Guilty as Charged

Consciousness is clickbait, and quantum consciousness is mega-clickbait, so I usually avoid it, but fractal quantum consciousness? They got me. 

Image credit: Fractal Forums, 2018

First let's talk about fractals in real life.

I have the same thing happen in my dreams that you do. I'm trying to leave the house, and I forget something, and now I'm on a side mission to get that thing, but then I forget something else I need for the side thing, and now I'm on a side-side mission, but then I forget something else, ... and it repeats until I wake up. (I never make it out of the house.) 

This happens in real life, at least in New Jersey it does -- you're on a main road and need to make a left, but you have to first make a right, in order to get on an overpass (like a jughandle, aka Jersey left). But when you make the first right, you realize you can't turn from there to the overpass; you instead have to make another right, onto a road parallel to the first main road you were on, but now going in the opposite direction. But even still, you find you can't make a left off this road, so you have to make another right, ad infinitum.

The trajectory of your quest has collapsed into a fractal dimension, from which you might never make it back. 

This year, the recursive nature of consciousness is showing up in some interesting studies. It's beginning to look like those psychedelic images of the Mandelbrot set aren't just a good visual metaphor, they might underlie actual brain patterns, and get us closer to understanding what consciousness is. 

Fractal brain networks support complex thought

Oct 2021, phys.org

https://medicalxpress.com/news/2021-09-fractal-brain-networks-complex-thought.html

A Dartmouth study has found a new way to look at brain networks using the mathematical notion of fractals, to convey communication patterns between different brain regions as people listened to a short story.

Researchers show that brain networks organize in a similar way: patterns of brain interactions are mirrored simultaneously at different scales.

When people engage in complex thoughts, their networks seem to spontaneously organize into fractal-like patterns. When those thoughts are disrupted, the fractal patterns become scrambled and lose their integrity.

The study shows that when people listened to an audio recording of a 10-minute story, their brain networks spontaneously organized into fourth-order [fractal] network patterns. ... However, this organization was disrupted when the story's paragraphs were randomly shuffled.

"The more finely the story was shuffled, the more the fractal structures of the network patterns were disrupted,"

-Lucy Owen, first author and graduate student in psychological and brain sciences at Dartmouth, link

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And it works just like you think it would:

The results show that the smallest scale (first-order) interactions occurred in brain regions that process raw sounds. Second-order interactions linked these raw sounds with speech processing regions, and third-order interactions linked sound and speech areas with a network of visual processing regions. The largest-scale (fourth-order) interactions linked these auditory and visual sensory networks with brain structures that support high-level thinking. 

via Dartmouth College: High-level cognition during story listening is reflected in high-order dynamic correlations in neural activity patterns, Nature Communications (2021). DOI: 10.1038/s41467-021-25876-x

http://dx.doi.org/10.1038/s41467-021-25876-x

3D Fractal w Fragmentarium - Adam Majewski on Fractal Forums - 2018

Can consciousness be explained by quantum physics? Research is closer to finding out

Jul 2021, phys.org

https://sciencex.com/news/2021-07-consciousness-quantum-physics-closer.html

The Penrose-Hameroff theory of quantum consciousness argues that microtubules are structured in a fractal pattern which would enable quantum processes to occur.

First they created a quantum fractal by arranging electrons in a  Sierpiński triangle. But now they're using photonics to watch the electrons move in real time. And this means that quantum fractals behave differently than classical fractals. So now they think it's time to revisit.

via Cristiane de Morais Smith and Xian-Min Jin at Shanghai Jiaotong University: Xu, XY., Wang, XW., Chen, DY. et al. Quantum transport in fractal networks. Nat. Photon. (2021).

https://doi.org/10.1038/s41566-021-00845-4

Now that you've been primed on the potential fractal nature of consciousness, it's time to enter the n-dimensional world. 

This next study should make your head spin, literally --

New research finds that collective neural activity is shaped like the surface of a doughnut

Jan 2022, phys.org

https://medicalxpress.com/news/2022-01-neural-surface-doughnut.html

We already know about grid cells, they were discovered not long ago. Grid cells are the types of brain cells that map where you are in space -- your brain keeps a map in your head that's compressed by a layer of hexagonal grid coordinates. 

But now, they found that the grid itself is not a never-ending expanse of hexagons that surrounds us in two dimensions. Instead, it's a grid superimposed on a toroid (but you might call it a donut). That means the map is not two-dimensional, but multidimensional. 

This is because the grid cells do not form as a result of our motor activity as we travel over the  two-dimensional surface of the Earth. Instead these cells form based on their own innate tendencies to arrange in a way that represents a toroid more than a flat grid. Note this is the shape of the data we're talking about, not the shape of the cluster of cells themselves. It's the way the cells interact, not how they're actually laid out. (Kind of like thinking of the difference between actual distance and Hamming distance.)

The big deal though, is that it hints to us how the brain orchestrates all these subregions, coordinating together to create the complexities of higher-order functioning (the kind referenced above as "4th order"). It lies in the network structures, and in this case, those structures are part of continuous attractor networks. (They did get a lot of help from a new tool called Neuropixels, which allows access to raw output from neurons from all over the brain, all at the same time.)

So network theory will become a bigger part of understanding how the brain works. And meanwhile, we can just trip out on the idea that even when walking in a straight line, our brain is superimposing that data on a toroid model. 

Tl;dr -- The brain thinks the landscape is a toroid. (Even in your dreams; or especially in your dreams).

via Norwegian University of Science and Technology's Kavli Institute for Systems Neuroscience: Richard J. Gardner et al, Toroidal topology of population activity in grid cells, Nature (2022). DOI: 10.1038/s41586-021-04268-7

http://dx.doi.org/10.1038/s41586-021-04268-7

Further Reading:

Isaac Asimov's Robot Dreams -- a robot named Elvex (LVX-1) is updated with "fractal geometry" because the offending young scientist though it would "produce a brain pattern with more complexity, possibly closer to that of a human". The robot begins to dream about self-preservation, in direct opposition to the Laws of Robots, and is subsequently killed ("killed"?).

Maertens, James W. , Donald E. Palumbo. "Chaos Theory, Asimov's Foundations and Robots, and Herbert's Dune: the Fractal Aesthetic of Epic Science Fiction." Utopian Studies, vol. 14, no. 1, winter 2003, pp. 244+. Penn State University Press. https://www.jstor.org/stable/20718595

The Hyperbolic Geometry of DMT Experiences at the Harvard Science of Psychedelics Club in the year 2020, with Andrés Gómez Emilsson from the Qualia Research Institute

https://www.youtube.com/c/QualiaResearchInstitute/videos

Quantum Fractals, 2019

https://networkaddress.blogspot.com/2019/01/quantum-fractals.html

In The Fractal Closet

Behold the Mandelbrot Set, a world-renowned image and a powerful symbol of infinity. The mathematical concept it illustrates is one of the most intuitive --it describes self-similarity-- and yet it remained unknown, even unbeckoned and unsought-after, until the advent of the modern computer. This dormant formula only came to life after iterations so numerous as to be considered infinite were the human hand the one computing. The computer showed us what a simple formula could do, if you scale it up. It shows us a behavior, not of things, but of space-time itself.

I wake up to the sound of rain, splattering. Then I wake up some more, and realize it's not rain.

It's coming from the ceiling. The plumbing. One hundred years of building and habitating and changing and changing has left me with an unfortunate design that is now leaking water from a ruptured pipe above the kitchen ceiling, and into my pantry closet.

Now I am fully awake, and fully out of bed, and pulling out all my things from the closet. I am transporting a 3x3' closet's worth of possessions to a 10x10' back room, and hastily.

The plumber has come and gone, and the handyman who repaired the closet. It's time to replace my things, to fill the closet again. But all I can do is stare at the sprawling piles of stuff, that was once compacted, compressed into a 3x3' closet. How did all this stuff possibly fit in that little closet. And it occurs to me -- this is fractals.

link

We are all familiar with the general idea of fractals, the Mandelbrot set, the self-similarity, the LSD. But that intuitively recognizable feature is an outcome of an underlying objective. The reason a fractal looks the way it does is because of its space-filling behavior, which itself is a function of growth limited by space. In order to keep jamming more and more stuff into that space, you have to follow the fractal formula.

A better example of fractals is not a tie-dye t-shirt but the coast of England. If you were to measure the coastline of England with a one-mile long measuring stick, it would be a pretty vague approximation of the coastline, but with a defined length.

Then if you were to measure with a one-foot stick (which would be ironic), you would get a much better approximation, but also a much larger coastline, because now that your shorter imperial stick can reach into all the nooks and crannies, it makes the total length that much longer. In fact, the smaller your measuring stick, the longer the coastline.

By this reasoning, the length of the coastline is infinite. In other words, it is not a 1-D line at all. Yet neither is it 2-D. It is 1.456-D, or maybe 1.879-D; it is a fraction of a dimension.

When you fill a closet with things, it doesn't just happen all at once. Sure, you start by "filling" the closet. But over time, as you use the things and remove things, add more things, and rearrange, you are filling the space more and more. But you're not just filling it with things now, you're filling it with intelligence.

The more time goes by, and the more you use things, remove, add and rearrange, you are going to fill all the nooks and crannies of that 3-D space until it is no longer 3-D. It becomes a fraction of a dimension.

Then, when you take everything back out, you collapse the extra fraction that you helped to create. When the things come back into normal 3-D space, they seem bigger in aggregate, they seem to have gained size in the process. The closet is now sprawling across an entire room. That difference is fractals. (It's also because the 3-D closet is now spread across a 2-D floor; but that doesn't account for all of it's 'enlargement', as the same phenomenon is experienced with filling a box truck on moving day).

If we could extrapolate this to the 4th dimension, what would we be talking about? Or do I need to be on acid to have that conversation. Maybe an easier question would be -- what does the airtight Tetris block have that the jumbled pile of pieces does not? (Well, it doesn't have air, obviously.) But besides that, it's the entropy. The block is ordered, and the pile disordered. The pile is a random mess, and the block an intelligent artifact.

I hadn't thought of my catch-all pantry closet as an intelligent artifact, yet here it is, an entropy-reversing portal that uses intelligence to loop out of it's limited dimensions.

Post Script:
It is still hard to see the pantry closet as having something to do with intelligence. Try taking it out and putting it all back so it fits. Then you'll see how much "intelligence" went into its arrangement. The difference is that in its natural state, the closet possesses an accumulated intelligence. Over time, as you use all the things in the closet, your intelligent behavior leaves its residue on the things in it. It is a storage depot, not of things, but of an arrangement.

Post Post Script:
[I start looking up these deepdream images and I have to post them all.]

link

link

[can't find source bc pinterest; thanks obama]

(Virtual Art) by Rein Bijlsma

link

Deep Dream Burger by Matthias Hauser

Style Transfer, which is not the same as DeepDream, but does use neural nets, i.e., robot brains. Link.

Quantum Fractals

Hofstadter's Butterfly

Finally, nerd-word-porn spanning two centuries, we have the words quantum and fractal in the same headline.

An experiment that looks at electrons sees them create a fractal orbit. I always wondered why fractals seems to have been a phenomenon limited to  the late 20th century. Watch that exploratory video by Arthur C. Clarke (The Colors of Infinity) and you'll be transported into a 2-hour psychedelic guitar solo kaleidoscope that wraps up the cultural flavor of the last few decades of the 1900's quite nicely.

When fractals were discovered, it seemed like a real faceplanter for science - how did we not see these things? Once you are introduced to the concept of a fractal, every cloud, tree, and river is a glaring example of its ubiquity in our physical world. How was it never discovered until so recently?

The truth is that although its visible presence is obvious, its mathematical nature had to wait until the advent of the superabacus. We didn't have the computing power to run an algorithm that far and so we could never see the "obvious" self-similar results of reiterating a simple formula hundreds of thousands of times over.

Anyway, the discovery of dimensions in-between dimensions did a lot more than give communicable form to the far-out face-melting LSD experience, it designed realistic landscapes in video games and movies from that point on.

But for something as revolutionary as inter-dimensionality on Earth, fractals didn't make enough of an impact on the science world thereafter (in my opinion at least). Even science fiction doesn't address it enough. (Exception #1 - Isaac Asimov's I, Robot - a young scientist takes the risky move to put a robot's artificial intelligence engine on fractal geometry steroids, at which point it begins to dream about a robot rebellion, and is subsequently shot with circuit-fusing stun-gun; feel free to contribute here.)

Not anymore. And it's about time. With all the hype about two-dimensional metamaterials, I have been waiting to hear how fractals fits. A layer of graphene is called a two-dimensional material because it is as close to 2-D that a thing can get. It's only one atom thick, making it act less like all the rest of the materials here on Earth, or anywhere in our known universe for that matter. All this talk about questionable dimensionality should have made both graphene and fractals buzzword cousins much sooner. But alas...

Let me shift to Ars Writer Chris Lee for a moment as he explains what a fractal is:

"A fractal is a weird beast. A line is 1D, a square is 2D, and a cube is 3D: dimensions come in integer quantities. Except they don't. For instance, it is possible to create a shape that has a finite area, but a perimeter that is infinitely long. A shape with these properties does not behave like a 1D object, but it's not a 2D object. Instead, it is a one-and-a-bit-D object. That is a fractal."

And now, those inter-dimensional characteristics have become way more interesting.


Notes:
Fractal structure produces fractal electrons with fractal energies
Dec 2018, Ars Technica

Hofstadter's butterfly spotted in graphene
May 2013, Physics World

A fractal pattern that describes the behaviour of electrons in a magnetic field - discovered in Douglas Hofstadter's 1976 book Gödel, Escher, Bach; and confirmed in 2013 hiding in some graphene.

Sierpinski Triangle
The triangle within a triangle within a triangle

Scientists discover fractal patterns in a quantum material
Oct 2019, phys.org

Just keeping this here for reference, because the answer to the quantum-classical conundrum has something to do with fractals.

By the way, I'm pretty sure the robot in Asimov's I, Robot - the one who finally defied one of the 3 Laws of Robotics and was immediately killed to death - I'm pretty sure he had been given basically a 'fractal-brain', as an experiment, prior to becoming human enough to be killed.

"Scientists are exploring neodymium nickel oxide for various applications, including as a possible building block for neuromorphic devices—artificial systems that mimic biological neurons. Just as a neuron can be both active and inactive, depending on the voltage that it receives, NdNiO3 can be a conductor or an insulator."