# Beautiful, intriguing, elegant ideas

### #1

Posted 29 December 2012 - 12:12 AM

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....!!

### #2

Posted 29 December 2012 - 04:49 AM

### #3

Posted 29 December 2012 - 10:28 AM

m® = 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® = m(o)*(1 - v^2/c^2)^-1/2

...using the binomial theorum to expand the above expression in a power series:

m® = m(0)(1 + 1/2(v^2/c^2) + 3/8(v^4/c^4) +....)

...which, when v is small, converges rapidly to:

m® = m(0) + 1/2 m(o)v^2(1/c^2)

...now, multiply both sides by c^2

m®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.

### #4

Posted 29 December 2012 - 02:48 PM

### #5

Posted 29 December 2012 - 04:31 PM

### #6

Posted 29 December 2012 - 08:55 PM

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

### #7

Posted 29 December 2012 - 09:43 PM

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.

### #8

Posted 29 December 2012 - 10:25 PM

### #9

Posted 30 December 2012 - 07:02 AM

### #10

Posted 30 December 2012 - 01:39 PM

Dave Mitsky

### #11

Posted 30 December 2012 - 03:31 PM

### #12

Posted 30 December 2012 - 03:39 PM

### #14

Posted 31 December 2012 - 10:41 AM

This is the one I was going to reply with too.e^(i*pi) = -1

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...

### #15

Posted 31 December 2012 - 01:34 PM

e^(i*pi) = -1

-i^2 like

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

-drl

### #16

Posted 31 December 2012 - 02:44 PM

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...

### #17

Posted 31 December 2012 - 04:12 PM

### #18

Posted 31 December 2012 - 08:26 PM

### #19

Posted 01 January 2013 - 04:24 PM

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

### #20

Posted 02 January 2013 - 11:50 AM

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.

### #21

Posted 02 January 2013 - 12:03 PM

Can we prove (or disprove) similar for, say, cubes?Demonstration of the Pythagorean Theorem using squares (actual squares)

### #22

Posted 02 January 2013 - 01:09 PM

magic square examples

### #23

Posted 02 January 2013 - 02:34 PM

That totally blew my mind! (which probably explains a few other things.)

### #24

Posted 02 January 2013 - 03:49 PM

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.

### #25

Posted 02 January 2013 - 05:30 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.

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