Watching two squirrels chase each other up, down, and all around a tree, a revelation came to me - why would we expect the "spatial organization" part of the brain to be limited to two-dimensional hexagonal tiles? These animals spend most of their time running around on trees, not on the ground. Their world is not flat. Their "surface" is a constantly curving tree trunk. Insects, same thing. For them, the basis of spatial dimension is not a flat plane, it's a curved plane. Up, down, and gravity in general are not constants. The default is a curved surface, not the other way around. And that's the more flexible model of the two, so why wouldn't it be the base model?
The idea that the spatial-brain is 2-D is a very human-based bias. It shouldn't be hard to accept the fact that not long after the hexagonal grid brain model was confirmed ("
grid cells" 2005), the
hyperdimensional toroid model came on the scene (2022). In other words, the brain thinks the landscape is a toroid.
What's happening in the rapid upheaval of our understanding of space (quantum gravity, Moiré lattices etc.) might be less disorienting if we considered this. I'm not sure how we do this, re-wiring our brains to perform in topological space, but I bet psychedelic mushrooms would help. Or DMT:
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
And finally, aside from the spatial implications, topology seems to be a fertile area of study for discoveries on the nature of all things quantum, as seen below.
Team discovers thousands of new transformable knots
Sep 2023, phys.org
They discovered thousands of new transformable knots including three novel shapes that the humble figure-eight knot can assume, doubling the number documented to date in scientific literature.
via EPFL Ecole Polytechnique Federale de Lausanne Geometric Computing Laboratory: Michele Vidulis et al, Computational Exploration of Multistable Elastic Knots, ACM Transactions on Graphics (2023). DOI: 10.1145/3592399
Molecular knots, left and right: How molecules form knots
Oct 2023, phys.org
They created a computational model for molecular knots, and mostly for optical topology.
via Max Planck Institute for Polymer Research in Mainz, Germany: Yani Zhao et al, Can Polymer Helicity Affect Topological Chirality of Polymer Knots?, ACS Macro Letters (2023). DOI: 10.1021/acsmacrolett.2c00600
Topologically structured light detects the position of nano-objects with atomic resolution
May 2023, phys.org
Just superoscillatory light, optical metrology, and topologically structured light.
via University of Southampton and Nanyang Technological University: Tongjun Liu et al, Picophotonic localization metrology beyond thermal fluctuations, Nature Materials (2023). DOI: 10.1038/s41563-023-01543-y
Researchers demonstrate that quantum entanglement and topology are inextricably linked
Jan 2024, phys.org
Skyrmion topology to be specific:
"We achieved this experimental milestone by entangling two identical photons and customizing their shared wave-function in such a way that their topology or structure becomes apparent only when the photons are treated as a unified entity"
In the realm of condensed matter physics, skyrmions are highly regarded for their stability and noise resistance.
"Our work presents a paradigm shift: the topology that has traditionally been thought to exist in a single and local configuration is now nonlocal or shared between spatially separated entities" says Ornelas.
Expanding on this concept, the researchers utilize topology as a framework to classify or distinguish entangled states. They envisage that "this fresh perspective can serve as a labeling system for entangled states, akin to an alphabet," says Dr. Isaac Nape, a co-investigator.
via Structured Light Laboratory in the School of Physics at the University of the Witwatersrand in South Africa, string theorist Robert de Mello Koch from Huzhou University in China, previously from Wits University: Pedro Ornelas et al, Non-local skyrmions as topologically resilient quantum entangled states of light, Nature Photonics (2024). DOI: 10.1038/s41566-023-01360-4
Quantum physicists develop robust and ultra-sensitive topological quantum device
Jan 2024, phys.org
They were the first to realize the topological skin effect on a microscopic scale in a semiconductor material. This quantum phenomenon was initially demonstrated at a macroscopic level three years ago—but only in an artificial metamaterial, not a natural one. This is therefore the first time that a tiny, semiconductor-based topological quantum device that's both highly robust and ultra-sensitive has been developed.
via Würzburg-Dresdner Exzellenzcluster for Complexity and Topology in Quantum Matter: Kyrylo Ochkan et al, Non-Hermitian topology in a multi-terminal quantum Hall device, Nature Physics (2024). DOI: 10.1038/s41567-023-02337-4
Classifying quantum secrets: Pendulum experiment reveals insights into topological materials
Mar 2024, phys.org
Interesting analog (quantum analog)
They built an array of 50 coupled pendula, with string lengths that slightly varied from one pendulum to the other. The strings of each neighboring pair of pendula were connected at a controlled height, such that each one's motion would affect its neighbors' motion.
The system obeyed Newton's laws of motion, but the precise lengths of the pendula and the connections between them created a magical phenomenon: Newton's laws caused the wave of the pendula's motion to approximately obey Schrödinger's equation. Therefore, the motion of the pendula, which is visible in the macroscopic world, reproduces the behaviors of electrons in periodic systems such as crystals.
via the Nuclear Research Center, Department of Biomedical Engineering, School of Mechanical Engineering, and School of Physics and Astronomy at Tel Aviv University: Izhar Neder et al, Bloch oscillations, Landau–Zener transition, and topological phase evolution in an array of coupled pendula, Proceedings of the Nation
'Tube map' around planets and moons made possible by knot theory
Apr 2024, phys.org
Knot Theory
In recent decades, space missions have increasingly relied on the ability to change the course of a satellite's path through space without using fuel by finding 'heteroclinic connections', usually calculated by using vast computing power to churn through one option after another or by making an 'intelligent guess' and then investigating it further. A new technique uses an area of math called knot theory to quickly generate rough trajectories.
via University of Surrey: Danny Owen et al, Applications of knot theory to the detection of heteroclinic connections between quasi-periodic orbits, Astrodynamics (2024). DOI: 10.1007/s42064-024-0201-0
