It is best to assume life exists in four and higher dimensions, because this happens to be true.
Really? How did you find that out? Can they talk?
One notes that stable circular orbits exist in all dimensions, but stable elliptical orbits exist only in three dimensions. In four and higher dimensions, one has things not reckoned with in three dimensions: fibulation.
According to Google, fibulation is spelt "fibrillation", and it's a medical term. Usually, when an object is almost in orbit, it will spiral in and out again. Sometimes when it comes out it never comes back, sometimes it keeps oscillating, tracing out a flowery pattern.
Atoms are indeed stable in four dimensions, because the electrons do not orbit the nucleus, the orbitals are solutions to standing waves in the vacinity of the nucleus. Electrons and protons are hardly kith and kin to planets and suns.
That doesn't mean the inverse square law doesn't pose a problem. In the Schrodinger's equation thread, I've mentioned that energy isn't quantised in 4D hydrogen. The reason is that the potential energy is k/x^2 instead of k/x.
In 3D, an electron is distributed by a Legendre function. This only converges if the energy is a multiple of 1/n^2, where n is an integer.
But in 4D, the distribution is a Bessel function, which always converges.
This means that you don't get separated orbitals. The more energy an electron has, the further out it will (probably) be.
The good news is that angular momentum is still quantised. We have 3 angular quantum numbers. I've yet to work out what conditions they have yet.
In 3d you can work out the periodic table by the sequence of orbitals:1s(2)2s(2)2p(6)3s(2)3p(6)4s(2)3d(10)4p(6)5s(2)4d(10)5p(6)6s(2)4f(14)5d(10)... etc.
This is spectroscopic notation. It was developed before the Schrodinger equation was solved, so I don't know why they still use it. I still don't understand it, even though it's in my exam next week.
In 3D, you have 3 quantum numbers: n, l, m<sub>l</sub>.
Each of these has to be an integer. Also, we have the conditions 0<= l <n, and -l <= m<sub>l</sub> <= l. There's actually a 4th, m<sub>s</sub>, that can have two values, -1/2 and 1/2, so you get twice as many possible states. Spectroscopic notation derives from these four numbers. The "s,p,d,f" relates to the l number. s=1, p=2, d=3, f=4. The number outside the brackets is n, and the number in the brackets is the number of electrons with this particular state. You never know what the m<sub>l</sub> value is, but there are only 2l+1 possible values, so once they get filled up, you go to another value of l, or a higher orbital.
So 2p(3) means n=2, l=2, and 3 of the 6 possible states are filled up.
Anyway, in 4D, there's no n, but we do have l, p and m. Instead of orbitals, there are "hyperspherical harmonics", the 4D analog of spherical harmonics. Mathworld has pictures of the spherical harmonics for different values of l and m.
http://mathworld.wolfram.com/SphericalHarmonic.htmlThe laws of physics is borne on the laws of geometry &c, and there are indeed a good deal of differences. But these differences mean for example, that vortices happen in some different way, and that there is possible for different kinds of fibulation-vortices (which swirl around a point), which we do not have in 3d.
I don't know what you mean, or how you could know this, but it might be interesting to look at some simple vector fields in 4D. That could give us some guidelines for developing 4D magnetism.