The Hofstadter Regime Converges Upon Us

AKA Fine-Tuning The Chaos Machine 

The above image is the perfect example of a moiré lattice and where it comes from. I noticed it while watching a lecture by a scientist cited below, for being the first to discover Hofstadter's Butterfly in real life. It's just the best image I've seen to explain what a moiré lattice is.

Credit: Graphene hBN Moire Lattices - taken from the presentation Bloch, Landau, and Dirac - Hofstadter's Butterfly in Graphene by Philip Kim - Kavli Inst 2018 [youtube link]

The perimeter of ignorance in science is also the front door of chaos theory. You could call it a lot of other things, most of which are listed in the tags for this post, but mostly, anything that's "too complicated" for us to understand right now, it's chaos-related. Discoveries in this field come from a bunch of different places, like interdimensional graphene, like topological operators, like epilepsy surgery.

Harnessing chaos: How the brain turns randomness into robust memory

Jan 2025, phys.org

https://medicalxpress.com/news/2025-01-harnessing-chaos-brain-randomness-robust.html

Previous work on brain-imitating artificial intelligence systems known as neural networks suggested that injecting random fluctuations into their activity could actually improve their performance as they learned to perform a task. 

Noise appears to increase the amount of time it takes for inhibitory neuron connections with other neurons to weaken. This slowing effect in turn stabilizes neural patterns of activity related to memories, helping them persist over time.

(This whole thing makes me think very differently about background noise, and maybe even the idea of 'functional music')...

via Columbia Engineering Systems Intelligence Laboratory: Nuttida Rungratsameetaweemana et al, Random noise promotes slow heterogeneous synaptic dynamics important for robust working memory computation, Proceedings of the National Academy of Sciences (2025). DOI: 10.1073/pnas.2316745122

https://dx.doi.org/10.1073/pnas.2316745122

How topology drives complexity in brain, climate and AI

Feb 2025, phys.org

https://phys.org/news/2025-02-topology-complexity-brain-climate-ai.html

Yes it does 

Transformative framework for understanding complex systems, using the new field of higher-order topological dynamics, and creating a connection between topological structures and emergent behavior. This comes from the field of information theory, and combines fusion of topology, higher-order networks, and non-linear dynamics.

via University of London: Ana P. Millán et al, Topology shapes dynamics of higher-order networks, Nature Physics (2025). DOI: 10.1038/s41567-024-02757-w

https://dx.doi.org/10.1038/s41567-024-02757-w

(Many body problem and higher order networks are the same thing - "interactions that extend beyond simple pairwise relationships".)

Most of us need to know: Hofstadter's butterfly (1976) was discovered before Mandelbrot coined the term "fractal" (1980), so he didn't know what to call it.

Hofstadter's butterfly: Quantum fractal patterns visualized

Feb 2025, phys.org

https://phys.org/news/2025-02-hofstadter-butterfly-quantum-fractal-patterns.html

"Our discovery was basically an accident. We didn't set out to find this."

This is the first time Hofstadter's butterfly has been directly observed experimentally in a real material.

It was found using a moiré lattice - they were investigating superconductivity in twisted bilayer graphene, and when you hear twisted layers, you know we're also talking magic angle sandwiches. They used a scanning tunneling microscope to image moiré crystals at atomic resolution and examine their electron energy levels. The microscope works by bringing a sharp metallic tip less than a nanometer from the surface to allow quantum "tunneling" of electrons from the tip to the sample.

via Princeton University: Kevin P. Nuckolls et al, Spectroscopy of the fractal Hofstadter energy spectrum, Nature (2025). DOI: 10.1038/s41586-024-08550-2

https://dx.doi.org/10.1038/s41586-024-08550-2

Fitness centrality: New tool finds critical points in everything from cybersecurity to ecological conservation

Jan 2025, phys.org

https://techxplore.com/news/2025-01-centrality-tool-critical-cybersecurity-ecological.html

The Vienna Complexity hub making waves

This approach is particularly good at finding nodes that, if removed, would isolate many other parts of the network—similar to a server failure interrupting the connection of many users in a communication network or a pump failure in a water supply network paralyzing the supply of water to districts.

Species in ecological networks, nodes in cybersecurity, roads in transportation networks. That's great. But it's people where this really has impact. Imagine trying to disable a social movement that could disturb the social fabric of a nation. You find the people, the nodes, at the center of the social network, and ... remove them, let's say. 

via Complexity Science Hub Vienna: Vito D P Servedio et al, Fitness centrality: a non-linear centrality measure for complex networks, Journal of Physics: Complexity (2025). DOI: 10.1088/2632-072X/ada845

https://dx.doi.org/10.1088/2632-072X/ada845

The Face of the Dimensional Liberator

The fractals are coming. 

New study reveals brain's fractal-like structure near phase transition, a finding that may be universal across species

Jun 2024, phys.org

https://phys.org/news/2024-06-reveals-brain-fractal-phase-transition.html

Now, a new Northwestern University study has discovered that the brain's structural features reside in the vicinity of a similar critical point - either at or close to a structural phase transition. Surprisingly, these results are consistent across brains of humans, mice and fruit flies, which suggests the finding might be universal.

Although the researchers don't know between which phases the brain's structure is transitioning, they say this new information could enable new designs for computational models of the brain's complexity and emergent phenomena.

Brain cells are arranged in a fractal-like statistical pattern at different scales. Self-similarity, long-range correlations and broad size distributions are all signatures of a critical state, where features are neither too organized nor too random. 

"These are things we see in all critical systems in physics"

The researchers were amazed to find that all brain samples studied—from humans, mice and fruit flies—have consistent critical exponents across organisms, meaning they share the same quantitative features of criticality. The underlying, compatible structures among organisms hint that a universal governing principle might be at play. 

via Northwestern University Weinberg College of Arts and Sciences: Helen S. Ansell et al, Unveiling universal aspects of the cellular anatomy of the brain, Communications Physics (2024). DOI: 10.1038/s42005-024-01665-y

https://dx.doi.org/10.1038/s42005-024-01665-y

Image credit: Cross section of European beach grass Ammophila arenaria leaf - Gerhard Vlcek Nikon Small World 7th Place - 2024 [link]

Can a computer chip have zero energy loss in 1.58 dimensions?

Jul 2024, phys.org

https://phys.org/news/2024-07-chip-energy-loss-dimensions.html

Many states without energy loss might exist somewhere in between one and two dimensions. At 1.58 dimensions - By growing a chemical element (bismuth) on top of a semiconductor (indium antimonide), the scientists in China obtained fractal structures that were spontaneously formed, upon varying the growth conditions. The scientists in Utrecht then theoretically showed that, from these structures, zero-dimensional corner modes and lossless one-dimensional edge states emerged.

"The fractals behave like two dimensional topological insulators at finite energies and at the same time exhibit, at zero energy, a state at its corners that could be used as a qubit, the building blocks of quantum computers. Hence, the discovery opens new paths to the long-wished qubits."

In follow-up research, the experimental group in China will try to grow a superconductor on top of the fractal structure. These fractals have many holes, and there are lossless currents running around many of them. Those could be used for energy efficient processing of information. They also exhibit zero-energy modes at their corners, thus combining the best of the one-dimensional and two-dimensional worlds. "If this works, it might reveal even more unexpected secrets hidden at dimension 1.58."

via the QuMAT consortium at Utrecht University and Shanghai Jiao Tong University: Canyellas, R., et al. Topological edge and corner states in bismuth fractal nanostructures. Nature Physics (2024). DOI: 10.1038/s41567-024-02551-8

https://dx.doi.org/10.1038/s41567-024-02551-8

Required Post Script:

If you didn't think you'd be getting an obligatory Isaac Asimov lesson here then I don't know where you think you are: Isaac Asimov wrote a story called Robot Dreams, and that's where we get our Laws of Robotics. Seriously, and this is why science and art need each other to continue to be relevant to humans. In the story, a robot named Elvex (LVX-1) is updated with "fractal geometry" because they thought 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").

Physicists explain how fractional charge in pentalayer graphene could work

Nov 2024, phys.org

https://phys.org/news/2024-11-physicists-fractional-pentalayer-graphene.html

Fractal Man, explained:

They found that the moiré arrangement of pentalayer graphene, in which each lattice-like layer of carbon atoms is arranged atop the other and on top of the boron-nitride, induces a weak electrical potential. When electrons pass through this potential, they form a sort of crystal, or a periodic formation, that confines the electrons and forces them to interact through their quantum correlations.

This electron tug-of-war creates a sort of cloud of possible physical states for each electron, which interacts with every other electron cloud in the crystal, in a wavefunction, or a pattern of quantum correlations, that gives the winding that should set the stage for electrons to split into fractions of themselves.

"This is a completely new mechanism, meaning in the decades-long history, people have never had a system go toward these kinds of fractional electron phenomena."

Note: Two other research teams - one from Johns Hopkins University, and the other from Harvard University, the University of California at Berkeley, and Lawrence Berkeley National Laboratory - have each published similar results in the same issue.

via MIT: Zhihuan Dong et al, Theory of Quantum Anomalous Hall Phases in Pentalayer Rhombohedral Graphene Moiré Structures, Physical Review Letters (2024). DOI: 10.1103/PhysRevLett.133.206502.

https://dx.doi.org/10.1103/PhysRevLett.133.206502

Keeping an Eye on the Fractal Frontier

Dimensions are not always whole numbers, they can be fractions too. Electrons are not always one single indivisible entity, but apparently they can be fractions of themselves. You too, the illusion of a persistent subjective self, fractions.

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states

Dec 2023, phys.org

https://phys.org/news/2023-12-fractal-photonic-anomalous-floquet-topological.html

Fractal photonics - stable carriers for high-capacity quantum information transmission by carry multiple topologically protected quantum chiral edge states

Chinese Academy of Science: Meng Li et al, Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states, Light: Science & Applications (2023). DOI: 10.1038/s41377-023-01307-y

https://dx.doi.org/10.1038/s41377-023-01307-y

Image credit: AI Art - Deep Within Synapses - 2023

Study finds no evidence for fractal scaling in canopy surfaces across a diverse range of forest types

Jan 2024, phys.org

https://phys.org/news/2024-01-evidence-fractal-scaling-canopy-surfaces.html

The way trees grow together do not resemble how branches grow on a single tree (called self-similarity or fractality).

Airborne laser scanning data from nine sites spread across Australia's Terrestrial Ecosystem Research Network. "We found that forest canopies are not fractal, but they are very similar in how they deviate from fractality, irrespective of what ecosystem they are in. Most ecosystems, like forests, will hit an upper limit—most likely determined by the maximum size of its organisms—beyond which their structure cannot vary freely anymore.

So it's scale, but not fractals. 

via University of Bristol: Fabian Jörg Fischer et al, No evidence for fractal scaling in canopy surfaces across a diverse range of forest types, Journal of Ecology (2023). DOI: 10.1111/1365-2745.14244

https://dx.doi.org/10.1111/1365-2745.14244

Electrons become fractions of themselves in graphene

Feb 2024, phys.org

https://phys.org/news/2024-02-electrons-fractions-graphene.html

"This five-layer graphene (pentalayer graphene) is a material system where many good surprises happen"

In very special states of matter, electrons can splinter into fractions of their whole. This phenomenon is known as "fractional charge", and known to physicists as the "fractional quantum Hall effect," is rare.

When five sheets of graphene are stacked like steps on a staircase, the resulting structure inherently provides just the right conditions for electrons to pass through as fractions of their total charge, with no need for any external magnetic field.

The results are the first evidence of the "fractional quantum anomalous Hall effect"

The pentalayer structure were aligned with hexagonal boron nitride (hBN) to produce a moiré superlattice that could slow electrons down in ways that mimic a magnetic field.

"The day we saw it, we didn't recognize it at first," says first author Lu. "Then we started to shout as we realized, this was really big. It was a completely surprising moment."

via Massachusetts Institute of Technology: Long Ju, Fractional quantum anomalous Hall effect in multilayer graphene, Nature (2024). DOI: 10.1038/s41586-023-07010-7.

https://dx.doi.org/10.1038/s41586-023-07010-7

Discovery of the first fractal molecule in nature

Apr 2024, phys.org

https://phys.org/news/2024-04-discovery-fractal-molecule-nature.html

The first regular molecular fractal in nature. They discovered a microbial enzyme - citrate synthase from a cyanobacterium - that spontaneously assembles into a pattern known as the Sierpinski triangle.

"We stumbled on this structure completely by accident and almost couldn't believe what we saw when we first took images of it using an electron microscope" 

via Max Planck Institute in Marburg and the Philipps University in Marburg: Franziska L. Sendker et al, Emergence of fractal geometries in the evolution of a metabolic enzyme, Nature (2024). DOI: 10.1038/s41586-024-07287-2

https://dx.doi.org/10.1038/s41586-024-07287-2

Foraging for Behavior Patterns and Other Levy Things

Read a disappointing book recently called The Drunkard's Walk -- only disappointed because I thought it would be about the drunkard's walk but it was just about randomness in statistics and probability. Thought it would be about Levy flights, foraging behavior, random walks, saccades, etc. Something very fractal about the nature of these phenomena.

Explore or exploit: How our brains make choices

Apr 2022, phys.org

https://medicalxpress.com/news/2022-04-explore-exploit-brains-choices.html

Basically, newer areas of the brain (evolutionarily speaking) such as the lateral prefrontal cortex, activated more when an unfamiliar choice was made while older brain systems including the amygdala and the orbitofrontal cortex showed increased activation while weighing the value of explorations vs. exploitation. This suggests that both the newer and older circuits in the brain worked together during the decision-making process rather than against each other as was previously thought.

via University of New Mexico: Jeremy Hogeveen et al, The neurocomputational bases of explore-exploit decision-making, Neuron (2022). DOI: 10.1016/j.neuron.2022.03.014

https://dx.doi.org/10.1016/j.neuron.2022.03.014

Typical movement behavior at large events increases risk of spreading infectious diseases

Sep 2022, phys.org

https://medicalxpress.com/news/2022-09-typical-movement-behavior-large-events.html

Burstiness (another word for foraging pattern behavior)

An earlier paper on this topic compared the movements of individuals in the crowd to the typical foraging patterns that were also present in our human hunter-gatherer ancestors.

The first author of the study, Ph.D. candidate Philip Rutten, says, "This [new study] shows that, if the infection probability is time-dependent, an intermittently moving but freely mixing crowd may present the highest level of transmission risk." 

Because infection is a probability game.

via University of Amsterdam: Philip Rutten et al, Modelling the dynamic relationship between spread of infection and observed crowd movement patterns at large scale events, Scientific Reports (2022). DOI: 10.1038/s41598-022-19081-z

https://dx.doi.org/10.1038/s41598-022-19081-z

Not-so private eyes: Eye movements hold clues to how we make decisions

Dec 2022, phys.org

https://medicalxpress.com/news/2022-12-not-so-private-eyes-eye-movements.html

"Unlike your arms or legs, the speed of eye movements is almost totally involuntary. It's a much more direct measurement of these unconscious processes happening in your brain."

In the study, the team asked 22 human subjects to walk on a treadmill then choose between different settings displayed on a computer screen: a brief walk up a steep grade or a longer walk on flat ground.

Researchers discovered that the subjects' eyes gave them away: Even before they made their choices, the treadmill users tended to move their eyes faster when they looked toward the options they ended up choosing. The more vigorously their eyes moved, the more they seemed to prefer their choice.

"Initially, the saccades to either option were similarly vigorous," Ahmed said. "Then, as time passed, that vigor increased and it increased even faster for the option they eventually chose."

via University of Colorado at Boulder: Colin C. Korbisch et al, Saccade vigor reflects the rise of decision variables during deliberation, Current Biology (2022). DOI: 10.1016/j.cub.2022.10.053

https://dx.doi.org/10.1016/j.cub.2022.10.053

Image credit: A laser beam (orange) creates excitons (purple) that are trapped inside the semicondcutor material by electric fields. Credit: Puneet Murthy / ETH Zurich

https://phys.org/news/2022-05-tunable-quantum-excitons.html

The Geometry of Information Space

New method to determine the dimensionality of complex networks through hyperbolic geometry

Oct 2022, phys.org

https://techxplore.com/news/2022-10-method-dimensionality-complex-networks-hyperbolic.html

"The intrinsic geometry of data sets or complex networks is not obvious"

Infer the dimensionality of complex networks with hyperbolic geometrics, which captures the complexity of relational structures of the real world.

The study provides a multidimensional hyperbolic model of complex networks that reproduces its connectivity, with an ultra-low and customizable dimensionality for each specific network. This enables a better characterization of its structure—e.g., at a community scale—and the improvement of its predictive capability.

(configurational geometric model or SD model)

The study reveals unexpected regularities:

• extremely low dimensions of molecular networks associated with biological tissues
• slightly higher dimensionality required by social networks and the Internet
• brain connectomes are close to three dimensions in their automatic organization

The model postulates a law of interconnection of the network elements (or nodes) that is gravitational, so nodes that are closer in a similarity space —of spherical geometry in D dimensions—and with more popularity—an extra dimension corresponding to the importance of the node—are more likely to establish connections."

In the study, the similarity and popularity variables are combined to give rise to the hyperbolic geometry of the model, which emerges as the natural geometry representing the hierarchical architecture of complex networks.

(It collapses the dimensions.)

"The Internet only requires D = 7 dimensions to be mapped into the hyperbolic space of our model, whereas this name is multiplied by six and scales to D = 47 in one of the most recent techniques using Euclidean space," says Professor Marián Boguñá.

via University of Barcelona: Pedro Almagro et al, Detecting the ultra low dimensionality of real networks, Nature Communications (2022). DOI: 10.1038/s41467-022-33685-z

https://pubmed.ncbi.nlm.nih.gov/36243754/

Image credit: AI Art - Archival Medical Deconstruction Chart of Icosahedral Phage - 2022

Fractals in the News

It's fractals all the way down -- if you've been waiting for fractals to have their moment, that time is coming. It took the advent of computers for us to discover fractals in the first place, but then it was used for video game graphics and psychedelic art, and that's it for about 40 years. Now we're using the Large Hadron Collider, the most advanced piece of scientific equipment created, to discover that fractals are not only found in all living things, but in the furthest depths of physics - the Bose Einstein Condensate (BECs are an integral part of quantum computing, metamaterials and intelligent matter, and will be a key technology in the future). 

Study proposes mathematical tool to help understand fractal structure of quark-gluon plasma

Jun 2022, phys.org

https://phys.org/news/2022-06-mathematical-tool-fractal-quark-gluon-plasma.html

When hadrons are accelerated to relativistic velocities and made to collide with each other their confinement is interrupted and the quarks and gluons scatter, forming a plasma that lasts only a tiny fraction of a second, but observation of it has produced important discoveries about the nature of material reality.

Quark-gluon plasma has a fractal structure. When it disintegrates into a stream of particles propagating in various directions, the behavior of the particles in the jets is similar to that of the quarks and gluons in the plasma. Moreover, it decays in a cascade of reactions with a pattern of self-similarity over many scales that is typical of fractals.

"Fractal theory explains BEC formation"

Deppman questions whether fractal structures could also be present in electromagnetism. This would explain why so many natural phenomena, from lightning to snowflakes, have fractal structures, as they are all governed by electromagnetic forces. It might also explain why Tsallis statistics are present in so many phenomena. "Tsallis statistics have been used to describe scale transformation invariance, a key ingredient of fractals," he said.

via Large Hadron Collider by the European Organization for Nuclear Research-research: E. Megías et al, Nonlinear Klein–Gordon equation and the Bose–Einstein condensation, The European Physical Journal Plus (2022). DOI: 10.1140/epjp/s13360-022-02511-2

https://dx.doi.org/10.1140/epjp/s13360-022-02511-2

Post Script:

Creating an ultrafast optoelectronic switch using a Bose-Einstein condensate of polaritons

Sep 2022, phys.org

https://phys.org/news/2022-09-ultrafast-optoelectronic-bose-einstein-condensate-polaritons.html

Terahertz polariton switch.

via Universities in Shanghai, Xiamen, Shandong, Nanjing, Shanxi and NYU: Fei Chen et al, Optically Controlled Femtosecond Polariton Switch at Room Temperature, Physical Review Letters (2022). DOI: 10.1103/PhysRevLett.129.057402

https://dx.doi.org/10.1103/PhysRevLett.129.057402

Image credit: AI Art - Fractal in a Jar - 2022

Pattern 240 - Half Inch Trim

It's a kind of bible of architecture, A Pattern Language came out in 1977, and in over 1,000 pages codifies the language of the built environment, or how it ought to be. It's a guide to livable cities and livable buildings. Maybe it's a bit Cali-centric, with all it's recommendations for outdoor space design, but it serves nonetheless as a builder's guide to making nice places to be in. 

The book is over 1,000 pages, and not until page 1,112 do we see Pattern 240 - Half Inch Trim. It is the most beautiful, simple, and airtight explanation in the whole book.

Pattern 240 - Half Inch Trim

Totalitarian, machine buildings do not require trim because they are precise enough to do without. But they buy their precision at a dreadful price: by killing the possibility of freedom in the building plan. 

A free and natural building cannot be conceived without the possibility of finishing it with trim, to cover up the minor variations which have arisen in the plan, and during its construction. p1112

The principle goes on to explain that it is essential that cuts be inaccurate within a half-inch or so, to avoid waste, but also to allow subtle adaptations.

And it goes on to explain why the modern practice of using precision building components is a bad idea:

"This one aspect of construction has by itself destroyed the builder's capacity to make a building which is natural, organic, and adapted to the site." p1113

But this is not the real argument for Half Inch Trim. The real argument is more deeply psychological, and underlies a quantitative, and universal, feature of aesthetic sensibility:

Our own bodies and the natural surroundings in which we evolved contain a continuous hierarchy of details, ranging all the way from the molecular fine structure to gross features like arms and legs (in our own bodies) and trunks and branches (in our natural surroundings).

We know from results in cognitive psychology that any one step in this hierarchy can be no more than 1:5, 1:7, or 1:10 if we are to perceive it as a natural hierarchy. We cannot understand a hierarchy in which there is a jump in scale of 1:20 or more. It is this fact which makes it necessary for our surroundings, even when man-made, to display a similar continuum of detail. 

Most materials have some kind of natural fibrous or crystalline structure at the scale of about 1/20 inch. But if the smallest building detail dimensions are of the order of 2 or 3 inches, this leaves a jump of 1:40 or 1:60 between these details and the fine structure of the material. ... [and 1/20 inch is 1:10 to a half inch] p1114-1115

Notes:
A Pattern Language - Towns, Buildings, Construction
Christopher Alexander, Sara Ishikawa, Murray Silverstein. Oxford University Press. New York. 1977.

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.

Right Angles in Nature

Tabular Iceberg, care of BBC

Although it looks like the future on a planet that is totally colonized by Megastructures and Ecumenopolises, this is a pretty common thing called a tabular iceberg. Nature is short on straight lines and definitely right angles, although Andy Goldsworthy would disagree.

Andy Goldsworthy made this. (Or did he?)

Ice, however, is good for straight lines. I do recall a drawing by a Japanese woodcut artist in a book about Modern Art. It was a frozen lake, just a bunch of straight lines, and predated the more abstract things to come out of the early 20th century.   

Notes:
Nasa photographs rectangular iceberg
Oct 2018, BBC

On Right Angles
Or why we tend to perceive right angles everywhere
Network Address, 2012

You Love Right Angles
AKA I'm Not a Right Angle You're a Right Angle!
Network Address, 2012

Comedy of the Commons

Balinese rice patties

Fractal patterns

Fractal planting patterns yield optimal harvests, without central control
Jun 2017, phys.org

Balinese rice farmers make some crazy patterns with their rice fields, but they don't do this on purpose. The rice fields plant themselves in this pattern, using the rice farmers. Just kidding, or not.

These farmers are all part of the same group, using the same resources, that being their rice patties. They plant their rice based on a whole bunch of variables, including the planting patterns of the other farmers who share the patties, and the amount of water flowing down the river. All of these variables are interdependent, such that the farmers in one area may change the amount of water in the river depending on when they plant, which in turn changes when other farmers will plant.

All of this decision-making, however, does not go through a centralized process, and although the farmers are making their own decisions, the final pattern of planted rice fields was not decided by them alone, but by the interaction and feedback of the system as a whole.

from the article:

"What is exciting scientifically is that this is in contrast to the tragedy of the commons, where the global optimum is not reached because everyone is maximizing his individual profit. This is what we are experiencing typically when egoistic people are using a limited resource on the planet, everyone optimizes the individual payoff and never reach an optimum for all," he says.

The scientists find that under these assumptions, the planting patterns become fractal, which is indeed the case as they confirm with satellite imagery. "Fractal patterns are abundant in natural systems but are relatively rare in man-made systems," explains Thurner. These fractal patterns make the system more resilient than it would otherwise be. "The system becomes remarkably stable, again without any planning—stability is the outcome of a remarkably simple but efficient self-organized process. And it happens extremely fast. In reality, it does not even take ten years for the system to reach this state," Thurner says.

notes:
The Tragedy of the Commons

Phase Change

Nothing at all happens to the individual molecules when phase change takes place, instead only a rearrangement of their positions and motions. AND the two phase can coexist (like ice floating in water) at any scale in the whole (little ice or big ice) – the pattern of the two phase are spread over a large range of scales. [like Fractals] (p169)

Lee Smolin, The Life of the Cosmos, 1997, Oxford