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Otto Piechowski
Pooh-Bah


Reged: 09/20/05

Loc: Lexington, KY
Re: Beautiful, intriguing, elegant ideas new [Re: Otto Piechowski]
      #5601124 - 01/02/13 11:50 AM

Demonstration of the Pythagorean Theorem using squares (actual squares) of the various sides of the various triangles created by connecting and extending lines of the sides of the original right triangle and connecting points of the original corners of the right triangle with various corner points of the original three squares or rectangles of the three sides, etc.

This demonstrated the truth of the proof not trigonometrically, or algebraically, or even with the use of the axioms and conclusions of geometry, but in a totally visual manner.


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Ravenous
professor emeritus


Reged: 11/14/09

Loc: UK
Re: Beautiful, intriguing, elegant ideas new [Re: Otto Piechowski]
      #5601136 - 01/02/13 12:03 PM

Quote:

Demonstration of the Pythagorean Theorem using squares (actual squares)



Can we prove (or disprove) similar for, say, cubes?


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Qwickdraw
Carpal Tunnel
*****

Reged: 03/03/12

Loc: Ann Arbor, MI
Re: Beautiful, intriguing, elegant ideas new [Re: Otto Piechowski]
      #5601236 - 01/02/13 01:09 PM

Magic squares have always intrigued me particularly the 6X6 sided where the sum of all rows,columns and diagonals add up to 111 and the the sum of all rows or columns is 666



magic square examples


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Rick Woods
Postmaster
*****

Reged: 01/27/05

Loc: Inner Solar System
Re: Beautiful, intriguing, elegant ideas new [Re: Qwickdraw]
      #5601357 - 01/02/13 02:34 PM

When I was small, my dad showed me a Mobius strip.
That totally blew my mind! (which probably explains a few other things.)


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ColoHank
Carpal Tunnel
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Reged: 06/07/07

Loc: western Colorado
Re: Beautiful, intriguing, elegant ideas new [Re: Rick Woods]
      #5601467 - 01/02/13 03:49 PM

The Golden Ratio:
a+b/a = a/b = 1.618033988...
and its reciprocal b/a = 0.618033988...

I used it quite a bit back in my cabinet-making days.


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Andy Taylor
Twisted, but in a Good Way
*****

Reged: 09/24/08

Loc: Epsom - UK
Re: Beautiful, intriguing, elegant ideas new [Re: ColoHank]
      #5601635 - 01/02/13 05:30 PM

Quote:

The Golden Ratio:
a+b/a = a/b = 1.618033988...
and its reciprocal b/a = 0.618033988...

I used it quite a bit back in my cabinet-making days.





Yup - calculated the most visually pleasing length of dew shield for my atm refractor with this ratio...


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deSitter
Still in Old School


Reged: 12/09/04

Re: Beautiful, intriguing, elegant ideas new [Re: ColoHank]
      #5605167 - 01/04/13 04:36 PM

Quote:

The Golden Ratio:
a+b/a = a/b = 1.618033988...
and its reciprocal b/a = 0.618033988...

I used it quite a bit back in my cabinet-making days.




The crazy thing is how this number emerges from the ruler and compass construction of a regular pentagon. G = (sqrt(5) - 1) / 2.


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CounterWeight
Postmaster
*****

Reged: 10/05/08

Loc: Palo alto, CA.
Re: Beautiful, intriguing, elegant ideas new [Re: deSitter]
      #5605383 - 01/04/13 06:58 PM

I believe that was why the pythagoreans chose the pentagon as their secret symbol?

+,-,/,*, arithmetic, algebra, geometry, Reimann(ian) geometry, trigonometry, exponential, root, log, functions,sigma summation, derivitive calc, integral calc, zymergy, DC electricity, AC and polyphase electricity, steam, periodic table, farming, floatation and boats, sailing and navigation, whoever figured when it was safe to eat oysters!, and that there was something good to eat in an artichoke (your going to try and eat that?), all the healing arts and medicine, most all semiconductor stuff, radio, RADAR, electromagnetic waves, satellites, space telescopes, Lagrange points, ... 'modern coffee apparatus'... the printing press, refraction and reflection of light - optics at large, pencils and erasers, musical instruments and sound (withing some limits for me personally). Periodic table. most things to do with chemestry especially the carbon cycle and the nitrogen cycle. Proof that pi is trancendental... are the same (though we dont know it completely) number when and where pops up (if you want to do the proof my hats off to you!) and all that came of that. Kelvins "On an Absolute Thermometric Scale", Libraries. Anything to do with maths for spacetime that is predictive and somehow testable.

I think all those are prtty interesting off the top on my head.


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llanitedave
Humble Megalomaniac
*****

Reged: 09/26/05

Loc: Amargosa Valley, NV, USA
Re: Beautiful, intriguing, elegant ideas new [Re: CounterWeight]
      #5605501 - 01/04/13 08:15 PM

I'd like to include plate tectonics in all this, but while it's certainly a beautiful and intriguing process, it's far from anything I'd consider "elegant". In fact, it's a chaotic mess.

If the devil is truly in the details, plate tectonics is one of the most devilish theories out there. Seems to be a bit whimsical on multiple scales.


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Qwickdraw
Carpal Tunnel
*****

Reged: 03/03/12

Loc: Ann Arbor, MI
Re: Beautiful, intriguing, elegant ideas new [Re: llanitedave]
      #5606217 - 01/05/13 09:50 AM

Speaking of Pi, I was watching the show "Person of interest" last night and the guy mentioned that Pi, having non-recurring infinite numbers has within it every conceivable number sequence possible and if you assign letters to the numbers also every conceivable word in any language. I have never really thought about it in that way before. It really is amazing to me.

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Skip
Starlifter Driver
*****

Reged: 01/23/08

Loc: Fort Worth, Texas, USA
Re: Beautiful, intriguing, elegant ideas new [Re: Qwickdraw]
      #5606384 - 01/05/13 11:30 AM

I saw that as well. After he said that, I paused it and thought on that for awhile. Amazing to me too.

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JohnMurphyRN
sage
*****

Reged: 09/09/12

Loc: Near St Louis
Re: Beautiful, intriguing, elegant ideas new [Re: Ravenous]
      #5606940 - 01/05/13 04:41 PM

Quote:

Quote:

Demonstration of the Pythagorean Theorem using squares (actual squares)



Can we prove (or disprove) similar for, say, cubes?




Fermat's Conjecture (Fermat's Last Theorem)


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Mike Casey

*****

Reged: 11/11/04

Loc: El Pueblo de Nuestra SeƱora l...
Re: Beautiful, intriguing, elegant ideas new [Re: JohnMurphyRN]
      #5607186 - 01/05/13 07:35 PM

(x)~(x=x)

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CounterWeight
Postmaster
*****

Reged: 10/05/08

Loc: Palo alto, CA.
Re: Beautiful, intriguing, elegant ideas new [Re: Mike Casey]
      #5607281 - 01/05/13 08:56 PM

is this what you were thinking?

Can we prove (or disprove) similar for, say, cubes?

Demonstration of the Pythagorean Theorem using squares (actual squares)


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Ravenous
professor emeritus


Reged: 11/14/09

Loc: UK
Re: Beautiful, intriguing, elegant ideas new [Re: JohnMurphyRN]
      #5609682 - 01/07/13 06:57 AM

Quote:

Quote:

Quote:

Demonstration of the Pythagorean Theorem using squares (actual squares)



Can we prove (or disprove) similar for, say, cubes?




Fermat's Conjecture (Fermat's Last Theorem)



Correct, and I think a few of you picked up on my sly (if inaccurate) attempt at mathematical humour

Cubes do not add up in the way squares do. There was an old proof for cubes, over a century old I think...

There was an older (unproved) assertion that in fact, it doesn't add up for all exponents bigger than squares. My understanding is it's this Fermat reckoned he'd proved.

So Otto's suggestion of Pythagoras' theorem is another one of those ideas that has a lot more to it than at first appears...

Edited by Ravenous (01/07/13 06:58 AM)


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JohnMurphyRN
sage
*****

Reged: 09/09/12

Loc: Near St Louis
Re: Beautiful, intriguing, elegant ideas new [Re: Ravenous]
      #5610354 - 01/07/13 02:42 PM

The Pythagorean theorem is just a special case of the Law of Cosines where it cancels to zero: a2 + b2 - 2ab(COS of angle opposite c side) = c2. (Sorry for the strange notation - it's as close as my phone can get it) When the angle opposite c is 90* then COS is 0 and the 3rd portion = 0. I found this very interesting when I originally learned it. In fact, I'll submit it to Otto's list of elegant ideas.

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Otto Piechowski
Pooh-Bah


Reged: 09/20/05

Loc: Lexington, KY
Re: Beautiful, intriguing, elegant ideas new [Re: JohnMurphyRN]
      #5611302 - 01/08/13 01:56 AM

In Minnesota there's this saying, John, which applies to the idea you shared with us, "She's a keeper...." Originally, as I learned it, this saying applied to a game fish, not just big enough to keep (Northern Pike or Walleye usually) but especially big or beautiful or special or some such thing. But, then a mother can use this same line to describe the girl her boy brings home to visit for the first time; after which you hope she says "She's a keeper." And the same applies to the grandchildren that come along, though they don't really count because every grandchild is a keeper.

Now, what makes your idea, John, a "keeper" is that the beauty and fascination and specialness you found in that idea came through your words. Thus, your idea is, in my opinion, "a keeper" even though I have to be honest and say I didn't understand what you were saying (and, if you are of a mind to do so, would love to have you spell it out in greater detail so I could understand why it struck you so.)

Having lauded praise (ah, there's a tautology) on your idea, I have to say I have been thrilled, tickled, touched by the ideas presented here and the feelings of awe and pleasure communicated in the telling of the first experiences of the same. For example, the golden mean...that really hit me when I first ran into it. And the mobius strip just took my breath away (OK, a bit of an exaggeration, but not much). The same with the time/length/mass dilation formulae of relativity, etc. etc..

Otto

PS I saw the Horsehead Nebula for the first time in my life this evening, visually through a telescope.


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JohnMurphyRN
sage
*****

Reged: 09/09/12

Loc: Near St Louis
Re: Beautiful, intriguing, elegant ideas new [Re: Otto Piechowski]
      #5611998 - 01/08/13 01:55 PM

Otto,

The pythagorean theorem is nice and useful, but it only works for right triangles. Wouldn't it be great if there were a similar equation that works not just for right triangles, but for ALL triangles? Enter the Law of Cosines, which is just that. It isn't limited to triangles containing a 90degree angle. Details:

http://en.wikipedia.org/wiki/Law_of_cosines


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Otto Piechowski
Pooh-Bah


Reged: 09/20/05

Loc: Lexington, KY
Re: Beautiful, intriguing, elegant ideas new [Re: JohnMurphyRN]
      #5612117 - 01/08/13 02:58 PM

Very interesting, John.

And now that you use that phrase, "law of cosines", I begin, but only begin to remember that phrase from long ago.

When I was in high school...a very very good high school...the only math we had was algebra and geometry. I found an old (early 1900s) trig book and taught myself trig. Step by step I went through it. Only when I hit the section on spherical trigonometry did my progress falter and eventually end.

I am curious, is the pythagorean theorem or some version of it applicable to spherical trig (triangles on the surfaces of spheres and other shapes)?

Otto


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Dave Mitsky
Postmaster
*****

Reged: 04/08/02

Loc: PA, USA, Planet Earth
Re: Beautiful, intriguing, elegant ideas new [Re: Otto Piechowski]
      #5616636 - 01/11/13 03:31 AM

If a Möbius strip is going to be included, shouldn't a Klein bottle be too?

http://www.kleinbottle.com/whats_a_klein_bottle.htm

Dave Mitsky


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