# para-geometry and imaginary gravity?

i is the square root of -1. An imaginary number line at right angles to a real nimber line is called the complex plane-used mainly in computing Mandelbrot and Julia sets, amongst other fractals.
Ive noticed that the number i also comes up in physics, both in relativity and quantum mechanics.
In relativity, a four dimentional continuum(Length, width, depth, and time) creates a field which, when distorted by mass, describes gravity.
Suppose you do the same with imaginary dimensions
( i * j * k * l ). Has anybody tried calculating such a continuum? And if so, what did they discover?

Quaterions are something like what you describe:
i[sup]2[/sup] = j[sup]2[/sup] = k[sup]2[/sup]
ij = k, jk = i, ki = j
j
i = -1, kj = -i, ik = -j
Note that multiplication is not commutative.

Chronos mentioned them once, but I can’t find the thread. He knows more about them than I. (I just posted everything I know about them.)

I ran into these things in my Abstract Algebra course, so I don’t know much about the applications of Quaternions. Mathematically, DrMatrix, I think you’ve got everything there, with the exception of one small detail that I think you just overlooked:

i[sup]2[/sup] = j[sup]2[/sup] = k[sup]2[/sup] = -1

I know Chronos has also tried calculating an 8-element group. I don’t remember what constraints he put on it, but I know he showed it to be inviable.

I know next to nothing about these things, but wouldn’t j*i=-k?

Achernar and Dijon Warlock are correct. “I just posted everything I know about them” and I had an omission and a mistake. It’s a good thing I don’t know a lot about them or I’d have really screwed it up.

Why am I reminded of unit vecors in all of this?

I’ve thought long and hard about that too, aynrandlover, but as best as I can tell, these can’t really be related to three-dimensional unit vectors, although that would make a good analogy to the continuum that enolancooper was mentioning. The second and third lines of DrMatrix’s listing there (with the correction made) look identical to the rules for vector cross product. However, there are some differences. Even though they don’t commute, Quaternions associate under multiplication; vectors do not associate under cross multiplication. That is, the statement:

(i × i) × j = i × (i × j)

is true for Quaternions but not for vectors. This is because in Quaternions, i × i = -1, but in vectors, i × i = 0.

Quaternions also form a mathematical group. I’m pretty sure that it’s impossible to make a group out of vectors under cross multiplication, there not being any identity vector. Maybe some clever redefinition of vector multiplication would make it all possible.

Well, you all have pretty much already summed up everything I know about them, too. They’re an interesting toy, but they don’t seem to be useful for much. The closest thing I’ve seen to them in physics is the Dirac realtivistic spin matrices (similar to the Pauli spin matrices, but 4x4), which have the same commutation relations, but you really do need the matrix properties with them, too.
Quaternions are already an 8-element group, of course, since you’ve got the negatives, too. When I was trying for a comparable 16-element group, the only constraints I set were that all of the elements (except for 1 and -1) had to square to -1, the commutation relations, and the symmetry between the elements (again, other than 1 and -1).

By the way, Achernar, good to see you back. Is this Spring Break week?

I actually do use quaternions from time to time, to calculate the action of PSL(2,C) on hyperbolic 3-space.

Chronos, have you ever heard of “octonions”? I don’t know much about them myself, except that they’re similar to quaternions but with 16 generators instead of eight (or eight generators and their negatives if you prefer), and they’re non-associative as well as non-commutative. But it sounds something like the 16-element group you were looking for…there’s a web page about them (and something called “sedenions” which I’ve never heard of) at http://www.geocities.com/zerodivisor/index.html, but that link seems a bit flaky at the moment.

That’s a neat link, Math Geek. It looks as though if you take out the requirement that the elements associate, you can find a 16-element group like the Quaternions. (Although, if they don’t associate, it’s not technically a group - I don’t know what the term is.) Odd, since I seem to remember there being some sort of simple obstacle to it, though what that might be has slipped my mind. So a 1-D universe associates and commutes. A 3-D universe only associates. A 7-D universe does neither. And it looks like a 15-D universe does some really weird stuff.

And yeah, Chronos, I’m on Spring Break. Thanks for asking. Nothing like leaving school and the harsh yoke of homework for a week to mess around on the Internet talking about Math.

Actually, Quaternions are sometimes used in modeling freely rotating bodies.
You can use euler angles, but the update equations are uglier .
you can use rotation matrices, but round-off error can push the matrix out of orthonormal.
quaternions have one extra variable (you use 4 numbers to represent an orientation), but the orthonormal constraint is only a single equation.
-Luckie

Probably just the requirement that it be a group, which I was assuming when I tried it. I did wonder, when first I heard of octernions, where I had gone wrong…

I’m wondering now, though… If they don’t commute and don’t associate, what use are they?

I was aware of the use of quats for rotating bodies, but Euler angles do that job just fine, I’d say. The update equations are only ugly if you choose the wrong coordinate system… If you choose the right coordinates, they become trivial, and you’ll almost always have the freedom to choose your coordinates.

Wow !

How is this math used ? Are these things used in your jobs or is it more of a hobby ?

Aside from the sheer curiosity factor, the only time I’ve ever seen them used is as a model space for manifolds. That is, you have normal (i.e. real) manifolds, then complex manifolds, then quaternionic manifolds, and then “octonionic” manifolds, which are usually called Cayley manifolds.

Of course that just begs the question, but I don’t know what Cayley manifolds are used for.