Are there such things as infinite-dimensional regular polytopes?

Question

The classification of the regular polytopes in any finite amount of dimensions is well known. In 2D, 3D and 4D, there are quite a few exotic shapes, but from $5$ dimensions up, every regular polytope lies in one of three infinite families:

• Simplexes, the analogs of tetrahedra.
• Hypercubes, the analogs of cubes.
• Orthoplexes, the analogs of octahedra.

What strikes me is that when we consider infinite-dimensional space $\mathbb{R}^\mathbb{N}$, all of these shapes still have clear (?) analogs. Here's the constructions I propose:

• For an $\infty$-simplex, we consider as vertexes the points $(1,0,0,\ldots),$ $(0,1,0,\ldots),$ $(0,0,1,\ldots),$ $\ldots,$ and we create a $k$-face out of the $k$-simplex formed by every $k$ vertices.
• For an $\infty$-hypercube, we consider as vertexes the points of the form $(\pm1,\pm1,\pm1,\ldots),$ and we create a $k$-face out of the $k$-hypercube formed by every set of $2^k$ vertexes in which all but $k$ coordinates remain constant.
• For an $\infty$-orthoplex, we consider as vertexes the points $(\pm1,0,0,\ldots),$ $(0,\pm1,0,\ldots),$ $(0,0,\pm1,\ldots),$ $\ldots,$ and we create a $k$-face out of the $k$-simplex formed by every set of $k$ vertexes in which the non-zero coordinate has a different index.

I have a few very related questions:

 a) Is there a formal definition of an "infinite-dimensional regular polytope"?

 b) Do my constructions satisfy it?

 c) Are there any other infinite-dimensional regular polytopes?

Any reference or any reasonable definition would suffice for a), and based on it, it'd be nice to have proofs for b) and c).

I've looked through the internet, found nothing related. I think that we could simply take the usual regular polytope definition and just not require a maximal element, but I have no idea of how we could talk about symmetries afterwards, when there's barely a distance notion. Also, I'd guess that no other regular infinite-dimensional polytopes are possible, since their possible facets (excluding low-dimensional ones) would be restricted to the three aforementioned families, which is kind of a strict condition. But again, I have no idea how to formalize my intuition.

Edit: For question a), we could slightly redefine isometries, so that they need to preserve distance between points at finite distance. That way, the normal definition of a regular polytope would work. I'm almost sure that, under this definition, we could answer question b) in the positive, by combining simpler transformations, but I'm a bit sketchy on those details. I'm still completely stuck on c).

Answer 1

Using unit edge sizes you can calculate various other measure properties of any (finite dimensional) simplex, orthoplex, and hypercube simply as a function of the dimension. E.g.

Circumradius of $D$-dimensional (unit-edged) simplex = $$\sqrt\frac D{2(D+1)}$$ Circumradius of $D$-dimensional (unit-edged) orthoplex = $$\frac1{\sqrt2}$$ Circumradius of $D$-dimensional (unit-edged) hypercube = $$\frac{\sqrt D}2$$

or

Inradius of $D$-dimensional (unit-edged) simplex = $$\frac1{\sqrt{2D(D+1)}}$$ Inradius of $D$-dimensional (unit-edged) orthoplex = $$\frac1{\sqrt{2D}}$$ Inradius of $D$-dimensional (unit-edged) hypercube = $$\frac12$$

or

Dihedral angle of $D$-dimensional simplex = $$\arccos\left(\frac1D\right)$$ Dihedral angle of $D$-dimensional orthoplex = $$\arccos\left(\frac2D-1\right)$$ Dihedral angle of $D$-dimensional hypercube = $$\arccos(0)=\frac{\pi}2$$

or

Volume of $D$-dimensional (unit-edged) simplex = $$\frac1{D!}\sqrt{\frac{D+1}{2^D}}$$ Volume of $D$-dimensional (unit-edged) orthoplex = $$\frac1{D!}\sqrt{2^D}$$ Volume of $D$-dimensional (unit-edged) hypercube = $$1$$

etc. And for all these terms you could try to evaluate the limit at $D\to\infty$. Thus e.g.

1. for the (unit-edged) simplex wrt. dimensional limit you get

   • Circumradius $\to\frac1{\sqrt2}$
   • Inradius $\to 0$
   • Dihedral angle $\to\arccos(0)=\frac{\pi}2$
   • Volume $\to 0$

2. for the (unit-edged) orthoplex wrt. dimensional limit you get

   • Circumradius $=\frac1{\sqrt2}$
   • Inradius $\to 0$
   • Dihedral angle $\to\arccos(-1)=\pi$
   • Volume $\to 0$

3. for the (unit-edged) hypercube wrt. dimensional limit you get

   • Circumradius (diverges)
   • Inradius $=\frac12$
   • Dihedral angle $=\arccos(0)=\frac{\pi}2$
   • Volume $=1$

Thus e.g. the orthoplex would become flat like an honeycomb! But still having finite size! - And the simplex, even so ultimately becoming right angled, still will become as flat as possible: with vanishing inradius!

$$\ $$

Probably you know of several of these things already. And therefore asking about some foundation.

You might look for Hilbert spaces in this context. Those are defined by their inner product, i.e. their scalar product. So, when considering some vector $\vec{v}=(v_1, v_2, v_3, …)$ wrt. its base, you would get its squared length by $<\vec{v}, \vec{v}>=\sum_{i=1}^{\infty}v_i^2$, i.e. it converges only if nearly all addends will (approximately) vanish. Therefore esp. any vertex coordinates of a (unit-edged) hypercube will NOT conform to a Hilbert space description. The simplex and the orthoplex however can be considered there strictly.

--- rk

Answer 2

I decided to write up my comments as an answer. I am not familiar with any generalization of polytopes to infinite dimensions, so I just write down what I can think of.

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First of all, there are two modern standard ways to define a (convex) polytope:

• as the convex hull of finitely many point, or
• as the intersection of finitely many half spaces.

Both are unsuited to generalize to infinite dimensions. In the first case, we obtain essentially a finite dimensional polytope, because the convex hull of finitely many points is always finite dimensional. In the second case, we obtain the "infinite dimensional" cylinder over a finite dimensional polytope. Still not too interesting.

Crossing out the "finite" in "finitely many points/hyperplanes" will make matters worse, as now we talk about all closed convex sets (as explained here).

But what might works is the following: allow for infinitely many isolated points (or go further, and exclude limit points). The examples you gave are polytopes under this definition.

What still persists is the problem that in a space like $\Bbb R^{\Bbb N}$ there is no direct way to talk about distance, inner products, angles, hence no direct meaning of isometries (distance preserving transformations) and symmetries. You might want to make your space smaller, e.g. work in the $\ell^2$-space in which an inner product exists. I have no idea what interesting symmetric sets of points (and polytopes) might live there.

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There seems to be another solution: abstract (regular) polytopes. Abstract polytopes are mostly studied in the case of highly symmetric polytopes (which is exactly what you want). They do not live in any Euclidean space per se, but they still have a well defined notion of dimension.

An abstract polytope is just a partially ordered set that satisfies some further axioms. Among these, the existence of a smallest and largest element. Drop the "largest element" axiom and we can probably talk about infinite dimensional polytopes. Note however, that in the abstract case, there are already more than five $3$-dimensional regular polytopes, and among these, some with an infinite number of vertices. So you will find much more than the three well known families.

I am not aware of any result listing the "infinite dimensional abstract regular polytopes".

Answer 3

This is not a full answer, but a pointer to narrow the field down a bit.

A flag of a polytope is a connected chain of incident elements, one of each dimensionality (vertex, edge, face, cell, ...), so a flag of an infinite-dimensional polytope is an infinitely long chain.

A regular polytope is most succinctly described as one whose symmetry transformations are transitive on its flags. That is to say, its flags are mapped onto each other under the symmetries of the polytope.

This illustrates that regularity presupposes the existence of appropriately dimensional symmetry groups. Every symmetry group can be used to generate regular polytopes via its kaleidoscope.

So you question is equivalent to asking, what are the infinite-dimensional symmetry groups, if any?

There are a few Lie algebras which fit the bill (but I don't know much about them).