The Aperiodic Table
blog.jgc.org
blog.jgc.org
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...
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).
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.
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.
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.
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.
> 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)
Sounds like the periodic table is already aperiodic.
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.
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.
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.
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.
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_...
For a popular exposition on quasi-crystals you can see
https://www.quantamagazine.org/quasicrystals-spill-secrets-o...
Also: when did everyone stop calling this quasiperiodic and start calling it aperiodic? I feel like the almost-but-not-quite translational symmetry was a useful distinction. Has it fallen out of favor?
I don't think mathematicians are going to change their term to "aperiodic". Almost everything is aperiodic. Quasiperiodic indicates something much more interesting is happening.
If I didn’t know that the author used AI, then I would have liked this way more. But that is because I would assume the author did this on his own and that would feel like a cool quirky thing to do. I just don’t care for a cool quirky thing if an AI made it.
While I understand the knee-jerk reaction, especially to writing, people make about AI, it's starting to have a counter-effect on me.
A brush can destroy (sweep) or create (paint) but it's still a tool.
For example, if I read a book which I thought was written by a human and loved it, why should my opinion change if I learned after the fact that it was written by AI and not a human? I can't un-laugh those laughs, and un-enjoy the enjoyment I received from it, you know what I mean?
In a very short period of time I got to try many different ideas and create the final site. The ideas were all mine, the implementation was Claude's. I view this as wonderful: I had an idea and was able to iterate an implementation very rapidly. I can't turn my back on a tool that helps me create more.
PS If it's any consolation, my blog posts are all hand written. I don't use AI for any of the prose; I do use a spell checker.
You made a cool thing, I like it. I don't care how you made it, the more cool things we have, the better for everyone. If you don't think this thing is cool, downvote and move on.
The parent post asked a more-or-less specific question about how this works and offered their working hypothesis (that it works by taking the rectangular periodic table and mapping each element to its nearest Penrose tile). I would also be curious about this, because e.g. it seems non-obvious to me that the resulting table would have no holes inside. Does there have to be any extra step to ensure that?
But your reply is not informative at all. If you wrote this yourself from scratch, you would be able to answer this kind of question (and report on anything interesting that came up while creating the program), but this way the post just hangs here to be forgotten, and I haven't learned a new thing about Penrose tilings. :(
So you've got 118 elements with ideal locations and a whole bunch of rhombi with known locations. The Euclidean distance (without bothering with the sqrt) is used as the "cost" function for the Hungarian algorithm (https://en.wikipedia.org/wiki/Hungarian_algorithm): the idea is to minimize the cost (distance from ideal) for each element.
You will sometimes see some gaps in the periodic table as this layout isn't "perfect" (in the sense of without isolated rhombi inside the table) and I did work on having a post-layout pass where these islands were identified and filled by moving adjacent rhombus assignments, but I wasn't happy with how this tended to break the layout of the periodic table (I was going for some visual "looks similar to the classic U shaped periodic table) and so decided to accept them.
I still find it curious how few holes there are (took a while to find one!), and finally figured out why: imagine a large square grid. It would probably have a different density than the rhombus grid, and it seems nontrivial to match it up. It seems that in the code this is done by each rhombus having edge length 50 while the periodic table elements are 38 x 42 pixels in size.
This, if I understand it correctly, means that this tiling is not just aperiodic but (in this regard) also anisotropic -- it's denser in one direction than another. And thus I have learned a new thing about Penrose tilings. :)
The other thing was the title. The code originally could have had two letters (e.g. P and E in APERIODIC) joined at a vertex and it looked odd (it looked like the word was broken) so there's specific code to make sure that doesn't happen.
But the simplicity of the shapes in the Penrose tiling has its own charm. And the way it rearranges itself when scrolling around the tiling is awesome!