[nilirl]

Even after reading the Wikipedia entry I couldn't intuit what Aperiodicity means.

Does it mean simply lack of pattern? But that doesn't seem to be the case, at least visually.

Just last month I wrote a post about how Mendeleev's real genius was in how we went looking for periodicity [0] and how that helped predict elements.

Does aperiodicity have any cool properties that help in specific domains?

[0] https://www.nair.sh/guides-and-opinions/communicating-your-e...

[shoxidizer]

> Does it mean simply lack of pattern? But that doesn't seem to be the case, at least visually.

Aperiodic tiling means that the whole thing doesn't repeat. If you overlay a copy of the tiling and move it around, you'll never find it match perfectly everywhere except for the one position you cloned it as. A grid of squares is periodic, you can translate it one unit to the side and it's the exact same, everywhere.

This is of course a different kind of periodic than is meant with The Periodic Table.

[skinfaxi]

It seems to be the same kind of periodicity unless I'm missing something.

> Periodicity, for our purposes, is a repetition of relationships in specific intervals. Mendeleev had found that the elements were periodic based on the relationship between atomic weight and valency.

(from parent link)

[shoxidizer]

Yeah, they're both about periodic repetition. But a periodic tiling repeats in its entirety, exactly, with a fixed period. The periodic table groups elements based on properties that repeat periodically, but the periods change as you move down the table. The tiling also has string of repeating patterns, like the elements, but a periodic tiling is something other than just that.

[indoordin0saur]

> but the periods change as you move down the table

Sounds like the periodic table is already aperiodic.

[xigoi]

Indeed, it’s not periodic in the strict mathematical sense.

[danbruc]

There exists no translation that leaves the tiling of the entire plane unchanged. The same local pattern [1] will of course occur repeatedly, there are only so many different patterns of any given size, but if you look at large enough scales, any translated copy will eventually diverge.

Maybe someone can make a version of this - if it not already exists - where you can move a semitransparent copy of the tiling around, maybe with a score for the alignment achieved.

[1] I am not sure if this is true, but as we have to tile an infinite plane, I could imagine that any pattern of finite but arbitrary size will occur infinitely often.

[shagie]

> Does aperiodicity have any cool properties that help in specific domains?

The proof of the aperiodic nature of the Wang tile set was showing that a Turing machine could be created out of Wang Tiles that tile the plane (periodic) if and only if the Turing machine does not halt. (I think I phrased that right)

https://en.wikipedia.org/wiki/Wang_tile

> In 1966, Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem.

...

This means that you can create tiles that solve specific computations. Andrew Glassner's notebooks has a chapter on aperiodic tiles ( https://archive.org/details/andrewglassnersn0000glas/page/18... ) Page 216 has a tile set that finds the minimum of two numbers.

There's also some "if you use Wang Tiles you can pattern textures without repeating groups" - https://www.researchgate.net/publication/2864579_Wang_Tiles_...

[mswphd]

Crystals are a form of solid that in some sense have "long range regularity". One way (the obvious way) to have this is to have a (translational) symmetry. For a while, it was thought this was the only way. Aperiodicity (in the sense of this post) was then discovered, leading to the discovery of a new way to arrange matter that maintains this "long range regularity" without having translational symmetry, namely quasi-crystals. This was the basis of the 2011 Nobel Prize in Chemistry.

For a popular exposition on quasi-crystals you can see

https://www.quantamagazine.org/quasicrystals-spill-secrets-o...

[adrian_b]

I do not know to which Wikipedia entries you have looked, but these give enough examples:

https://en.wikipedia.org/wiki/Aperiodic_tiling

https://en.wikipedia.org/wiki/Quasicrystal

https://en.wikipedia.org/wiki/Aperiodic_crystal

A couple of days ago there have been 2 HN threads about quasicrystals, one being about quasicrystals that are found in some very rare natural minerals, which form in special conditions, like meteorite impacts.

Some people before Mendeleev have thought about periodicity of the chemical properties with the atomic mass, but the genius of Mendeleev consisted mainly that he had much more trust in the idea that periodicity must exist.

So while his predecessors were discouraged by the discrepances between periodicity and the known chemical properties, Mendeleev assumed that periodicity is true and any facts that appear to contradict it must be caused either by earlier experimental mistakes that have produced wrong values for some chemical properties of the known elements or by the fact that some chemical elements have not been discovered yet, so empty spaces must be reserved for them in the periodic table.

Nonetheless, the periodic table that comes from Mendeleev has remained somewhat misleading until today, because it was based mainly on the periodicity of valence, which was indeed the most important chemical property for the chemical researchers of the 19th century, which were interested in laboratory experiments made for the discovery of new chemical substances and elements and for the investigation of their properties.

For practical applications, e.g. for the modern chemical industry or metallurgy, valence, which determines the ratios in which elements may combine to form substances, is only one of the properties of interest. The chemical behavior of elements is mainly determined by 3 characteristics, valence, i.e. the number of electrons on the outer layer, atomic size and electronegativity. All 3 properties are approximately periodic, but the quasi-periods vary slightly and a big cycle that goes between 2 noble gases is frequently segmented in 2 or more minor cycles within which properties vary monotonically, but they jump at boundaries. For example, the electronegativity grows from alkaline metals until Cu/Ag/Au, then it jumps down to Zn/Cd/Hg, then it grows again until the noble gases, after which it jumps downwards again.

The result is that for each of the 3 essential properties of a certain chemical element there may be different chemical elements in the next "period" that resemble best with it and only one of those is located in the same "group".

The classification of chemical elements in "groups" is only partially useful, because to really understand chemistry you must also understand the other kinds of similarities between elements, which group the elements in a different way than the periodic table of Mendeleev.

For instance, given the 3 elements carbon, oxygen and sulfur, it is impossible to say which pair of them contains more similar elements, so they can be grouped together. Oxygen and sulfur are in the same Mendeleev group, differing from carbon. However, carbon and sulfur have almost the same electronegativity, differing from oxygen, which causes a lot of similarity between many of their chemical compounds, e.g. between carbonates and sulfates. Moreover, carbon and oxygen have closer atomic sizes, differing from sulfur, which also explains many chemical properties, e.g. why the carbonate ion is CO3, while the sulfate ion is SO4.

A similar discussion can be done about almost any chemical elements, e.g. for some properties silicon resembles germanium and selenium resembles tellurium, because they are in the same Mendeleev groups, while for other properties silicon resembles selenium and germanium resembles tellurium.

In conclusion, the periodic table of Mendeleev provides only a fraction of what must be known about the periodicity of properties among chemical elements.

[kkylin]

In addition to the above links on quasicrystals etc, it may help to have a bit of context on periodic tilings, which have very precise mathematical properties: https://en.wikipedia.org/wiki/Lattice_(group).

The structure of the periodic table itself can also be understood, to leading order anyway, in symmetry terms. See for example https://mathstodon.xyz/@johncarlosbaez/112540366778806757 and references there.

[nilirl]

Thank you for writing this, it flowed well and clearly took time and thought to write.

I think I still haven't fully understood the significance of aperiodicity, even if I learnt some of its properties.

And thank you for elaborating on Mendeleev.

I wrote that post on Mendeleev because I'd actually come across his example in a book about library science.

I was interested in this story (a rumor) that he used physical cards to sort and reorganize the elements in front of him till he found a periodic organization.

[FarmerPotato]

I appreciate your contribution.

In high school chem/physics, we had an assignment to arrange element flashcards (no atomic number!) in some kind of periodicity, and then defend our choice. I imagine we were exploring dead ends that didn't persuade Mendeleev.

[Melatonic]

Have you heard of anyone using Machine Learning and/or some kind of AI algos to look at potential new quasicrystal structures?

[s0rce]

I always enjoy seeing the HR-TEM of the quasicrystals after learning about them in grad school.

[keeganpoppen]

> Does aperiodicity have any cool properties that help in specific domains?

well, it helps as the basis for all life on the planet as we know it (DNA, RNA, et al.). which is pithy, but i think actually has fairly deep implications (ie is not _just_ pithy).

[gilleain]

DNA is one-dimensional aperiodicity, which is ... fairly common. This sentence is one-dimensionally aperiodic!

Most biological structures are amorphous in some way, rather than strictly aperiodic. There are structures (honeycombs, for example) that are extremely regular, and especially on smaller scales (like virus capsids). Other biological structures are loosely fractal (self-similar across a narrow range) - lungs, blood systems, and so on.

I don't know of any biological structures that could accurately be described as aperiodic - like a quasicrystal - but given I'm just learning about Defense-associated reverse transcriptases, it would not surprise me if there is something out there.

[TacticalCoder]

> well, it helps as the basis for all life on the planet as we know it (DNA, RNA, et al.)

And how much of that would we have understood if not for Mendeleev allowing us to discover new elements thanks to the periodicity of its table?

Elements in Mendeleev's table are at a lower level than DNA / RNA.