Geometry of arbitrarily large dimensions

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Geometry of arbitrarily large dimensions

Postby quickfur » Fri Mar 08, 2024 7:43 pm

As humans we love to generalize. As we speculate on the nature of 4D space, we can't help thinking also about 5D, 6D, ... and pretty soon we're making the mental leap to ∞D. Unfortunately, a space with literally ∞ dimensions is fraught with fundamental, unresolvable problems that, to this day, precludes any meaningful generalization of n-dimensional geometry to infinite dimensions that doesn't come with either contradictions or crippling restrictions.

For example, an ∞-cube, if such a thing were to exist, would have an uncountable number of vertices, whereas the number of facets would be countable. Since these two quantities are clearly distinct (they lead to a contradiction otherwise) that implies that we have to extend our number system not just to include infinite quantities (the dot product of the ∞-cube's vertex coordinates with itself, for example, cannot be finite number), but multiple infinite quantities of distinct cardinalities. It also immediately leads to problems with non-converging sequences. For example, an omnitruncated ∞-cube would have coordinates of the form <±1, ±(1+√2), ±(1+2√2), ±(1+3√2), ...>. Pick two vertices such that not all terms are the same sign. What's their dot product? Well that depends. If the combination of signs is such that it's a conditionally convergent, then the order of summation will matter. But a dot product that depends on the order of summation has no sane geometric interpretation. (All this ignoring the fact that the dot product will in all likelihood be infinite, which leads to another problem: which of the numerous infinities would it be? since we've established that our number system would have to include multiple.)

This is just the tip of the iceberg of the numerous problems that arise when you deal with a space with an actual infinite number of dimensions. There are a host of other problems that I'm sure you smart people can easily come up with. IOW, dealing with a space of actual infinite dimensions leads to all sorts of paradoxes, contradictions, and other problems that preclude us from having fun speculations about it.

The only way to tame the beast is to impose crippling restrictions on it, such as using a Hilbert space: where vectors are required to have a metric that converges. This does give us a sane, consistent ∞-dimensional geometry, but unfortunately it doesn't contain many objects that we might be interested in. Such as the ∞-cube, which cannot be in any Hilbert space because it contains points of infinite norm. So that's not interesting either -- we have no consistent way of defining an ∞-polytope.

//

Fortunately, all is not lost.

Thanks to the nature of finite numbers, we actually can get a glimpse of what geometry in ∞-space might look like by investigating a finite n-dimensional space: but for a sufficiently large value of n. The key here is "sufficiently large". The thing about set of finite numbers is that it includes numbers which totally defy our intuitive grasp of what a finite number is. Take a look at Graham's number sometime, and have your intuition of infinity exploded several times over. In fact, not just several times over, probably a Graham's number of times over. :lol:

Graham's number is actually rather small, in the grand scheme of things. :P If you consider its generating function and its place in the fast-growing hierarchy, it sits only slightly above the Ackermann function, which is only a couple of levels above the Grzegorczyk hierarchy. You can go much higher up the fast-growing hierarchy and generate numbers of such enormity that it would utterly defy any intuitive conception of "infinity", and yet they are finite. :lol:

OK, the point of this post isn't to gawk at inconceivably huge finite numbers. The upshot of all this is that if we pick one of these crazy-huge numbers, we would, for the most part, be unable to distinguish it from an actual infinite quantity. Call such a number N. What's the magnitude of N/2? Well, it's more-or-less equal to N, because at that magnitude, it far, far, far exceeds any "normal" everyday number that we might encounter, so anything that we might want to compare it with would pale in comparison. Similarly, what's N/3? or for that matter, N / (1010000000)? or N / (googolplex^googolplex^... (googolplex times))? When N is a number like Graham's number, even dividing it by a power tower of a googolplex exponentiated to itself a googolplex times would hardly change its overall magnitude; the resulting number would be less than Graham's number, of course, but still so insanely huge that you couldn't tell the difference.

IOW, the difference in magnitude between N and N / (googolplex^...(googolplex times)) is indiscernible according to everyday arithmetic operations. They are definitely distinct, obviously, but we couldn't tell them apart using everyday arithmetic operations without invoking N itself.

What I'm trying to get at here is that N, for all intents and purposes, behaves as if it were an infinite number, even though it's actually finite. In any algebraic expression involving N and the usual everyday operations (+, -, *, /, ^, √, etc.), the magnitude of N would totally overwhelm everything else in the expression that doesn't also involve N, such that you could make simplifications like:

(1 + N) / (N^2 - N + 10^100000) ≈ (1/N + N/N) / (N - 1 + 10^100000/N) ≈ (0 + 1) / (N - 1 + 0) ≈ 1/N ≈ 0

whereas

(1 + N^2) / (N + 10^100000) ≈ (1/N + N) / (1 + 10^100000/N) ≈ (0 + N) / (1 + 0) ≈ N

It's not an actual equality, of course, but the huge magnitude of N makes the difference totally negligible.

//

Now consider a space of dimension N. The number of dimensions is so large that we could hardly differentiate it from ∞, even though it's actually still only finite. Furthermore -- and here's a key point -- it hardly matters what the actual value of N is; all that matters as far as everyday geometric operations are concerned, is that it's a very large number. N could be Graham's number, or 2^(Graham's number), or, for that matter, some number generated by a function far more powerful than the one that generates Graham's number -- but from the POV of everyday geometry, it hardly makes a difference. We couldn't tell the difference, and it doesn't change the nature of the geometric results we obtain.

IOW, we could use N as a finite stand-in for ∞, and the results we obtain would be very close to what we might get had we used an actual infinite number. So close that, apart from some pathological corner cases where the difference becomes pertinent, it might as well be indistinguishible.

Furthermore, since it doesn't matter what the actual value of N is, we can leave it unspecified besides the postulation that it be sufficiently large that any conclusions we might draw do not change with the choice of N. I.e., N is arbitrarily large.

So now we could talk about the properties of an N-cube, and it would be pretty close to how an ∞-cube might behave. Since N is arbitrarily large (albeit finite). We could draw conclusions about its behaviour compared to an N-sphere, for example. Since N is arbitrarily large, any expression that involves N in the denominator is essentially 0, and any expression that involves N in the numerator might as well be considered infinite, for all practical purposes. When N appears in both, we can safely use algebraic methods to reduce it without fearing of running afoul of issues involving actual infinite quantities. As long as we take care to indicate when an equality is approximate and when it's exact, we would avoid running into contradictions. So while the result of some calculation might not actually be 0, it might be sufficiently close to 0 that we could treat it as 0 (or, if you want to split hairs, like an infinitesimal). Similarly, when the result of a calculation results in some expression involving N, we could treat it as if it were "equal to infinity" (in the approximate sense).

Thus, we could speculate on the nature of N-dimensional space, and our conclusions would not be too far off from how things would behave at the limit when N→∞, even if an actual infinite-dimensional space may exhibit pathological behaviour not present in the case of finite N.

Best of all, using a finite value of N we avoid the need of invoking arithmetic of infinite quantities, and all the problems that would entail. We could talk about the convex hull of N-dimensional polytopes without issues, we could enumerate the coordinates of the omnitruncated N-cube, we could talk about the norm of the N-cube's vertices, etc., take dot products, perform various geometric operations, and wouldn't run into the troubles we'd run into if N were an actual infinite number. And our conclusions, being independent of the choice of N, would generally apply to all very-high-dimensional spaces, and thus would be indicative of what kind of behaviours to expect from an actual ∞-dimensional space.

//

Here are some simple examples.

An N-cube's vertices would have coordinates <±1,±1,...,±1>; their norm would be √N -- which is a number less than N, but from our POV still so large that it's approximately the same order of magnitude as N. So √N ≈ N. I.e., the N-cube's vertices are "almost infinitely distant" from its center.

Similarly, an N-cross would have vertices of the form <±1, 0, 0, ..., 0>, <0, ±1, 0, ..., 0>, ... <0, 0, ..., 0, ±1>. It would have 2^N facets -- again a larger number than N, but "approximately equal" to N. (We could, in an abuse of terminology, replace "N" with "∞" in the previous sentence and we wouldn't be too far off.) The centroid of the facet perpendicular to <1,1,...,1> would be <1/N, 1/N, ..., 1/N>, which is a point that isn't equal to the origin, but, since N is so large, we could abuse the terminology and say it's "infinitesimally" close to the origin. Its norm is √(N*(1/N^2)) = 1/√N.

What about the N-sphere? It would consist of all N-vectors whose norm equals its radius. Let's say it's radius is 1. Then it would inscribe the N-cross, and interestingly enough, its intersection with the line parallel to <1,1,...,1> is <1/√N, 1/√N, ..., 1/√N>, which is also infinitesimally close to the origin, but further away that the centroid of the N-cross's facets (because √N < N, so 1/√N > 1/N). :lol: The point <1/√N, 1/√N, ... 1/√N> is 1 unit away from the origin, whereas the centroid of the N-cross' facet is 1/√N units, which is "infinitesimally" small. But both pale in comparison with the N-cube's vertices, which are √N units away from the origin, i.e., "infinitely" distant. (Or, if you like, "N-finitely distant". :lol: :lol:)

Many other such conclusions can be drawn using similar logic. Basically, you just plug in N for the number of dimensions, treat it as an extremely large number, and simplify / approximate accordingly.

Some simple geometric results: the facets of an N-polytope are (N-1)-polytopes. But since (N-1) is indiscernible from N from our small-numbers POV, we could say that the facets are "essentially of the same dimension" as the N-polytope itself. They really aren't, of course, but since N is so large, the number of dimensions in an N-polytope vs. the number of dimensions an in (N-1)-polytope is practically the same from our POV of small everyday numbers. (Of course, we have to be careful in how we use such results later -- we can't actually treat the (N-1)-polytope as equal to an N-polytope otherwise we'd run into contradictions. But intuitively speaking, they're "more-or-less" the same.) But the point is that this is a pretty good approximation of the ideal case when N=∞ where we'd expect that the facets of an ∞-polytope would still be an ∞-polytope. (Actually working this out in an actual ∞-dimensional space may lead to problems, though. :lol:)

//

Anyway, I think the idea should be clear by now.

What interesting conclusions can you draw using this scheme? :?:
quickfur
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