Geometric geekery

[spudtater]

Non-algebraic proofs of Pythagoras' theorem:
- First an animation. Although it might not be strictly a proof, it's a great demonstration. (Alternate local copy.)
- Then a more thorough proof by tiling. A little less accessible, though.

I want to do geometric proofs! For example,

Is this a proof?

c d a ac ad b bc bd a+b (a+b)(c+d) c+d

Oh, and blah blah Pope blah.

Comments

[figg.livejournal.com]

The first one is a proof.
:D

[spudtater.livejournal.com]

Yay!
It's great, isn't it?

[galaxy-girl00.livejournal.com]

Large pat on the hand for that one!

[sigmonster.livejournal.com]

Yes, but how do I know that really and truly is a tesselation? Maybe it doesn't work always. You only know that it's an accurate representation if you already know an awful lot of Euclidean geometry; so I don't think it's a proof without a little bit more annotation. (Sum of angles in a triangle is enough, which depends on opposite angles in intersecting lines, which is proved from the axioms, iirc.)

(Euclid, thou shoulds't be living at this hour!)

The algebraic proof is fine. I actually mark this stuff from first years, and the only thing missing is the axioms that justify each step - commutative addition, multiplication distributive over addition, and commutative multiplication.

Can you prove that there is only one 1? It's such a pretty thing...

[spudtater.livejournal.com]

Well, if you're constructing a system of graphical proofs, you'd expect to build up and use a fairly high level of geometric proficiency.

I have been thinking about the uniqueness of 1, but just can't figure it out. Please put me out of my misery!

[sigmonster.livejournal.com]

A defining property of 1 (the identity in an algebraic system) is 1 \times r = r \times 1 = r, for any object r in the system. (Even for non-commutative systems, like matrix multiplication, the identity always commutes.)

So if there are two identities, 1_a and 1_b, we can write

1_a = 1_a \times 1_b = 1_b

and they are the same!

[spudtater.livejournal.com]

That proves that 1_a and 1_b have the same numeric value. It doesn't prove that they are identical.

[sigmonster.livejournal.com]

Hoh yus it does, sunshine. I'm using the strongest possible value of equals (in the given system), *both in the definition and in the equation*, which is identity (in the given system). Ought to be the triple parallel line symbol, really.

Otherwise you'd be saying that 1 \times r and r are not identical. As a matter of typography, that's trivially true - but there is no way at all to distinguish them by the algebraic rules, so we call them identical in that system.

We're getting into some deep waters, here...

[sigmonster.livejournal.com]

Just had another quick thought - this is also the reason 1 isn't a prime; you can tag on 1 \times 1 \times 1 ... and still have the identical integer, so you lose unique prime factorisation. In larger fields, you specify unique prime factorisation *up to units* (where a unit has a multiplicative inverse, basically).

[spudtater.livejournal.com]

Er... oh... okay.
Guess I wasn't taking 'defining property' as literally as I should have been.

[sigmonster.livejournal.com]

It's all consistent, and it all fits the real numbers and their extensions. It doesn't have to make sense.

See also: Proof by Intimidation.