[HN Gopher] The Johnson Solids (2019)
___________________________________________________________________
The Johnson Solids (2019)
Author : twoodfin
Score : 50 points
Date : 2023-12-23 12:48 UTC (10 hours ago)
(HTM) web link (www.qfbox.info)
(TXT) w3m dump (www.qfbox.info)
| noelwelsh wrote:
| So many new dice options for your TTRPGs.
| swayvil wrote:
| You could do nonuniform probabilities.
|
| Of course the same could be done using multiple dice (ex 3d6)
| but yes, options.
| bediger4000 wrote:
| The wikipedia article
| (https://en.m.wikipedia.org/wiki/Johnson_solid) is arguably
| better, giving a concise definition of "Johnson Solid" and better
| visualizations.
| doubloon wrote:
| 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.
| magnatiler wrote:
| This is such a cool site.
| olooney wrote:
| 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%7
| B%22...
|
| [3]: https://www.wolframalpha.com/input?i=PolyhedronData%5B%7
| B%22...
| mcphage wrote:
| 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.
| JKCalhoun wrote:
| 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. :-)
| magnatiler wrote:
| My little one has "magna-tiles" that have triangles and
| squares. I think you can expand to pentagons with an extra set.
| JKCalhoun wrote:
| 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.
| mcphage wrote:
| I think you can get pentagons and hexagons for the 2 major
| styles of magnet tile, but they're rarer.
| dylukes wrote:
| 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.
| quailfarmer wrote:
| Maybe it's because I just woke up, but why does a hexagonal
| pyramid not satisfy the definition? (Or any n-pyramid?)
| mcphage wrote:
| Are the triangles in a hexagonal pyramid regular?
| quailfarmer wrote:
| Ah...
| dr_dshiv wrote:
| 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.
| 082349872349872 wrote:
| 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?)
| yportne wrote:
| 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
| dylukes wrote:
| 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...
| dylukes wrote:
| 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.
___________________________________________________________________
(page generated 2023-12-23 23:02 UTC)