Here's something interesting I discovered yesterday. I was looking at the ∞-simplex (as one of the ∞-cross's facets):

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` < 1, 0, 0, 0, ...>`

< 0, 1, 0, 0, ...>

< 0, 0, 1, 0, ...>

...

and one of

its facets:

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` < 0, 1, 0, 0, ...> `

< 0, 0, 1, 0, ...>

< 0, 0, 0, 1, ...>

...

Obviously this facet, let's call it F, must be a simplex of some sort. The most obvious choice is that it's equal to the ∞-simplex itself, since, after all, it has an infinite number of vertices, and there's an obvious isomorphism here mapping <0, x, y, z, ...> to <x, y, z, ...>, preserving the face lattice structure. There's a slight problem, however. In finite dimensions, if two polytopes are identical, there must be an isometry, i.e., a rigid motion, that translates one polytope into the other. However, the mapping <0, x, y, z, ...> → <x, y, z, ...> is

not a rigid motion, because if applied to another polytope that has a non-zero first coordinate, it drops that coordinate. I.e., it's a

projection rather than a rigid motion. But if the only thing that could map F to the original ∞-simplex is a projection, then they can't possibly be identical?

However, there's something more interesting going on here. Consider the reflection that maps <0, 1, ...> to <1, 0, ...>, i.e., it interchanges the first two coordinates. Obviously, this is a rigid motion. We can compose it with another reflection that interchanges the 2nd and 3rd coordinates. And compose the result with a 3rd reflection that interchanges the 3rd and 4th coordinates. And so on. At each stage, we have a compound reflection through a series of mirrors, that shift the first n coordinates left by 1 position, pushing the first coordinate to the n'th place. Obviously, these are all rigid motions. They are permutations of coordinates (no coordinates are "missing" after the transformation, unlike the projection case above). Interestingly enough, though, if we take the limit as n→∞, we get a transformation that shifts all the coordinates of the point left by 1 position, but the 1st coordinate is pushed to infinity, i.e., the limiting transformation is no longer a rigid motion, but a

projection.

IOW,

the limit of this series of left-shifting reflections is a projection.

This has very interesting implications, because it implies that there's a connection between rigid motions and projections, namely, the latter is a limit of infinite sequences of the format. Since we're in ∞-dimensional space, it would seem logical that we could perform an infinite sequences of reflections to a polytope. And it appears to be the case that such a limit transformation is required in order to map the ∞-simplex's facets onto itself. Or, to put it another way, the ∞-simplex consists of ∞-simplex facets, but infinitely reflected such that there is no rigid motion between them, but a projection / lifting (lifting = immersing in a space 1 dimension higher than the original one, by prepending a coordinate like in this case to map the ∞-simplex to F -- only, in this case, this higher-dimensional space happens to be identical to the original space).

In finite-dimensional space, the dimension of a polytope P is greater than the dimension of its projection, i.e., dim(proj(P)) ≤ dim(P), and the dimension of a facet of P is less than the dimension of P, i.e., dim(facet(P)) ≤ dim(P). In this case, however, the ∞-simplex is a projection of its facet F; which seems to imply that dim(∞-simplex) ≤ dim(F). But since F is a facet of the ∞-simplex, we also have dim(F) ≤ dim(∞-simplex). So we have dim(∞-simplex) ≤ dim(F) ≤ dim(∞-simplex), meaning that dim(∞-simplex) = dim(F).

(Note that this also implies that (∞-1)=∞, because the facets of the ∞-simplex, by dimensional analogy, ought to be (∞-1)-simplices, but since these (∞-1)-simplices

project to the ∞-simplex, they cannot be of a smaller dimension than ∞.)

All of this implies that in ∞-dimensional space, there are (at least) two kinds of motions: rigid motions, that behave like their finite-dimensional counterparts, and limit motions, which are projections that, under certain circumstances, can map an ∞-polytope into a "different" copy of itself such that it can become facets of another ∞-polytope or vice versa. This "different" copy is isomorphic (in fact, isometric) to the original polytope, but oriented in an infinitely different way such that there does

not exist a rigid motion that maps it back to the original polytope. Furthermore, there's an infinite hierarchy of these "different orientations" (consider, for example, the facets of F and their respective sub-facets), all disconnected from each other in terms of rigid motions, but connected via limit motions.

The corollary is that if you reflect/rotate/etc. an ∞-polytope, you can "change" its dimensionality.

Infinite-dimensional space is

weird.

(The same argument can be applied to the ∞-cube: its facets are also ∞-cubes, but oriented in a way that's infinitely different from the original ∞-cube such that no rigid motion can transform one to the other. But they are isometric under projection / lifting, i.e., under limit motions.)