That to assign dimensional relevance must have invited views within those compacted dimensions?

So what would emerge from that flash? Layman here scratches his head.

There still has to be a better discription in concernt with dimensions as with Clifford if corrected, if you can spell this out better, which numbered/geometric system would you use?

If we deal with quantum gravity what kind of geometry will emerge? ONE WOULD HAD TO ASSUME THAT SOMETHING ALWAYS EXISTED, BUT WHERE?

Uncertainty reigns large and like any bubble that emerges, it had to emerge from space, and we accept this space is “not empty”, right?

If one did not accept “hyperdimenisonal views” then none of this would matter would it?

The image of the binomial triangle showed up just fine in the preview, but doesn’t in the submitted post. You can see it here:

Binomial Expansion

You know, I never saw that movie. One day I’m going to go rent the DVD, but let me give you some food for thought, since you seem to be bright and have a keen interest in these things: The negative numbers took centuries to be accepted (see the “crying jag” story posted by Sean), and the imaginary numbers had to be invented to cope with them. Now, we wonder why Bott’s periodicity seems to pop up everywhere (see John Baez’s “spooky facts” story), but it was Clifford, taking his clue from Hamilton and Grassmann, that was the first to recognize the significance of the fact that two interpretations of number are possible, the “quantitative” and the “operational” (see Hestenes GA).

It turns out that, if we define number as an operational quantity by the ratio of two natural numbers, we form the rational number system, which in turn forms the integer number system, thus integrating the three number systems, the natural, the rational and the integer, into one number system with “negative” and “positive” quantities that make sense (no “crying jag”), and without the need for an imaginary number, since the square root of n/m = 1 is the same as the square root of m/n = 1. On this basis, the negative sign is only an indication of which of the two inverses we’re dealing with. It’s not mysterious at all.

Now, when we add dimensions to these “signed” integers, we get fields of “signed,” n-dimensional, magnitudes and 0 is eliminated altogether! The best way to see this is with the triangle of the binomial expansion:

Remember, we are interested in the binomial because we have two “directions” or “polarities” of signed integers, operationally interpreted from the rational numbers, formed from the natural numbers. Now, the rows show the composite set of the transformed numbers as the dimension, n, progresses. The 1st row has 1 type of number in it, the scalars, that some want to identify with the reals. The 2nd row has 2 types of numbers in the set, the reals and the complex type, some say. The 3rd row has 3 types, the third of which is identified with the quaternions, and finally, the 4th row has 4 types, the fourth of which is identified with the octonions.

Interestingly, Bott’s periodicity seems to be tied with this pattern, because the next row (row five) seems to just be the beginning of a repetition of this pattern, and no matter how far you expand the binomial, the pattern just repeats itself, augmenting the original sequence, like a never-ending spiral. Well, since the pattern is equivalent to a scalar and three vectorial directions, what does this tell you? The unreasonable effectiveness of mathematics in physics strikes again, limiting us to 3 + 1 dimensions!

However, here’s the really interesting part: the operational interpretation of number sheds a whole new light on this curious expansion of two “directions.” Indeed, it shows that the concept of a counting number, that Euclid so carefully kept separated from magnitudes, can be generalized to a concept of n-dimensional magnitudes with two “directions,” negative and positive, if you will; that is, a scalar magnitude with no “directions;” a 1D magnitude with two “directions;” a 2D magnitude with two “directions;” and a 3D magnitude with two “directions,” geometrically corresponding to a point, a line, a plane and a volume, each with two “directions:” a point with two potential, unrealized, “directions,” a line with two ends, a plane with two sides, and a volume with a center point and a surface.

With the unreasonable effectiveness of mathematics in physics and the incredible symmetry this expansion exhibits, and the fact that the only known relationship of space and time is the reciprocal relation of motion, s/t, you’d think that we could get the attention of some of these practicing physicists by pointing this out. So far, however, we haven’t found any interest. They cling tenaciously to the 1D definition of motion, the function x(t) in n-dimensions of space, in spite of the fact that they could entirely eliminate space and time as a background structure, if they would pay attention to this slightly different way of understanding the mathematics of reciprocal systems.

Now, Hestenes is one smart fella. Recognizing that Gibbs had side tracked everyone from the young Clifford’s brilliant discoveries that Hamilton and Grassmann were on to something profound, he put together Geometric Algebra, based on Clifford’s C3 algebra (based on row four above). Trouble is, not being able to escape the definition of motion of Newton’s system, he ends up with a confusion of direction and “direction” and gets stuck with artifacts as a result: the non-commutative vector product! Turns out, it’s not so bad from a practical point of view, because it helps to make sense of imaginary numbers, but had he not taken this misstep, he might have broken everything wide open and then maybe he would have called Weinberg to tell him that he found the way to unstick theoretical physics (Weinberg says their “stuck” for the same reasons we’ve been discussing; that is, the incompatibility of the two definitions of coordinate systems JoAnne refers to).

Anyway, Elliot, it seems that your original question, about the dimensions of motion, go right to the heart of the huge crisis in theoretical physics. Go figure.

Doug

Just a follow up on multi-dimensional motion. Would this be properly called “Superball” theory?

Seriously though, what if at the planck level all that occured was expansion and contraction of planck scale spheres and all phenomena were function of this action?

Be kinda like the Matrix?

Elliot

Doug is right – there are different definitions of coordinate systems which may or may not be useful in solving a particular problem. However, let’s be clear on one point. All currently observed physical phenomena in the universe is described by 4 dimensions (3 spatial + 1 time), no more and no less, no matter what coordinate definition is used.

Yes, that’s true IF we exclude the standard model’s masses and coupling constants from the “observed physical phenomena,” but let’s remember that mass, charge, magnetic fields, etc. are also physical phenomena, and can’t be described by any number of dimensions regardless of the definition of the coordinate system.

Scientists, wishing to describe the matter in the universe, draw a particularly convenient grid with which to describe the matter content. The point being that the grid doesn’t have any real, physical existence, but is a mathematical abstraction made by someone wishing to have simple equations with which to describe cosmology.

Couldn’t find a spot for it on cosmic variance, so I hope you won’t mind it here JoAnne.