The Johnson Solids (2019) | Hacker News Reader

The Johnson Solids (2019)

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55 pointstwoodfin22 comments

There are applications of the Johnson solids to characterizing patterns that occur in nature at the nanoscale. You wouldn't be able to see them with the naked eye, but they are very clearly there!

For example, we observed nested J27 "shells" in the structure of the Au_146(p-MBA)_57 nanoparticle [1]. In particular, take a look at the attached .mpg video to get a clear picture of the "shells" inside this particular nanoparticle [2]. We observed three nested (two complete, one outer incomplete, corrupted by the surface protectant p-MBA) J-27 shells.

Nanoparticles like this exhibit interesting surface plasmonic effects. For smaller particles, a long standing theory was that they behave as "super-atoms", with gold atoms taking the place of neutrons and protons, and metals in the protectant shell taking the place of electrons.

While I don't subscribe to that theory, this particle in particular occupies a sort of partial transition point between the regime in which it was previously hypothesized and the regime of bulk gold where it clearly does not hold.

Disclaimer: I am a first author on this paper and produced this visualization, as well as many of the figures shown in the paper. I think the video in particular is quite neat :).

[1] https://pubs.acs.org/doi/abs/10.1021/acs.jpclett.7b02621

[2] https://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.7b02621/s...

Addendum: the positions of the atoms shown in that video and other figures are NOT smoothed or snapped to a grid in any way. They are directly drawn from the experimental results! They really are just that clean.

This also extends to the gorgeous rotational symmetry in the "imperfections" of the outer incomplete shell, which is perpendicular to the reflective symmetry of the inner "perfect" J-27 shells.

It's very neat to zoom so far in on reality and see such a well-ordered structure.

The wikipedia article (https://en.m.wikipedia.org/wiki/Johnson_solid) is arguably better, giving a concise definition of "Johnson Solid" and better visualizations.

There is an really impressive recent site here https://polyhedra.tessera.li/ with a 3d viewer, it even has interactive animation where you can see how they transform into each other, for example an icosahedron gets changed into a diminished icosahedron and back.

This is such a cool site.

That site is amazing. I love the animated "operations" transitions.

It's not on the same level as the above, but Wolfram Alpha has a list of the Johnson solids which shows each one unfolded into its 2D net[1].

Wolfram Alpha can also generate a 3D model[2] or list of vertices[3] for any Johnson solid using the `PolyhedronData[{"Johnson", n}]` dataset.

[1]: https://mathworld.wolfram.com/JohnsonSolid.html

[2]: https://www.wolframalpha.com/input?i=PolyhedronData%5B%7B%22...

[3]: https://www.wolframalpha.com/input?i=PolyhedronData%5B%7B%22...

This site gives coordinates of the vertices, which I don’t think Wikipedia does. That’s nice if you’re looking to render it in code.

They have some polygonal magnetic tiles that you can assemble into geometric solids at the Omaha Luminarium (like the Bay Area's Exploratorium). They're fun to play with — perhaps you could make all the Johnson solids with them.

I'm having a hard time though finding a set you can purchase. Something called MAGFORMERS was the closest I could find on Amazon. Most similar products consisted though only of squares and equilateral triangles.

Of course you could easily 3D print these — leaving a void along the center of each vertex suitable for inserting a long cylindrical magnet (which is how they generally appear to hold together). It's hard to be a the look of injection molded plastic though. :-)

My little one has “magna-tiles” that have triangles and squares. I think you can expand to pentagons with an extra set.

Yeah, no hexagons apparently. The expansion set that gives you pentagons also comes with diamonds though (three of which would make a hexagon – an improvement over using 6 triangles).

PicassoTiles appears to be another set... also not perfect.

Maybe I can have some laser-cut from acrylic — leaving a notch along each vertex where I can glue in a cylindrical magnet.

I think you can get pentagons and hexagons for the 2 major styles of magnet tile, but they’re rarer.

I am reminded of Magnetix, though they were ball and stick based, and I don't believe included "solid" polygons.

Maybe some kits beyond the ones I saw did.

For cheapo, can carve up Dollar Tree foam core boards and tape together.

In Plato’s Timaeus, atoms of the elements were proposed to be the simplest possible geometric forms — the Platonic solids. Furthermore, it was proposed that the geometry of the forms determined the material properties of the elements. For instance, cubes were earth because of how well they stack together.

However, atoms don’t have Euclidean geometry. For instance, a hydrogen atom can be described with spherical harmonics. However, it seems that the intuition seem to bear out:

1. that the elements are composed of variations in the simplest geometrical forms.

2. that the properties of the elements are derived from their geometric forms.

Curious if anyone has a good piece of evidence for or against this platonic perspective.

A related point: medieval philosophers often divided things into 'male' and 'female', or 'male', 'neuter', and 'female', but it seems like they weren't attempting to convey anything gender-based, but rather found that the most intuitive way (grammar-based?) to make bi- or tri-partite distinctions.

That made me wonder why so much of the world is easily modeled by black-white (or black-white-red) distinctions — I eventually came to the conclusion that if 'n' is the real number of species within a given genus, after one has taken even subtle differences into account, well: for any given finite horizon there are many more n's that are divisible by 2 than by 3, and by either of those than larger primes...

(The chinese philosophers, who loved to stuff things into 5 categories, made things difficult for themselves by this model. Then again, other people loved to make 7-way categorisations, so maybe they all just thought binary splits were too easy to show their erudition?)

STL models of the Johnson solids ready for 3D printing. (click on an stl file and github will even render the solid for you) https://github.com/gecrooks/shapes2stl/tree/main/shapes

Maybe it’s because I just woke up, but why does a hexagonal pyramid not satisfy the definition? (Or any n-pyramid?)

Are the triangles in a hexagonal pyramid regular?

Ah…

The regular hexagonal pyramid is a degenerate example, at least.

So many new dice options for your TTRPGs.

You could do nonuniform probabilities.

Of course the same could be done using multiple dice (ex 3d6) but yes, options.

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