TL;DR: A mathematical, neuroscientific hypothesis about why humans hallucinate fractals and hyperbolic space artifacts, when under the influence of DMT, LSD, etc. In short: visual processing is done by cortical columns in the visual cortex. The perception of 3D arises as "constraints" between cortical columns, and these constraints are exactly those that convert free groups (free modules) into Cartesian spaces. (The "constraints" are the presentations, the quotients of the modulo quotienting operation.) This is a special case of the history monoid/trace monoid of parallel computing, where mutexes/semaphores give rise to a semi-commutative monoids. In the visual cortex, they are fully abelianized, giving rise to 3D cartesian space. If these constraints are lossened, one gets not cartesian space, but e.g. elliptic curves, assorted hyperbolic spaces, hyperbolic tilings, etc. These are assorted half-way points between free groups (free modules) and flat Cartesian space. The hypothesis is that hallucinogens disrupt communications between cortical columns, loosening the constraints that project onto pure Cartesian 3D space, and instead generate the various intermediary group/module structures, which are all fratcal-ish to varying degrees.
In mathematics, one gets 3D Cartesian space as a certain "presentation" (technical term) of "free modules". The presentation is an "abelianization"; informally it is a constraint that says "if you take one step forward and then a step to the left, you'll end up in the same place as a step to the left, and then a step forward." (This kind of constraint is called a "presentation"; all groups are presentations of the free group. All spaces are presentations of the free modules, etc. it's a fairly general thing.)
The free group looks like a fractal, as do various partial abelianizations and presentations of it. Assorted hyperbolic spaces (viz MC Escher prints) arise in similar ways.
I want to map this to the processing that cortical columns do in the visual cortex; but first, a short aside. In theoretical computer science, when you have a bunch of machines computing in parallel, you may want to send messages from one to the other, to synchronize data across them, or perhaps to interlock computations for certain resources (mutex, semaphores, etc.) The abstract theory for this gives you the "history monoid" and the "trace monoid" (see wikipedia) which are free monoids, which occasional abelianizations of bits and pieces here and there; ("semi-commutative monoids") the mutexes between threads/cpus do this; they're a kind-of "presentation")
Suppose I wanted to build an electronic model of a visual cortex out of a collection of single-pixel processors. Well, first, there are the "obvious" constraints: something that says this or that pixel is to the left/right, above/below this other neighboring pixel (and here, I am imagining "pixel" == "cortical column") Next are the "non-obvious" constraints, that invert the 2D (stereographic) image back into the 3D space we perceive. Here, I imagine interconnects between the cortical columns, implementing *exactly the same* presentation constraints that convert a free module to a 3D cartesian space. That is, the history/trace monoid of the cortical columns is abelianized into 3D cartesian space. It is "relatively clear" how one might do this mathematically/computationally: my hypothesis is that the visual cortex does it this way too. Presumably, it's more or less the simplest possible way that this could be done.
So here's the kicker: if you loosen some of these constraints, you no longer get flat, plain, old boring 3D space, you get various hyperbolic spaces and/or fractal structures. My hypothesis is that DMT/LSD disrupts the connectivity between cortical columns, removing some of the constraints, directly exposing more weakly coupled cortical columns, still doing their parallel processing, but now less tightly bound together. So, viola, fractals! Add the conventional observation about Indian paisley fabric prints, Islamic tiling patterns, etc. as quasi-hallucinatory visual patterns inherited from cultural tradition. So this is where they come from, this is the neurological "explanation" of this kind of visual perception/hallucination.
Of course, you know that I know almost nothing about neuroscience, so perhaps this is already well-known. (Or perhaps it's just wrong.) But it is very much an "a hah!" moment for me.
Background: I'm studying the geometry of the deep-learning neural nets,, transformers, the large-language models, StableDiffusion, etc. and I currently understand them as "explainable" by the geometry of extremely high-dimensional (cartesian) spaces. (it is, in some ways, 'just like' ordinary 3D space, but it's also very very bizarrely different. Weird non-intuitive stuff happens in high dimensions.) But I'm also studying syntactic extensions of this, where the cartesian-ness of it is loosened, the constraints are now syntactic (grammatical). So I was already walking down this path; the new insight is "gee golly, but this is exactly why humans hallucinate paisley and fractal visual artifacts."
I can explain the math above in greater detail; I'm developing it for unrelated purposes, anyway. I plan to do nothing with the neuroscience of this; I just thought you might be entertained.
Linas Vepstas 14 March 2023.