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William_Thompson
2005-Dec-25, 10:34 PM
When I was a kid, one of my most favorite things to do was to solve mysteries. If there was a magician on television, I got more enjoyment figuring out how the trick was done than being amazed by the trick itself.

There was elation in figuring something out and I always wanted to share it with people.

When I learned of the Pythagorean Theorem I was taught that it was true simply because regardless of the dimension of the right angled triangle, the hypotenuse squared always equaled the sum of the other two sides squared. But when asked why, the teacher simply said that it just happens to always work out that way and that there wasn’t really any proof written.
http://www.completetranslations.com/christmas_math/one.JPG
Well, I took it upon myself to find one. And I did. And as a gift I am going to post it here now.

There are two interesting observations about a right angle triangle. obviously, you can make a rectangle with two identical right triangles put together at their hypotenuse and the area of this rectangle would be a∙b.
http://www.completetranslations.com/christmas_math/two.JPG
What is less obvious is that if you take the two identical rectangles from the first observation and make a square by using the hypotenuses as the parameter and the area of this square would be c². This square form the second observation would have a square inside of it whose side would measure b-a and so its area is (b-a)².
http://www.completetranslations.com/christmas_math/three.JPG
http://www.completetranslations.com/christmas_math/four.JPG

No matter what proportion of right angled triangle you use, the result is always that you can form a perfect square with four identical triangles using their hypotenuse and you will always have a little empty square in the middle. I find it interesting conceptualizing this as you go through the possibilities, the little empty square in the middle spins and collapses like the iris on a camera.

Regardless of the size of the empty square formed in the middile of the square whose sides are of lenght "c". and regardless of the proportions of the 4 identical triangles, the length of each side of the inner square will always be b-a.

I call the area of the whole square, "Area C" where each side of the square is made of the hypotenuse of 4 identical triangles.

And the square area of the empty space inside this square I will call "Area B".

And the area formed by the two rectangles to be called "Area A".

The proof is simple algebra and substitution using the equations determined:

http://www.completetranslations.com/christmas_math/five.JPG

teri tait
2005-Dec-25, 11:24 PM
Thank you for sharing in easily digestible terminology.

[this could be the one missing truth always wondered deep inside my obscure pan]

Love and thamks again!

Teri Tait

paulie jay
2005-Dec-25, 11:40 PM
What has always sat uncomfortably in the back of my mind about Pythagoras' theorem is that if we use it to determine the hypotenuse of a triangle where the sides are 1cm x 1cm, our answer for the hypotenuse is an irrational number, even though the hypotenuse must have a specific length. Don't get me wrong, I'm not disputing Pythagoras, I just don't like to see finite measurements expressed with irrational numbers.

Nowhere Man
2005-Dec-26, 12:16 AM
You're in good company. The irrationality of the square root of 2 really got under Pythagoras's skin as well, to the point that he swore his cronies to secrecy on the subject, and they killed one of their own for blabbing.

Think about the circumference of a circle, as well. If the radius is 1, the circumference is 2*pi, another irrational. They're everywhere.

Fred

peteshimmon
2005-Dec-26, 01:41 AM
Reminds me of a little bit of scribbling I
was doing some years ago. All possible right
angled triangles could be represented up to
the 45 degree angle. The side also
represented the loci I thought. Then The
family of all triangles where the sides were
equivalent to a cubic was represented with
the loci an "s" shape going up. All higher
powers were shifted from this curve a bit
until the infinite power relation between the
sides which was the same as the square
relation. So why could we not get integer
solutions on the curved loci? It stumped
me and Fermats last theorem remained
unsolved:) Hope you follow me.

Tensor
2005-Dec-26, 02:56 AM
When I learned of the Pythagorean Theorem I was taught that it was true simply because regardless of the dimension of the right angled triangle, the hypotenuse squared always equaled the sum of the other two sides squared. But when asked why, the teacher simply said that it just happens to always work out that way and that there wasn’t really any proof written.
http://www.gelsana.com/christmas_math/one.JPG
Well, I took it upon myself to find one. And I did. And as a gift I am going to post it here now.

Well, not to spoil it (you did find it independently, which counts for something) but this (http://www.cut-the-knot.org/pythagoras/index.shtml) site has 54 proofs.

Lance
2005-Dec-26, 02:59 AM
Well, not to spoil it (you did find it independently, which counts for something) but this (http://www.cut-the-knot.org/pythagoras/index.shtml) site has 54 proofs.
Hmmm...

I wonder if he included a gift receipt then...

montebianco
2005-Dec-26, 09:11 AM
Think about the circumference of a circle, as well. If the radius is 1, the circumference is 2*pi, another irrational. They're everywhere.

Well, most real numbers are irrational :D

Irrational numbers don't bother me at all, and I wonder how long this demeaning and insulting term will continue to be used :D

HenrikOlsen
2005-Dec-26, 10:42 AM
But when asked why, the teacher simply said that it just happens to always work out that way and that there wasn’t really any proof written.
This is an atypical example of a bad teacher accidentally triggering something good.
BTW, I checked your proof, it may be because you posted a simplified version, but it looks to me that you only proved the case where one of the sides is 0, and missed the step where you prove that the iris'ing always gives the same result.



Well, most real numbers are irrational :D

Irrational numbers don't bother me at all, and I wonder how long this demeaning and insulting term will continue to be used :D
Probably for a very long time.
The term is neither demeaning nor insulting in the mathematical context.

montebianco
2005-Dec-26, 11:00 AM
This is an atypical example of a bad teacher accidentally triggering something good.
BTW, I checked your proof, it may be because you posted a simplified version, but it looks to me that you only proved the case where one of the sides is 0, and missed the step where you prove that the iris'ing always gives the same result.

I haven't checked the proof, but I do know that I did not write what you have attributed to me.


The term is neither demeaning nor insulting in the mathematical context.

Yes, I know, just note the :D

parallaxicality
2005-Dec-26, 12:20 PM
I think all irrational numbers prove is that geometry is not applicable to the real universe. You cannot construct a square of exactly one cm by one cm, and you cannot draw an utterly perfect circle. Thank God for that, or nothing could physically exist.

Maksutov
2005-Dec-26, 12:33 PM
It is wonderful that William_Thompson would take time out from his transcendent endeavors to stoop so low in order to provide us underlings a bit of his knowledge: a gift, as it were.

Thank you, William_Thompson, for that contribution. I am now overwhelmed by the brilliance discerned therein and am sure the rest of the BAUT realizes the incredible genius they are in the presence of. Congratulations on moving from fractions to decimals (decimals being where things aren't quite so convenient).

Meanwhile Carl Sagan was spot on re his opinion of Pythagoras. The concealment of knowledge is always a bad, anti-scientific thing.

N C More
2005-Dec-26, 01:42 PM
It is wonderful that William_Thompson would take time out from his transcendent endeavors to stoop so low in order to provide us underlings a bit of his knowledge: a gift, as it were.

But the question remains, will those of us who are wallowing in mediocrity (http://www.bautforum.com/showpost.php?p=626863&postcount=1) be able recognize this "pearl" cast before us, mere "swine"? http://www.cosgan.de/images/smilie/tiere/c015.gif

William_Thompson
2005-Dec-27, 12:10 AM
It is wonderful that William_Thompson would take time out from his transcendent endeavors to stoop so low in order to provide us underlings a bit of his knowledge: a gift, as it were.

Thank you, William_Thompson, for that contribution. I am now overwhelmed by the brilliance discerned therein and am sure the rest of the BAUT realizes the incredible genius they are in the presence of.

I almost expected and wondered if people would find reason to disagree and argue.

I think I will make such purely mathematical posts for a while until my blood pressure drops.


Congratulations on moving from fractions to decimals (decimals being where things aren't quite so convenient).
Would you care to elaborate?


Meanwhile Carl Sagan was spot on re his opinion of Pythagoras. The concealment of knowledge is always a bad, anti-scientific thing.
Would you care to elaborate?

William_Thompson
2005-Dec-27, 12:17 AM
This is an atypical example of a bad teacher accidentally triggering something good.
BTW, I checked your proof, it may be because you posted a simplified version, but it looks to me that you only proved the case where one of the sides is 0, and missed the step where you prove that the iris'ing always gives the same result.


http://www.completetranslations.com/christmas_math/four.JPG

The image on the top shows this iris'ing effect.

The image at the bottom is if we started with 45 degree angled right triangles.

In fact, I never used an example where any triangle had a side with length zero.

OK, I went back and added some more explaination in bold.

William_Thompson
2005-Dec-27, 12:20 AM
But the question remains, will those of us who are wallowing in mediocrity (http://www.bautforum.com/showpost.php?p=626863&postcount=1) be able recognize this "pearl" cast before us, mere "swine"? http://www.cosgan.de/images/smilie/tiere/c015.gif

no




:razz:

01101001
2005-Dec-27, 12:35 AM
This being the day after Christmas, I'd like to exchange the slightly irregular line:

c2 = 2(a.b) + ((b - a)(b - a)2

for:

c2 = 2(a.b) + (b - a)(b - a)

William_Thompson
2005-Dec-27, 12:38 AM
This being the day after Christmas, I'd like to exchange the slightly irregular line:

c2 = 2(a.b) + ((b - a)(b - a)2

for:

c2 = 2(a.b) + (b - a)(b - a)

unclosed paren.. Good call.

DOH! :doh:

c2 = 2(a.b) + ((b - a)(b - a)2
should be
c2 = 2(a.b) + ((b - a)(b - a))
or
c2 = 2(a.b) + (b - a)2

I must have been interrupted while jotting this down. It is a family holiday.

HenrikOlsen
2005-Dec-27, 09:25 AM
http://www.gelsana.com/christmas_math/four.JPG

The image on the top shows this iris'ing effect.

The image at the bottom is if we started with 45 degree angled right triangles.

In fact, I never used an example where any triangle had a side with length zero.

OK, I went back and added some more explaination in bold.
Ah, the iris'ing is irrelevant for the proof, it's just a nice extra.
You confused me by the bottom image, which isn't actually used in the proof.

William_Thompson
2005-Dec-27, 06:47 PM
Ah, the iris'ing is irrelevant for the proof, it's just a nice extra.
You confused me by the bottom image, which isn't actually used in the proof.

I think I will add this (in red above):

Regardless of the size of the empty square formed in the middile of the square whose sides are of lenght "c". and regardless of the proportions of the 4 identical triangles, the length of each side of the inner square will always be b-a.

William_Thompson
2005-Dec-27, 06:51 PM
Well, not to spoil it (you did find it independently, which counts for something) but this (http://www.cut-the-knot.org/pythagoras/index.shtml) site has 54 proofs.

I glanced over the list and did not find one like mine. Maybe mine is unique. Maybe I should clean it up and submit it to that website.

I think mine is easier to grasp too. Maybe it should be shipped off to grade schools.

William_Thompson
2005-Dec-27, 06:52 PM
Hmmm...

I wonder if he included a gift receipt then...

I didn't see a proof like mine nor one that is as easy to grasp.

So it is still a gift, Lance.

Merry Christmas.

Nicolas
2005-Dec-27, 06:57 PM
I think all irrational numbers prove is that geometry is not applicable to the real universe. You cannot construct a square of exactly one cm by one cm, and you cannot draw an utterly perfect circle. Thank God for that, or nothing could physically exist.

I lost you there I'm afraid. Why?
upto the molceular dimensions of the ink or material used we can in theory.


btw William_Thompson, our high school teacher gave us a proof of Pythagoras which we could all understand (the first one in the linked site). I forgot how it went, but it had something to do with drawing squares on each side IIRC (i checked the site: indeed it did :)). Anyway it was part of the standard lesson subjects, and easily understandable for all of us (16 yo). Too bad your teacher failed to show your class the validity of Pytahgoras' theorem. But anyway already six years ago (and I'm sure tens of years before that as well) we had easy proofs of Pythagoras readily available at schools here, so there's no need to rush to schools I think. In fact we used that proof n°1 because we needed other high school knowledge (similar triangle rules such as SAS) which was good for educational purposes. We saw that proof not when we learned Pythagoras but when we learned the SAS etc rules. We saw proof n°3 (the same as your proof) when we learned Pythagoras which was some years before that.

That does not reduce your feat of finding a proof yourself of course. If I see it correctly, your proof is proof n°3 on the site. Maybe you missed it in your glance.

btw you've got a ² instead of a ) in line 6 of your maths.

HenrikOlsen
2005-Dec-27, 07:10 PM
I forgot how it went, but it had something to do with drawing squares on each side IIRC.
Sounds like it could well be the one from Euclid's Elements, which is definitely at a level where highschoolers will get it.
Looking back, I would have gotten it if presented with it when I was 10, it was around that age I found my fathers copies of Martin Gardner's first two book's of Mathematical Puzzles and Diversions.

ToSeek
2005-Dec-27, 07:25 PM
I glanced over the list and did not find one like mine. Maybe mine is unique. Maybe I should clean it up and submit it to that website.

I think mine is easier to grasp too. Maybe it should be shipped off to grade schools.

Proof #3 seems to be essentially the same as yours.

William_Thompson
2005-Dec-27, 09:51 PM
btw William_Thompson, our high school teacher gave us a proof of Pythagoras which we could all understand (the first one in the linked site). I forgot how it went, but it had something to do with drawing squares on each side IIRC (i checked the site: indeed it did :)). Anyway it was part of the standard lesson subjects, and easily understandable for all of us (16 yo).



That was an example and not a proof, right?

We all saw those squares that we could add up.

It was just an example.




Too bad your teacher failed to show your class the validity of Pytahgoras' theorem. But anyway already six years ago (and I'm sure tens of years before that as well) we had easy proofs of Pythagoras readily available at schools here, so there's no need to rush to schools I think. In fact we used that proof n°1 because we needed other high school knowledge (similar triangle rules such as SAS) which was good for educational purposes. We saw that proof not when we learned Pythagoras but when we learned the SAS etc rules. We saw proof n°3 (the same as your proof) when we learned Pythagoras which was some years before that.

That does not reduce your feat of finding a proof yourself of course. If I see it correctly, your proof is proof n°3 on the site. Maybe you missed it in your glance.

aw :(

story of my life (luckily not always, tho')




btw you've got a ² instead of a ) in line 6 of your maths.

We got that already.

Will I be forgiven if I go back and clean up the graphic?

William_Thompson
2005-Dec-27, 09:53 PM
Proof #3 seems to be essentially the same as yours.

oh well
:(

Nicolas
2005-Dec-27, 10:10 PM
That was an example and not a proof, right?

We all saw those squares that we could add up.

It was just an example.



The squares proof (proof n°1 on the linked site) is the one I meant, and it is as much proof as a proof can be. It proves pythagoras purely by the use of mathematical formulas and goniometry, so it is a pure proof



We got that already.

Will I be forgiven if I go back and clean up the graphic?
Oops, my bad. I missed the posts where that one was corrected :). Of course you will be forgiven :).



btw I have some questions on your grand unification theory, which I don't know how to interpret.

HenrikOlsen
2005-Dec-27, 10:47 PM
The squares proof (proof n°1 on the linked site) is the one I meant, and it is as much proof as a proof can be. It proves pythagoras purely by the use of mathematical formulas and goniometry, so it is a pure proof

It is the proof from book 1 of the Elements.
Since it's all about "if this triangle is the same as that triangle and each of these rectangles are double the area of one of those triangles, then these rectangles have the same area", it doesn't use formulae at all, just geometry.

Incidentally, there's no goniometry either. No angles where measured in the course of the proof, they where just proven equal.

Nicolas
2005-Dec-27, 10:52 PM
"Goniometric formulas" was what we called them (like the SAS equality, Pythagoras' Theorem etc). If that aren't formulas officially, then indeed no formulas were used :).

About goniometry: it depends on what you call "measuring". If measuring needs to include quantification, nothing is measured. Angles were put equal, so they were measured relatively to each other one could say. But here too, I'm not really into the strict definition of the term "goniometry" so I don't know whether it is applicable here.

I'm not good in the definitions of these terms (formula, goniometry, geometry) so I don't know about that.

but anyway I strongly feel that this proof n°1 from the site is a strong proof, not a inductive conclusion from inaccurate measurements or a single example or something like that.

William_Thompson
2005-Dec-27, 11:06 PM
btw I have some questions on your grand unification theory, which I don't know how to interpret.

Some of my cards I am holding until The Library of Congress returns my request for a copyright. This is one of them.

HenrikOlsen
2005-Dec-27, 11:09 PM
It's a very strong proof, especially when you remember that the way it was "originally" presented was as the last of a series of proofs, each building on the previous. I.e. what you call the SAS equality can be used (in Proposition 47) because it has already been proven (in Proposition 4).
Here's (http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html) a good presentation of the whole book.

ToSeek
2005-Dec-28, 03:05 PM
I have a t-shirt somewhere with Euclid's diagram from I.47 on it. (My "Great Books school" (http://www.sjca.edu/asp/home.aspx) started out the math curriculum with Euclid.)

William_Thompson
2005-Dec-29, 06:54 PM
I just got an email from Nature urging me to write a cover letter. I guess that means they are not un-interested.
But this is OT.
I am just saying I might have more fun things to post soon.

ZaphodBeeblebrox
2006-Jan-05, 07:44 PM
I just got an email from Nature urging me to write a cover letter. I guess that means they are not un-interested.
But this is OT.
I am just saying I might have more fun things to post soon.
GOOD For you ...

May you Only!!!!

Nicolas
2006-Jan-05, 09:26 PM
I just got an email from Nature urging me to write a cover letter. I guess that means they are not un-interested.
But this is OT.
I am just saying I might have more fun things to post soon.

What is Nature planning to do with this proof that, while being really nice has been covered many times in the past both in books and on the internet?

*********
I'm looking forward to the other things! Are they math related as well?

Oh btw how's the patent application going?

HenrikOlsen
2006-Jan-05, 10:12 PM
I don't think the Nature thing has anything to do with the proof shown here.:)

Nicolas
2006-Jan-05, 10:23 PM
Oh was it about the grand theory?

Fram
2006-Jan-06, 08:24 AM
Yep, the one he presented on this board (well, the conclusion), then linked to in a separate new thread, and then refused to explain or discuss because he waits for his copyright notification from the library or congress, but which he has supposedly sent to Nature yet...