See here
Could someone explain what this means?
For example, can you tell me within which two dimensional plane embedded in the three dimensional world these “energy fluxes appear to align?”
If you took out two more dimensions, you’d have scalar energy!
Interesting, thanks for bringing this up. I’m aware of speculations (originating from causal dynamical triangulations, an idea of uniting quantum theory and general relativity distinct from the more well-known string or loop quantum gravity approaches) of space having a lower dimensionality at high energies/short length scales (picture a piece of yarn being knitted, and then crumpled up into a ball – you might, from some distance, observe a 3D object, then, looking closer, you see a folded-up 2D sheet, and then, looking yet closer, the knitted 1D yarn), but I’m not sure how that would cause cosmic rays to align in a plane. The experimental signature I’d expect would be for decay products of particle collisions to be preferentially aligned in some plane relatively to each other. I’ll have a look into this…
From a quick skim of the first part of the paper, it seems that they are really talking about the cascade of decay products produced when cosmic rays are incident on atoms in the atmosphere, not the cosmic rays themselves. So, a cosmic ray hits an atom at a high enough energy for the 2D nature of spacetime to be visible, and the products of this collision are arranged in a plane with respect to one another. So it’s not the case that cosmic rays lie preferentially in a plane, which would have been a strange thing indeed; in order to preserve Lorenz invariance (i.e. satisfy special relativity), the planes in which the decay products of cosmic rays lie must be aligned randomly to one another.
Thanks for your two posts.
By way of both misunderstanding the analogy and taking it too far, I’ll point out that it seems to me if the three dimensions I appear to inhabit are actually like the wadded up yarn you mentioned, that still doesn’t make it seem to me that the world is really one wadded up dimension–since somehow, apparently, things are making a leap from one bit of the dimension to one-dimensionally very-far-separated points along that dimension–and that appears to require the existence of a three dimensional manifold the one dimension is embedded in.
Though it occurs to me these leaps could be considered to be of a kind with a sort of instantaneous teleportation–where the laws governing the teleportation would be absurdly complicated if expressed using just one dimension, but become more simple if expressed using three dimensions.
Yeah, though, like I said–that’s just me taking the analogy too seriously I’m sure.
I suppose things aren’t going to get any clearer if I mention that CDT’s basic building blocks (quite literally – spacetime is produced by gluing them together) are 4-simplices, or four-dimensional tetrahedra. 
To build intuition, take ordinary triangles – they’re 2D, but you can build a line from them by just gluing them edge to edge, with the orientation reversed. In CDT, to ‘finish’ the spacetime, the limit of vanishing tetrahedron-size is taken, so that what’s left in the end is truly a line of ‘infinitely small triangles’; you need one number, one coordinate, to specify a point on this line. Now consider a tessellation of the plane with triangles; in the small triangle limit, this is again just an ordinary plane, where you need two numbers to specify any given point. However, another view of this plane is to think of it as obtained from the original line by ‘twisting’ it into a plane-filling curve.
Intuitively, you might think that every point on that line, and hence, the plane thus generated, could be identified by a single number, and hence, the whole geometry should still really be 1D. However, the length of the curve obtained through the iterative process outlined in the wiki is infinite, and in fact, any finite way you might go along it does not get you measurably far from the origin. I.e. if the line starts in the lower left corner of the square containing it, and you wish to, say, specify the coordinate of the upper right corner, there is no finite number corresponding to the length of the curve up to that point. So even though all we really have is a 1D line that’s bunched up a certain way, one coordinate isn’t sufficient to describe it. This doesn’t derive from the fact that the curve is embedded in a 2D-manifold, but rather, from the curves’ fractal nature; at the bottom, a curve, a plane, a space and whatever else are just sets of points, whose dimension is intrinsic to them.
But the dimension used in CDT is yet a bit of a different beast, commonly called, I think, the spectral dimension. Roughly, it’s the dimension that a particle undergoing diffusion – i.e. moving about randomly – sees. It’s intuitively obvious that this process is sensitive to the dimension of space; constrained to two dimensions, there are much less possibilities for the particle to move than there are in three. Now, as it turns out, in the small regime, a particle diffuses as if spacetime were 2D; in the large scale limit, it diffuses as if there were four dimensions.
I believe that was very clear, so thanks!
Are you saying this last bit is an observed fact, or a hypothetical consequence of CDT?
Also: if the 2-d space is a twisted up line in the way you describe, are there necessarily two orderings involved–one defining an ordering on the 2-d space, one defining it in the 1-d space? Or is it possible, rather, for the two ordering relations to somehow define all the same pairs of points as having the same distance measure?
It’s an observed hypothetical consequence of CDT. 
Most of the work in CDT is carried out via numerical simulation, due to the equations involved generally being intractable, and those simulations point to small-scale 2D behaviour. There’s no observations (setting aside the possible cosmic ray thing) suggesting anything like it in the real world, to the best of my knowledge, sorry for not being clear on that.
Hmm, I believe I led you to mix up a few things. The plane-filling curve isn’t really a 1D set in any sense anymore, I was talking a bit carelessly. It’s a fractal of dimension 2. I merely brought it up as an example to illustrate how ‘bunching up’ can change dimensionality. In particular, it’s 2D on any scale, unlike the CDT spacetime.
I just read through some of the papers for the co-planar cosmic shower claims. For what it’s worth, I have come away unconvinced by the experimental claims. (To be sure, I didn’t read up on all of the claims, though, since some are in obscure cosmic ray journals that I wasn’t about to go chase down in hardcopy form.)