
Is There a Crystal Lattice Possessing Five-Fold Symmetry? (2012) [pdf] - wolfi1
https://www.ams.org/notices/201201/rtx120100022p.pdf
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cbkeller
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](https://en.wikipedia.org/wiki/Icosahedrite)

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

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

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

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zwsxedcrfvtgb
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,.

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willis936
Could you theoretically have 5-fold symmetry in time crystals, with the 4th
dimension being time?

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zwsxedcrfvtgb
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).

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willis936
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.

