In the past few weeks I have been trying to understand the eigenvalue problem (time-independent Schrödinger equation)

–𝑢'' + λ (cos 𝑥 + cos τ𝑥) 𝑢 = 𝐸𝑢

where λ is a parameter, 𝐸 is the eigenvalue (blame the physicists for the notation), τ is the golden ratio and the problem is posed on the infinite line. The motivation comes from quasicrystals.

Some solutions are localized around a minimum of the potential, but the none of the corresponding eigenvalues are isolated.

At higher energies, solutions spread out over the whole line, giving rise to the absolutely continuous spectrum which is a Cantor set.

This is wild, at least for me, but partially supported by my own computations and functional analysis results. But I am not fully confident of the former and struggling to understand the latter, so I am not sure whether this picture is complete or even correct.

The more I look into it, the less I understand ... any pointers are appreciated.

#FunctionalAnalysis #quasicrystal #SchrodingerEquation

#PhysicsJournalClub
"Direct observation of colloidal quasicrystallization"
by Y. Gao, B. Sprinkle, DWM Marr, and N. Wu

Nat. Phys. (2025)
doi.org/10.1038/s41567-025-02859-z

Quasicrystals are weird. When you solidify something it tends to get into a high-order state: a crystal. If you cool it down too fast so it doesn't manage to make a monocrystal it will form a polycrystalline state or, worse case scenario, something completely amorphous like a glass.
But quasicrystals are weird. They are ordered structures that lack a periodicity and making them is not easy.
In this paper the authors show how paramagnetic colloidal microspheres (i.e. big enough to be clearly visible under a microscope) subject to an electromagnetic field spontaneously arrange themselves into a quasicrystal.

This is 100% not my field, but the ability to create quasicrystals on demand looks so cool!

#Physics #Chemistry #Quasicrystal

#HowToThing #022 — Quasi-random lattice generator using https://thi.ng/lowdisc and https://thi.ng/geom-voronoi

These infinite and parametric low-discrepancy sequence generators can be used to produce n-dimensional samples/points and are useful for a whole range of graphics (and more generally sampling related) topics and generative design techniques. The example also shows basic usage of the 2D Delaunay/Voronoi mesh, incl. obtaining cells which can be automatically clipped against a user-provided convex boundary polygon.

Demo (reload to randomize):
https://demo.thi.ng/umbrella/quasi-lattice/

Source (more comments than code 😎):
https://github.com/thi-ng/umbrella/tree/develop/examples/quasi-lattice/src/index.ts

Further reading:
https://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/
https://en.wikipedia.org/wiki/Low-discrepancy_sequence

#ThingUmbrella #Voronoi #Quasicrystal #Generative #Graphics #LowDiscrepancy #Sampling #Geometry #TypeScript #JavaScript #Tutorial

Referenced link: https://phys.org/news/2023-05-liquid-quasicrystal-dodecagonal-tiling-pattern.html
Discuss on https://discu.eu/q/https://phys.org/news/2023-05-liquid-quasicrystal-dodecagonal-tiling-pattern.html

Originally posted by Phys.org / @physorg_com: http://nitter.platypush.tech/physorg_com/status/1656303086028111875#m

Researchers discover liquid #quasicrystal with dodecagonal tiling pattern @NatureChemistry https://www.nature.com/articles/s41557-023-01166-5 https://phys.org/news/2023-05-liquid-quasicrystal-dodecagonal-tiling-pattern.html

https://www.eurekalert.org/news-releases/982581

"We showed that a magnetic quasicrystal lattice can host states that go beyond the zero and one bit states of classical information technology,” Lopez-Bezanilla said. “By applying a magnetic field to a finite set of spins, we can morph the magnetic landscape of a quasicrystal object"

#magnetism #qubit #quantumComputing #quasicrystal #informationTechnology #magneticField #spin #quantumSpin #spinStates #morphing

#genuary2023 #Moiré Just some screensnaps from the physics sim example I'm working on, here with more rigid springs, the dot patterns are creating some interesting (animated) interference patterns, some of them defo in #quasicrystal territory...

(Ps. Probably only properly visible at full resolution...)

The Infinite Pattern That Never Repeats | Veritasium

https://www.youtube.com/watch?v=48sCx-wBs34

#veritasium #Penrose #tilings #quasicrystals #golden_ratio #quasicrystal