Big Magical Numbers

[tomandlu]

This isn't really a FS question, but it's for FS, so...

Are there any nice big (preferably integer) numbers as magical as Pi? I.E. constant across all bases? (so I guess it would be a proportion)

[gmalivuk]

What do you mean "constant across all bases"? Any constant is constant across all bases, but all of them (except 0 and 1) have different digit representations in different bases.

If you want it to be "as magical as pi", you'll need to specify more than that.

[sevenperforce]

Perhaps by "across all bases" you mean "across all units"? If so, then yes. The proton-electron mass ratio is 1836.15, though it's not actually an integer. The fine structure constant is roughly 1/137 at low energies and 1/127 at high energies. Euler's number, 2.71828, is marginally more important to science than pi, though it's not an integer.

If your FS is set on Earth, there are some constants related to the ratios of masses and distances in the solar system that might prove interesting. For example, the ratio of the Sun's mass to the Earth's mass is about 330,060:1 while the ratio of their diameters is 109:1. The ratio of the Earth's orbit around the Sun to the Moon's orbit around the Earth is about 373:1.

[Hypnosifl]

Perhaps by "across all bases" you mean "across all units"? If so, then yes. The proton-electron mass ratio is 1836.15, though it's not actually an integer. The fine structure constant is roughly 1/137 at low energies and 1/127 at high energies. Euler's number, 2.71828, is marginally more important to science than pi, though it's not an integer.

If your FS is set on Earth, there are some constants related to the ratios of masses and distances in the solar system that might prove interesting. For example, the ratio of the Sun's mass to the Earth's mass is about 330,060:1 while the ratio of their diameters is 109:1. The ratio of the Earth's orbit around the Sun to the Moon's orbit around the Earth is about 373:1.

Here is a nice list by physicist John Baez of all the known dimensionless (unit-independent) physical constants--note that when he refers to various particle masses, he's really talking about the ratio between the particle's mass and the Planck mass, since this ratio doesn't have any units even though the masses do. It's not really clear if this is what the OP was asking about though, since pi is not a physical constant at all, but a purely mathematical one (though of course it does appear in many physics equations). Here is a list of common mathematical constants--though the very notion of a "mathematical constant" is fairly qualitative, defined on the page as a real number that is "significantly interesting in some way".

[tomandlu]

Perhaps by "across all bases" you mean "across all units"?

Indeed I do - and apologies, gmalivuk - I should have said something like "independent of any units of measurement" or something. Brain fart.

The math constants are interesting but too small, but the particle ratio looks about right (and I quite like it having a fractional part, since that makes even less sense in context).

Tnx all.

[sevenperforce]

The math constants are interesting but too small, but the particle ratio looks about right (and I quite like it having a fractional part, since that makes even less sense in context).

Tnx all.

The various particle ratios (proton:electron, neutron:electron, W boson:electron, Higgs:electron) are neat because they're going to be the same on every world in every solar system in every galaxy, and you can begin to figure a few of them out using chemistry alone (though for the more complicated ones you're going to need a particle accelerator). I actually have a tattoo that incorporates a bunch of particle ratios along with some other fun stuff like pi, tau, the alpha constant, Euler's number, and so forth.

[lorb]

The math constants are interesting but too small

graham's number is a not-small number

[ThirdParty]

Does it have to be a proportion? Maybe you could count something instead. For example, 624 = maximum number of quarks that can be found in a non-radioactive atomic nucleus. (Admittedly, what counts as "non-radioactive" depends somewhat on perspective. But only somewhat.)

[>-)]

The order of the monster group is a nice one:
808017424794512875886459904961710757005754368000000000

[jewish_scientist]

The golden ratio is another good choice. It is Φ = (1 + √5)/2. Its most famous property is if you draw a rectangle that is 1 by Φ and then remove a 1 by 1 square from it, the left over rectangle is similar to the original rectangle. Repeating this over and over results is that spiral people add to a bunch of images for some reason. In addition Φ = √(1 + √(1 + √(1 + √(1 + √(1 + √(1 + √(1 + ..., Φ-1 = 1/Φ and the ratio between two adjacent numbers in an Lucas sequence (such as the Fibonacci sequence) approaches Φ as the number grow larger. The Lucas Numbers is the Lucas sequence that starts with 1,3; the nth number in this sequence is equal to Φ^n rounded.

[Xanthir]

I really, truly love that φ¹ + 1 = φ², and φ¹ - 1 = φ^-1.

Also, if you use the resulting recurrence that φⁿ = φ^n-1 + φ^n-2 to break down φⁿ to a sum of a*φ + b, a and b will be successive Fibonacci numbers (which shouldn't be surprising, given the form of the recurrence, but still).

[thunk]

Of course, there's also the e^(pi*sqrt(163))= about 640320^3+744 thing...

The math behind that I don't really understand. I guess I'm just too tired to try most of the time.

[PsiCubed]

The order of the monster group is a nice one:
808017424794512875886459904961710757005754368000000000

Or if you need a smaller number:

60 - the order of the smallest nonabelian simple group.
7920 - the order of the smallest sporadic simple group.