorem at cut-the-knot

tag: inductive. math. proof. example. divide. step. intermediate. auxiliary.

my notation
a + b : a and b put together.
a = b : a and b are equidecomposable.

the proof of the theorem combines various simple equidecomposability results in these steps:
1. polygon = triangle + triangle + ...
2. triangle = rectangle = square
3. square + square + ... = big square
4. (from 1&2&3) therefore polygon = square
5. (finally) polygon = square = another polygon.

1&3 are examples of "divide and conquer", in particular, decomposing the mathematical object into simpler pieces.

2 are examples of "step by step" or "intermediate object": where rectangle is the intermediat object.
4to5 is an example of introducing "intermediate object", where square is the intermediate object for the proof.

the following three simple equidecomposability results are used in the proof:
* square = rectangle
* rectangle = triangle
* square + square = square

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That's not the only way: the largest example
link: Pythagorean Theorem and its many proofs at cut-the-knot

54 proofs of the pythagorean theorem.

lesson: there are always other ways to prove a given theorem.

tag: math. alternative. proof. many.


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