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# Beautiful, intriguing, elegant ideas Topic Page: 1 2

I would like to know what scientific ideas you have found to be stunningly beautiful, breath-taking, elegant, intriguing, inspiring.

I'll start. Kepler's Second Law of Planetary Motion. This is the one which says if the orbit of a planet around a heavier body (i.e. sun) is an ellipse, and if one considers the distance the planet travels along the ellipse in the same amount of time, but at two different parts of the ellipse; one span being near perihelion and another near aphelion, the areas "covered" will be equal.

The idea I am trying to convey, which most of the readers here already know, is much more clearly stated with an accompanying diagram. Also, some of you probably know how to just use language better to express his seminal idea more simply and clearly.

Back to the point; I think this idea is stunning. The geometry of euclidean space is such, and the nature of gravity within that space is such, that it sweeps out equal areas. Who would have thought....!!

Pesse ( E=MC^2 ) Mist

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relativistic mass:
m(r) = m(0)/sqrt(1 - v^2/c^2)

Also very elegant is how e = mc^2 is directly derived from the above expression for relativistic mass; first restate the equation for relativistic mass as:

m(r) = m(o)*(1 - v^2/c^2)^-1/2
...using the binomial theorum to expand the above expression in a power series:
m(r) = m(0)(1 + 1/2(v^2/c^2) + 3/8(v^4/c^4) +....)
...which, when v is small, converges rapidly to:
m(r) = m(0) + 1/2 m(o)v^2(1/c^2)
...now, multiply both sides by c^2
m(r)c^2 = m(o)c^2 + 1/2m(0)v^2
...the last term on the right side is ordinary kinetic energy, the left term on the right side is the intrinsic energy of a body at rest. The term on the left is usually encapsulated as simply 'e', which incorporates both the intrinsic "rest" energy and kinetic energy expressions on the right.

*thanks to Feynman's "Lectures on Physics", pp 15-8 through 15-11 for providing this clear, and surprisingly straightforward mathematical explanation.

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Og...inventing the wheel...

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The Mayan calendar...

--------------------- --------------------- "Nothing exists but atoms and empty space. Everything else is opinion." Titus Lucretius Carus 99-55 B.C.
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FirstSight:

I would say that the beauty of your example lies in the mathematics, not in the science, and that the line between the two is in itself interesting. I cannot think, offhand, of a scientific idea that is beautiful without it receiving a mathematical formulation. This is one of the main points of Penrose's The Road to Reality, section 34.2 (page 1014). I can think of beautiful experiments--Newton's prism--but scientific ideas...? On the other hand, mathematics is filled with beauty---IMHO. Thanks for your example.

Dark skies.

Jack

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Making fire burn downwards.

In his Nicomachean Ethics, Aristotle asserted that persons are not born morally good or bad but are taught good behavior and bad behavior. This idea was the birth of all subsequent moral education in the western world. If good behavior could be learned, it could then be taught.

To make this point Aristotle contrasted moral values with things that happened "by nature". He pointed out that things which occurred "by nature" always happened the same way regardless of how many attempts were made to change their "way"; their "habit". One example he used was that although a person could be taught to be generous, a flame could never be taught to burn downwards.

However, in his 1869 Christmas lectures to children at the Royal Institution of Great Britain entitled The Chemical History of a Candle, Michael Faraday described and executed an experiment in which the fire of a candle was induced to burn downwards.

This novel idea, experiment and description in no way affect Aristotle's ethical ideas. However they do evidence an improvement of modern science compared to the more deductive methodology of ancient science; the importance of testing assumptions such as, fire always burns upwards.

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If true, the discovery of the Higgs boson.

Mike (tVA) Is a universe that contains self-awareness aware?
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NOBODY would be here discussing this if Ice didn't float....

Tony

"After the Laws of Physics, everything else is opinion" -Neil deGrasse Tyson

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Two equations from Sir Isaac certainly got the ball rolling: F=ma and F=Gm1m2/d^2

Dave Mitsky

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Natural Selection is one.

- Rick

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"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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e^(i*pi) = -1

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Quote:

e^(i*pi) = -1

This is the one I was going to reply with too.

The Euler identity. A simple looking equation but it involves three fundamental constants - e, pi and i. (And 1 if you count that as a fundamental constant.)

I don't understand most of the maths behind it, but it's one of the most thought-provoking single lines of mathematics known...

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e^(i*pi) = -1

-i^2 like

You can argue, this is the most important formula in all of science and engineering.

-drl

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You can argue, this is the most important formula in all of science and engineering.

I always thought that honor belonged to the formula for beer...

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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the square root of a negative number..

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.

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And there was light.

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Quote:

Quote:

You can argue, this is the most important formula in all of science and engineering.

I always thought that honor belonged to the formula for beer...

Zounds, I had forgot the beer.

-drl

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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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Demonstration of the Pythagorean Theorem using squares (actual squares)

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

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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

“The first gulp from the glass of natural sciences will turn you into an atheist, but at the bottom of the glass God is waiting for you.” â€• Werner Heisenberg

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

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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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.

--------------------- --------------------- "Nothing exists but atoms and empty space. Everything else is opinion." Titus Lucretius Carus 99-55 B.C.
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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...

<Insert droll quotation or ridiculous equipment list here>

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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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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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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.

"Scientists aren't perfect, just peer reviewed."
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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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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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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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(x)~(x=x)

Mike (tVA) Is a universe that contains self-awareness aware?
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is this what you were thinking?

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

Demonstration of the Pythagorean Theorem using squares (actual squares)

Clear skies - Jim Crazy PNW imager !

A-P Mach1 on 12" concrete pier - ROR structure.

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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...

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

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

Dave Mitsky

A man is a small thing, and the night is very large and full of wonders.

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