74. More Reasons Why the First "High Dimension" is Six or Maybe Five

(This will make much more sense if you read "11. Why the First "High Dimension" is Six or Maybe Five" first. As before, epistemic status: morally correct, in the mathematician’s sense; here to give flavor and intuition without too much rigor. Frankly, I'm going to move even faster and more cursorily than last time. With thanks to IL, and to everyone I've tried to shill 4D Golf to.)

After talking with IL a bit more and doing some thinking for myself, I realized with creeping delight that the long heuristic argument that I gave for why we should consider the first qualitatively high dimension to be six or maybe five is not remotely the only one. In fact, a shocking number of strong heuristic arguments all converge on the same figure: very specifically, "six or maybe five". That is: five is marginal, and six is definite.

To give a very quick recap of the heuristic argument from the original post, the idea is that we can operationalize what it means for a space to be high-dimensional by looking at what kinds of intersections pairs of unit disks which are relatively close in space but which have random facings are likely to have. The conclusion was that in 5D, there's a tiny but maybe real chance of the overlap being a point, but in 6D and up, the two disks will pretty much never intersect at all. That is: they're generically skew to each other in 5D, and overwhelmingly so in 6D. But what more is there?

Most straightforwardly, there's the argument about how hard it is to locate the sun in the daytime sky. If you're a 3D creature, then you know that this is really quite easy to do in the 2D sphere of a 3D sky. You align your face to be normal to a single plane, pretty much, and you're done. The tricky thing is, to even find the sun, you need to align an additional plane for each two dimensions you go up. In 4D, you need to align a single plane and then do a little more twiddling around, but in 5D, you need to align two planes simultaneously, and you need to do precise double rotations just to have any hope of locating the sun. In 6D, you need to do that, but your work still isn't done - after all that you still have some postprocessing to do. I'd give up if I were you.

 

There's a related argument around the (number-theoretic) partition function, of all things. For a brief definition, the partition function \(p(n)\) is the function that tells you how many distinct ways there are of splitting up a natural number \(n \in \mathbb{N}\): the first few values go \(p(0) = 1, p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, p(6) = 11, p(7) = 15, p(8) = 22...\) where notably the function starts to subtly accelerate around (wouldn't you know it) 5 or especially 6. The connection to high-dimensional spaces is a little more subtle: here, we might operationalize "high-dimensional" as "there are lots and lots of ways of slicing up or factoring apart any given geometric object of codimension 0 in the space"; those ways correspond to partitions of the dimension number.

 

Another argument relies on the behavior of random walks in \(d\)-space. It's a theorem that if you take two random walks starting at different points in \(\mathbb{Z}^d\) for \(d \leq 4\), the two wandering paths will encounter each other infinitely often, but for \(d \geq 5\), it's all but certain that the two paths will only ever cross finitely many times... if they ever cross at all. In a similar vein, consider that if you lose something on the shore of a 2D lake in a 3D world, all you need to do is walk around it once and you'll find it. In 4D, the shore is now 3-dimensional, so your task is going to take some effort and care but is still fundamentally doable - you're now searching something like an entire (2-)sphere rather than a circle. But the pattern continues - in 5D, you now have to search across an entire 3-sphere, and now the sad fact that a random walk in \(\mathbb{Z}^d\) will only be certain to return to the origin eventually in \(d \leq 2\) becomes relevant: you might well wander forever about the shore and never find your lost trinket.

Mathematicians have already picked up on this, naturally. Not only are topology departments frequently divided into low-dimensional topology (3D and often 4D) and high-dimensional topology (5D and up), geometric group theory, my old field, has an even sharper and more rigorous argument. You'll probably want to read through some of the Thesis Hike series if you want to understand it. (There's a tag for it.) The argument goes like this: with a lot of effort, cleverness, and lore, you can prove that the fundamental group of any 3-manifold at all must be residually finite, which you can think of as essentially being nice and well-controlled in some finite-flavored way. On the other hand, the fundamental group of a 4-manifold can be literally anything finitely generated at all. We can even explicitly construct examples for this. Let \(G = \langle L | R \rangle \) be any finitely generated group at all with letters \(L\), relations \(R\), and with \(|L| = n < \infty\). To construct a manifold \(M\) with \(\pi_1(M) = G\), start by taking the wedge of \(n\) circles - that is, take one circle for every letter \(l_i \in L\), give them each an arbitrary orientation - a direction to move along them that counts as positive, with the opposite being negative - and join them all together at a single point. Then, for each relation \(r_j \in R\), attach a 2-cell - a solid disk - to the circles such that the boundary of the disk spells out \(r_j\) in the letters \(l_i\), where attaching one full time in the positive direction picks up a positive factor of \(l_i\) and likewise attaching one full time in the negative direction picks up an inverse factor of \(l_i\). Since the boundary of the disk spells out \(r_j\), we now have an object whose fundamental group has a generator for every letter of \(G\) and in which a path corresponding to each relation of \(G\) can be shrunk to a point, pulling tight along one of those disks we just glued on. But won't those disks intersect each other? No! The key is that there's only finitely many of them, and by the argument from the earlier post, those disks will effectively always miss each other completely. So what we have - modulo some subtleties around thickening \(M\) up a little to a tubular neighborhood to avoid sharp bends and local tangles while still retracting cleanly to \(M\) itself - is a 4-manifold with the desired fundamental group. And of course, while a 3-manifold generally wants to live in 4D or maybe 5D space - with a few thorny exceptions demanding 6D, but no more - a 4D manifold will want to live in 5D space at the bare minimum, and more often 6D or even up to 8D! 3-manifolds are fundamentally well-behaved in a way that 4-manifolds are extremely not, and 5D and up - especially 6D and up - is where those 4-manifolds want to live.

From a totally different angle, we might contemplate the volume of a unit sphere in any given dimension. From this perspective, it'd be extremely strange if the volume of a unit sphere - which by all accounts should be some kind of standard measure - were to start decreasing rather than increasing. And yet - as a recent excellent video by 3blue1brown exquisitely displays, and as IL pointed out with surprise, this not only happens - which I already knew - but it happens very specifically just past \(d = 5\)! A full account of precisely why this happens is outside the scope of this post - and I encourage you to go looking for yourself - but the quick reason is that the formula of the volume of \(S^n\), \(V(S_n) = \frac{\pi^\frac{n}{2}}{\frac{n}{2}!}\), depends directly on the angular content of \(\mathbb{R}^n\) - roughly, how much "volumeness" there is to the sphere - and inversely on a normalization factor superexponential in the number of dimensions - roughly speaking, how many planes' worth of directions there are to point in, in some order. At \(d \geq 5\), there now begins to be a more powerful normalization cost to those extra directions than the sphere's raw volume can keep up with. Just like with the original post's argument, we notice that \(d = 5\) is the point at which the orthogonal complement of a plane in the ambient space begins to outweigh the plane itself.

 

For one last angle, consider the number of compact convex regular polytopes there are in each dimension. A regular polytope in \(n\) dimensions is a kind of shape that's maximally symmetric. It's bounded by some number of \(n-1\)-dimensional pieces which themselves must be regular, and so on down the chain; additionally, it has to be true that you can rotate any such highest-dimensional fragment onto any other while keeping the resulting shape looking the same. (Compact just means that they're only finitely large, and convex means that the pieces are all normal and don't cross through themselves; you don't need to worry about that part.) For a concrete example, regular polygons like squares and hexagons are all regular polytopes in 2D: they're bounded by 1D pieces - their sides - which themselves are bounded by 0D points. If the sides are all the same length and the angles are all equal, you can rotate any side to any other while keeping the final shape looking the same. The five Platonic solids are the regular 3D polytopes. So to count up the regular polytopes by dimension, there's infinitely many of them in 2D - they're all the regular polygons, from triangles on up. There's precisely 5 of them in 3D; the Platonic solids. It turns out that there are exactly 6 regular polytopes in 4-D: analogues of the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, and one last self-dual one called the 24-cell. The 24-cell has no equivalent in 3D space: the closest we can come are the cuboctahedron and the rhombic dodecahedron, and neither one is an especially close fit. Infinity, 5, then 6: how many regular polytopes might we expect there to be in 5D, or 6D? It turns out that there are just 3 of them - an equivalent of the tetrahedron, an equivalent of the cube, and an equivalent of the octahedron - and no others, and this pattern holds up forever. From 5D on up, there are just 3 flavors of regular polytope.

 

Basically, all of these varyingly rigorous arguments say about the same thing: that the codimension of a plane in an ambient space matters quite a lot, and thus \(d \geq 5\) is where our ordinary intuitions about how geometry is supposed to work, how nice and well-behaved it's supposed to be, break down. Our geometric intuition breaks down much like bankruptcy comes upon one: gradually, and then suddenly.