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Is There a Crystal Lattice Possessing Five-Fold Symmetry? (2012) [pdf] (ams.org)
45 points by wolfi1 4 days ago | hide | past | web | favorite | 8 comments





For a definition of crystal that requires translational symmetry then no, but more colloquially speaking there are natural mineral "crystals" like Icosahedrite [1,2] (technically quasicrystals [3]) that have space-filling tilings with five-fold rotational symmetry.

The first discovery this type of symmetry (in synthetic Al-Mn alloys in 1982) led to the 2011 Nobel Prize in chemistry [4]

[1] https://en.wikipedia.org/wiki/Icosahedrite

[2] https://doi.org/10.2138/am.2011.3758

[3] https://en.wikipedia.org/wiki/Quasicrystal

[4] https://www.nobelprize.org/prizes/chemistry/2011/summary/


TL;DR:

Crystals require translational symmetry, therefore, no, not in 3-D space.

In 4-D you can have 5-fold symmetry, and a projection of these onto 3D space makes an arrangement with 5-fold symmetry which make quasi-crystals.

My own experience:

These have been observed in nature in nanoparticles, and are even the energy lowest state (!!). As they grow, however, there is a an energy strain penalty due to lacking translational symmetry that destabilizes them,.


Could you theoretically have 5-fold symmetry in time crystals, with the 4th dimension being time?

Time crystals are not an equilibrium state of matter.

You can have ‘driven’ Floquet time crystals which are mildly interesting in their own right. I wonder if 5-fold symmetries exist in those.

I'm not a PhD in physics (my interest in quasi-crystals stem from having to consider their existence and their energies in some simulations), but I wouldn't think so. Not stable, anyway.

Mathematically, yes, but atoms are on a discreet lattice locations. If you used time for the fourth dimension, you'd have to jump in time discreetly. That's kinda weird because where would the energy go? A photon? Then how do you recapture the energy? Do you "borrow" the energy (exploiting some uncertainty principle)? These are all high energy processes -> not a crystal (crystals are low energy states).


I’m no expert in this field either, but perhaps on sufficiently small time and size scales the discrete jumps in space and time would line up. It might be that for fermion matter it isn’t possible. I know incredibly little about more exotic matter, but I think I may have heard of theoretical crystals formed in non fermion matter in violent celestial scenarios.

I’d be interested to see a simulation that’s able to make a time crystal that can have symmetry between spatial and time dimensions, even if the physics are tuned for them rather than our universe. Just to see what it would look like. I’d take any new imagery to aid in understanding solid state physics.


>but atoms are on a discreet lattice locations

This is where it starts getting interesting. Where is the "location" of an atom? Is it the nucleus? Is it the electrons? Is it just the electrons which are localized to particular nuclei? Etc.

Typically the "location" has some uncertainty, both from thermal jitter and from the nature of orbitals as probability densities.




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