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Beauty in mathematics

 
 
 
phands

phands

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One of the things that keeps me teaching maths and science is the sheer gorgeousness of maths....

I love Godel's Incompleteness, although Godel numbers give me a headache, adore Calculus i general, but the most beautiful thing I ever saw is Euler's Identity.

What's your maths turn-on?

Euler-Identity1.png
 
Today and yesterday I've been thinking of distance estimation functions for thin objects, so that I can write code and generate flowers (petals and leaves can be quite thin).


Besides that, this one isn't really mathematical because it's more indicative of deliberately crafted jokes in the the English language, and it's definitely not a turn on, but I like the fact that natural log of #2 and pi appear at different "ends" of a generating function (sic) that uses infinite nested roots (sic: recursive roots).



For limit n-->infinity x=x^n-1 (positive real root, so x^n ~2) k-->infinity nestings
\(\sqrt{log(2)}= \lim_{n\to\infty} \,\, \sqrt{\left(nx^{n-1}\right)^k \,\, \times \,\, \left(x-\sqrt[n]{1+\sqrt[n]{1+\sqrt[n]{1+\dots}}} \right)}\)

if you set n=2, and x=x^2-2 (positive real root)
\(\frac{\pi}{2}= \,\, \sqrt{(4)^{k} \,\, \times \,\, \left(2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}} \right)}\)

 
Kharakov said:
Today and yesterday I've been thinking of distance estimation functions for thin objects, so that I can write code and generate flowers (petals and leaves can be quite thin).


Besides that, this one isn't really mathematical because it's more indicative of deliberately crafted jokes in the the English language, and it's definitely not a turn on, but I like the fact that natural log of #2 and pi appear at different "ends" of a generating function (sic) that uses infinite nested roots (sic: recursive roots).



For limit n-->infinity x=x^n-1 (positive real root, so x^n ~2) k-->infinity nestings
\(\sqrt{log(2)}= \lim_{n\to\infty} \,\, \sqrt{\left(nx^{n-1}\right)^k \,\, \times \,\, \left(x-\sqrt[n]{1+\sqrt[n]{1+\sqrt[n]{1+\dots}}} \right)}\)

if you set n=2, and x=x^2-2 (positive real root)
\(\frac{\pi}{2}= \,\, \sqrt{(4)^{k} \,\, \times \,\, \left(2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}} \right)}\)

Click to expand...

Oh my, oh my!

Roots al the way down?

What's really infinite?

\(\sqrt{Turtle}\)


EB
 
A bit tangential from pure math, but I often have this feeling about programming.

Take an ape-species that votes in a guy like Trump and believes in religion. Then take a subset of that species who is really good at logic to an extent that almost looks like a superpower. Then have those people solve complex logic problems on an electrified rock in order to transmit critical information across internet networks. Now take the profit from that activity and go on a cruise in the Greek Isles.

That is an aspect of my profession that I really like a lot.
 
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
 
gy=c

I first saw that in a science fiction novel, T.J. Bass' The Godwhale. It was supposed to be a formula to identify human-habitable planets; acceleration due to gravity at the surface times number of seconds in a year equals the speed of light. It also was supposed to have been precisely accurate for the Earth about the time human beings evolved, but I think that isn't really the case.

Bass wrote that long before we knew of any exoplanets. It might be interesting to apply it to some of the plainly uninhabitable exoplanets we've discovered.
 
My calculation: gy/c = 1.032

It is unlikely that the length of the year has varied by much. This is because the Earth's orbit does not have any strong resonances with its neighboring planets' orbits. A strong one would be something like a 2:1 resonance. Venus and the Earth are close to a 13:8 resonance, but that's about it.

One must consider g and y separately for exoplanets.

For Earthlike ones, a lower limit is from being unable to hold onto an atmosphere, while an upper limit is from accumulating too thick an atmosphere from being able to hold onto hydrogen and helium. Those limits are roughly:

R = 0.5 Re, M = 0.1 Me, g = 0.4 ge
R = 2 Re, M = 10 Me, g = 2.5 ge

For the year, we need the major axis. It is determined by the stellar flux being close to the Earth's value: F = L/(4*pi*a^2), giving a ~ L^(1/2).

The period P ~ (a^3/M)^(1/2) ~ L^(3/4)/M^(1/2)

From  Main sequence, the year lengths in Earth years and Earth days:
  • F5: 1.74 - 637
  • G0: 1.13 - 414
  • G2: 1 - 365
  • G5: 0.87 - 317
  • K0: 0.57 - 208
  • K5: 0.30 - 111
  • M0: 0.18 - 67
  • M5: 0.058 - 21
  • M8: 0.018 - 5.5
 
phands said:
One of the things that keeps me teaching maths and science is the sheer gorgeousness of maths....

I love Godel's Incompleteness, although Godel numbers give me a headache, adore Calculus i general, but the most beautiful thing I ever saw is Euler's Identity.

What's your maths turn-on?

View attachment 16590
Click to expand...

Mathematicians: We’re #FED67E92-FE20-44F4-9C04-95A46817A391.jpeg
 
Kharakov said:
Artemus said:
There are 10 types of people, those who understand binary numbers and those who don't.
Click to expand...
That's a weird second post.
Click to expand...

I find a certain beauty in the fact that our base-ten system is so arbitrary and the same number can be represented in so many ways. (For those who missed the joke, 10 in binary = 2 in decimal. Our computer engineering group sells t-shirts that say it. Confuses all the non-geeks on campus.)
 
6b2bdb20e9b3f3bb0b09035eacff69dc6ff55767
 
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