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The following is what a Penrose tiling looks like:

enter image description here

I know the verticies of a dodecahedron can be grouped into 5 groups each of which are the verticies of a tetrahedron. So I thought of the idea that maybe a covalent network of carbon atoms each of which makes 4 unstrained bonds and has the form of something like a Penrose tiling but is dodecahedral is mathematically possible.

My question is

Has there been a discussion of whether that is mathematically possible according to the Dalton model or something like that, which can easily be found?

That was my question. The rest is just facts I want to share.

Diamond has 4 weak cleavage planes. Lonsdalite has 1. In either case, the cleavage plane is where there is only single bonds going through it perpendicular to it. If that form of diamond existed, it would probably have no such plane and therefore would probably have a higher fracture toughness when you apply tension in the direction perpendicular to the weakest cleavage plane or something like that. I think English is a bit flexible and they don't make the definition of fracture toughness that precise.

I read on the internet about other covalent networks of carbon atoms that do have strained bonds, which included on that resembled a tessellation of the truncated octahedron. I already know about those discussions and am not interested in an answer that talks about that or any form other than the very specific form I was asking about here.

Also, according to the Wikipedia article Quasicrystal, quasicrystal lacks periodic symmetry. It also says there are some quasicrystals in nature. I'm not sure they really are truly aperiodic.

This:

enter image description here

shows that you can take part of the Penrose tiling and turn it into a primitive cell. Maybe nature also takes part of what a true dodecahedral quasicrystal would look like and makes it into a primitive cell, but has the primitive cell large enough to include 5 subparts that have the relative orientation to one another like the 5 tetrahedra whose verticies form the verticies of a dodecahedron. The Wikipedia article Silicon nitride has an image with a description saying it is the $\alpha$ phase of $Si_3N_4$. It also looks disordered. If you take that unit cell and turn it into a torus, maybe it can be split into 1 or multiple primitive cells each of which could be part of a quasicrystal and has 5 small parts that have the relative orientations of 5 tetrahedra that make a dodecahedron.

Sources:

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There are two related structures.

Dodecahedrane is known - $\ce{C20H20}$ in which the carbon atoms form the lattice you described, with hydrogen atoms on the outside.

Beyond that, $\ce{C20}$, a fullerene is known, and you can find a great variety of similar structures, from $\ce{C20}$ to $\ce{C720}$.

As for your discussion of quasi-crystals, I think it's well established that there are indeed, aperiodic structures. The Nobel Prize in Chemistry in 2011 was awarded to Dan Shechtman for their discovery (and battle to ensure recognition).

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  • $\begingroup$ I guess there is no harm in this answer but it really doesn't answer the question at all. Maybe that there was a substance that could theoretically have a dodecahedral quasicrystal shape is a bit of use. The information about dodecahedrane is of no use. It's just a single molecule that has the dodecahedral shape. But if you had an answer on how a primitive cell of a certain substance has been shown to be a large enough part of a dodecahedral quasicrystal that it has 5 subparts that are the relative orientations of the 5 tetrahedra that make a dodecahedron, it may be useful. $\endgroup$
    – Timothy
    Dec 27 '20 at 3:58
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    $\begingroup$ @Timothy It's an excellent answer by one of the major experts in theoretical chemistry, and you're not likely to get a better answer so quickly, so I would just suggest being a bit more thankful to the user that spent the time writing an answer to your unanswered question, in the middle of his winter break period between academic terms. $\endgroup$ Dec 27 '20 at 4:03
  • $\begingroup$ @NikeDattani I'm not stressed. I guess that's just my style that feels normal to me. I guess I felt like the normal method I didn't have evidence for was give ideas and have them taken as ideas for debate rather than orders. I guess it felt normal to me to make attempts to get things done without stress about whether it ends up successful. I had trouble becoming a good Stack Exchange contributor quickly so I'm glad I didn't get stressed and let it happen as a very sluggish discovery process and got some question blocks in the slow process and didn't worry about it. $\endgroup$
    – Timothy
    Dec 27 '20 at 4:08
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    $\begingroup$ @Timothy - I realized you were asking about a solid-state quasi-crystal as you described. That's not known, but there are clearly examples (as I mentioned) of that kind of symmetry. It's certainly possible to investigate further whether such a solid-state phase could exist .. that seems more like a few-month research project than a Stack Exchange answer. $\endgroup$ Dec 27 '20 at 20:18
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    $\begingroup$ You don't have to accept my answer if you don't feel it addresses your question - no worries. Perhaps someone will come with a better answer. In the meantime, ask good questions - that's the key to good science! $\endgroup$ Dec 27 '20 at 20:19

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