

What made you realize that math is beautiful? - mzahir
http://math.stackexchange.com/questions/323334/what-was-the-first-bit-of-mathematics-that-made-you-realize-that-math-is-beautif/323455#323455

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lutusp
Many of the examples in the linked article equate "beauty" with visual beauty,
but when Bertrand Russell famously described mathematics as beautiful, he was
talking about something a bit deeper:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a
beauty cold and austere, like that of sculpture, without appeal to any part of
our weaker nature, without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as only the greatest
art can show. The true spirit of delight, the exaltation, the sense of being
more than Man, which is the touchstone of the highest excellence, is to be
found in mathematics as surely as poetry."

In fact, as one can see, Russell goes out of his way to specify that he's not
speaking of visual beauty, but another kind. I know of many examples of
mathematical beauty -- here's one I've been investigating recently:

A planet or comet in an elliptical orbit conserves energy -- its kinetic
energy (based on its velocity) and its potential energy (based on its distance
from the parent body) always sum to a constant. In his Second Law, Kepler
noted that an elliptical orbit sweeps out equal areas in equal times, which is
true, but Kepler didn't actually know why. It turns out that his Second Law
demonstrates the conservation of energy.

Two equations describe the energy in an elliptical orbit -- one describes the
orbit's kinetic energy:

    
    
        e_k = 1/2 m v^2
    

The other describes the orbit's potential energy:

    
    
        e_p = G m1 m2 / r
    

In the kinetic equation, velocity governs the dynamic change in kinetic
energy, and in the potential equation, the object's distance from the parent
body (r) governs the dynamic change in potential energy. Amazingly, if you
create a model of an elliptical orbit, the orbiting object speeds up and slows
down in a way that forces a perfect constant sum of potential and kinetic
energy -- and, as a side effect, shows perfect agreement with Kepler's Second
Law.

Read more here:
[http://arachnoid.com/conservation_of_energy/index.html#Plane...](http://arachnoid.com/conservation_of_energy/index.html#Planetary_Orbit_Example)

And make sure to scroll down to see the orbit animation.

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mzahir
I think the responses have been skewed towards visual presentations since the
questioner intends to write a children's book. Thanks for the link - simple,
powerful and beautiful.

