2781: The Six Platonic Solids

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The Six Platonic Solids

Title text: Plato made the solids, and five were gifted to the mathematicians. But in secret Plato forged a sixth solid to rule over all the others.

Explanation

This comic imagines an alternate reality where mathematicians discover a new Platonic solid beyond the five proven to exist in three-dimensional space. In three dimensions there are 9 regular polyhedra (though depending on who you ask, there could be as many as 48 of them). A regular polyhedron is a solid figure with all faces being congruent regular polygons with the same number of alike faces arranged around each vertex. While the most familiar, Platonic solids, are referenced in the comic, there are also 4 Kepler–Poinsot polyhedra. In four dimensions, there are six regular polytopes, five of which are analogous to the five Platonic solids in 3-D space, and a sixth which is analogous to the rhombic dodecahedron.

In the comic, Randall reveals the discovery of a new Platonic solid, called the "jorb", which appears to be a roughly conical shape with a round base, a triangular tip, and a rectangular extension at the bottom. One of its surfaces also seems to have parallel grooves or ribs, which may indicate curvature. The jorb does not meet the criteria for a Platonic solid, in that the faces must all be regular polygons of the same shape, and each vertex must join the same number of edges. Also, Platonic solids were discovered by Plato, who they are named after. A new-found regular polyhedra would not be called a Platonic Solid because it was not discovered by him^{[citation needed]}. This could be a reference to the fact that many regular polyhedra have only been discovered recently, most of which do not fit the naive understanding of a regular polyhedron, having irregular concave external faces, or being infinite or self-intersecting.

The title text references the Lord of The Rings, in which the "One Ring to Rule Them All" was forged in secret by Sauron to control the wearers of three magic rings given earlier to elves, seven given to dwarves, and nine given to humans, primarily by allowing him to know their location, letting him visualize the wearers and their surroundings, and by allowing him to impose his will on the wearers (which for arcane reasons only worked reliably on the rings given to humans, worn by the nine Nazgûl.[1]) The joke is that Plato forged a sixth Platonic solid, the jorb, to rule the five he "gifted" to mathematicians, similarly to how Sauron tried to rule the other magic rings' wearers in Middle-earth with his One Ring.

Transcript

[Six geometrical shapes are shown. All have gray surface areas with different shading to reflect their orientation. There is one shape in the middle with the other five arranged around it roughly in a pentagon. With two at the top, two just below the central and one directly below the central shape. Each shape has a label. The five above the bottom one are names after the platonic solids, and are drawn to look like them. The last one at the bottom has a roughly conical shape with a round base, a triangular tip, and a rectangular extension at the bottom. It surface also seems to have parallel grooves or ribs. Here the labels in reading order with the four rows mentioned above used.]

Cube

Dodecahedron

Icosahedron

Octahedron

Tetrahedron

Jorb

[Caption below the comic:]

Mathematicians long believed there were only five platonic solids, all regular polyhedra, until this year's discovery of the Jorb.

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Discussion

Does he know about Homestar Runner? 172.70.131.137 06:02, 27 May 2023 (UTC)

Yes. http://www.hrwiki.org/wiki/Webcomic_Sightings#xkcd Trogdor147 (talk) 01:07, 29 May 2023 (UTC)

Why Jorb? Only thing I can find is Jorb on wikitionary just meaning spelling of bad pronunciation of Job. And yes the episode of Homestar Runner A Jorb Well Done comes up. Also this episode that is the top meaning of jorb on Urban dictionary. Would really like there to a better idea than that Plato did a great Jorb making a sixth solid to rule the mathematicians. --Kynde (talk) 07:18, 27 May 2023 (UTC)

I agree. We should ask for our money back. -- Dtgriscom (talk) 17:19, 27 May 2023 (UTC)

What if there're much more of them, like a Ď̩̰odec̭ähedron, but our minds can't properly comprehend their shape?

There are a bunch of other regular polyhedra besides the Platonic solids. Most notable are the triangular, square, and hexagonal tilings (which are planar and infinite) and the four Kepler-Poinsot polyedra (which are nonconvex). And there are dozens more if you don't require faces to be planar. 172.70.178.234 09:44, 27 May 2023 (UTC)

See https://youtu.be/_hjRvZYkAgA for an overview of every regular polyhedron in Euclidean 3-space. 162.158.146.40 09:59, 27 May 2023 (UTC)

I had never seen this channel before, and I'd very much like to thank you for introducing it to me. 162.158.167.8 21:39, 29 May 2023 (UTC)

Some of the proofs of the theorem that there are exactly five platonic solids do not require our minds to "comprehend their shape", because they only rely on their algebrical properties. In fact, the Group theory proof works in any dimension (≥3), despite our minds being very bad at picturing what stuff looks like in higher dimensions. In fact, it's a bit of the opposite: lower dimensions (2 and 3) are "special cases", because all other dimensions have exactly 6 such platonic solids. Jthulhu (talk) 15:41, 27 May 2023 (UTC)

I think this is a reference to how the Utah Teapot is nicknamed “the sixth Platonic solid” due to its presence beside real Platonic solids in demonstrations of 33D rendering. 172.68.118.133 08:52, 27 May 2023 (UTC)

...yeah, but you need to render that 33D shape on a proper 32D monitor, ideally, because even on a 31D monitor the two different forced perspectives/projections you need to collapse the extra dimensions down tend to look confusing. 172.70.162.229 10:46, 27 May 2023 (UTC) *insert winky-face as necessary*

Thanks, that predates my claim in https://arxiv.org/abs/2103.00090, so I won’t claim priority:

‘By this conception, the claim that there is a Russell Set is simply wishful thinking, analogous to wanting the phrase “sixth Platonic solid” to have a referent, while not actually intuiting which shape this might be.’

— FlashSheridan (talk) 12:33, 31 May 2023 (UTC)

Should we think about Jorb, perhaps, as "J orb," which might lead us to think about (i,j) coordinates, i.e. notational systems where j is the square root of minus 1? (blah blah engineering vs. mathematics, what does i mean, &c., &c., &c.) Maybe not! JohnHawkinson (talk) 10:41, 27 May 2023 (UTC)

When I naively imagine further platonic solids, my approach to cheating is usually to imagine arbitrarily non-euclidean space. It’s somewhat pleasant that the jorb looks a little like that —- it’s only platonic if the coordinate system it’s in bends and stretches to make it so. I imagine the name and depiction are simply to be intentionslkynvague or confusing about what it actually is. It’s a lot harder to do real innovation nowadays and tasks like inventing a jorb can readily crop up to nerds (the nerd snipe kind) who are looking for sufficiently hard and harmless puzzles. A sixth platonic solid seems to mostly happen outside the norms of what people usually consider in a pretty significant way. 162.158.155.99 02:30, 28 May 2023 (UTC)

I asked GPT-4, "Why could Sauron control the Nazgûl when not in possession of the One Ring?" The response:

Sauron was able to control the Nazgûl even when he didn't possess the One Ring because of the nature of the rings they received and the relationship these rings had with the One Ring.

Sauron, disguised as Annatar, the "Lord of Gifts," helped the Elven smiths of Eregion, led by Celebrimbor, forge the Rings of Power. Unknown to the Elves, Sauron had instructed them to forge the rings in such a way that they could be controlled by one master ring - the One Ring. He secretly forged the One Ring in the fires of Mount Doom, uttering the famous verse: "One ring to rule them all, one ring to find them, one ring to bring them all and in the darkness bind them."

When he gave the nine rings to Men, these rings corrupted them over time and turned them into the Nazgûl, also known as the Ringwraiths. Because these nine rings were still under the influence of the One Ring, Sauron was able to control the Nazgûl even when he did not possess the One Ring himself. The One Ring's power was inherent and didn't need physical proximity to exert control. The Nazgûl were bound to the will of Sauron as their wills were dominated by their rings, which were in turn dominated by the One Ring.

The link and dominance were so strong that even when the One Ring was lost, the Nazgûl remained under Sauron's control. It's also worth noting that Sauron held the Nine Rings the Nazgûl originally received, which could also contribute to his ability to control them.

It's a complex network of influence and control established by Sauron, making him a powerful Dark Lord even in the absence of his master ring.

Well. I'm glad someone something understands that. Lol. 172.69.22.145 06:11, 28 May 2023 (UTC)

Looks suspiciously like a bicycle saddle to me.  And as anyone who's parked their backside on one for any distance or time, it is DEFINITELY a solid. RAGBRAIvet (talk) 07:25, 28 May 2023 (UTC)

(Well, there's all kinds of saddles, and the golden era of leather saddles were actually not that bad (that link states a 10 year guarantee for the current output, but I personally know of a couple used regularly for 60+years and were still doing well), though with the advent of more synthesised materials they've had to go creative with the texture/covering to give it back the 'give'. Which no doubt makes more an 'impression' on the more modern saddle purchaser, in more ways than one.)

Reminds me of one of the more abstract 'elemental' Henry Moore pieces, or possibly one of his contemporaries/'inspired-by's. And you could certainly find painters and other art-form creators who have gone into 'abstract blob' shapes, either to represent something real in a novel way or to deliberately represent nothing real at all, and I suspect that's Randall's aim... (Like "xkcd" made to not look enough like anything else, or so intended). It certainly shows no sign of any symmetry at all. 172.70.86.32 12:29, 28 May 2023 (UTC)

Based on the shape, ribbing, and a different definition of "platonic" I think this could be referring to the Roman artefact that was recently re-classified as a dildo. Platonic solid? 172.70.111.111 16:06, 28 May 2023 (UTC)

Not gonna lie, this feels like a Cow Tools comic to me. Sometimes mathematicians discover new things, wouldn't it be weird if they discovered something impossible? End of joke. Everything else regarding the shape and name is an inkblot test (until the title text which is actually a LOTR reference of course). 172.70.82.59 17:04, 28 May 2023 (UTC)

Is it worth mentioning that the dodecahedron and isocahedron have their names switched? C.h.ninnymuggins (talk) 21:00, 28 May 2023 (UTC)

It would be if they were (a common error), but... they aren't. (See here.) 172.69.79.196 21:33, 28 May 2023 (UTC)

Ok whoever added that jan misali link that's super cool of you nice 172.71.82.147 03:50, 29 May 2023 (UTC)Bumpf

I feel that there's a joke missing here, especially with the LOTR connection made in the alt-text. After all, Plato might have gifted the solids to the mathematicians, but thanks to Gary Gygax, it was the gamers who found a use for them.... 172.70.86.7 06:37, 29 May 2023 (UTC)

Plato did a a great jorb ruling the mathematicians by forging the impossible... plato got a gold star at his job review... was that really it?--4til7 (talk) 20:33, 29 May 2023 (UTC)

I kind of feel sorry for the platonic solids. Condemned to the friend zone for eternity; or at least until the end of this universe. These Are Not The Comments You Are Looking For (talk) 22:54, 29 May 2023 (UTC)

You've all been nerd sniped. 172.70.42.43

Also the Gömböc comes to mind from the name, the shape and the novelty of Jorb. 162.158.182.115 18:51, 2 June 2023 (UTC)

Added link to "Gömböc", as that is immediately what I thought of when seeing this comic. Randall's Jorb looks designed to have Gomboc-like properties: if starting on the long thin facet seen on the top edge, it would be gently pulled down by the large round egg-like "end" and then presumably roll onto one side or the other i.e. rotating around its long axis until reaching the position shown. If the Jorb were indeed a mono-monostatic shape, then like the known Gömböc design it would be a "polyhedron" with one (albeit curved) "facet", and like the Platonic solids it can rest on any one of its facets, trivially true as there is only one facet. Mrob27 (talk) 05:50, 12 June 2023 (UTC)

So, I think this explanation is under-focused on the actual joke. The core humor here is the idea that "Platonic Solid" means "Solids Plato made" and not regular polyhedra at all. "Mathmetitions long believed there were five platonic solids, all regular polyhedra" So the jorb is explicitly a platonic solid that is not a regular polyhedra. Our explanation should point out then, things like the fact that the platonic solids predate Plato.

There is no doubt that the cube is the most common, but today a strange force made a lot of tetrahedron objects appear. This happens every year. ConlangGuide (talk) 06:23, 12 July 2023 (UTC); 06:44, 11 July 2023 (UTC); 09:07, 9 July 2023 (UTC); 06:07, 8 July 2023 (UTC); 06:17, 7 July 2023 (UTC)

They get either eaten or thrown into the water. ClassicalGames (talk) 04:53, 12 July 2023 (UTC)

I'd like the answer to two questions:

1. What is CG even talking about?
2. What is the person who keeps removing CG's comment even complaining about? (Something about Cultural Misappropriation, but which culture(s)?)

...maybe the answer to one will answer the other, but I've a feeling they're entirely unconnected. Perhaps one (or both?) just trolling. Yet, unless 7/7 is International Dungeons&Dragons™ Day, or something, neither even close to what I'm imagining it's about. 172.71.242.96 10:50, 9 July 2023 (UTC)

That post first appeared in June. 172.71.154.17 23:45, 9 July 2023 (UTC)

Stop your games. And they clearly are games, as we already know about the historic link between the .XXX and CG-styled accounts, which you (singular/plural, I don't care) have now used to 'edit war between yoursel(f/ves)'. I don't generally care if the comment is there (whatever it even refers to, but with no obvious 'illegality') or not, but useless reverts/de-reverts/re-reveets/etc should not be tolerated. 172.70.91.53 08:06, 11 July 2023 (UTC) Ha, CG started to fight against himself! Hee-Ha! Hung-Huh-Ha-Hey! 162.158.166.205 23:01, 11 July 2023 (UTC)

I wonder if this is relevant: http://www.hrwiki.org/wiki/A_Jorb_Well_Done 162.158.62.141 (talk) 3:47, 2 July 2023 (please sign your comments with ~~~~)

Looks like an ocarina to me 172.70.86.186 19:18, 20 July 2023 (UTC)

It does kinda feel like you all are oblivious to the fact that lots of people won't have a blind clue what any of this actually means. Can someone please translate the explanation from mathematical gibberish to language a lay-imbecile such as myself could understand? What actually *is* a platonic solid might be somewhere to start, in words like 'edges' and 'faces', instead of this 'convexual polyhydra' nonsense... Thanks. 141.101.99.145 09:28, 23 January 2024 (UTC)

It's all there in the links. And nobody says "convexual polyhydra" on the page, except you.

But... Take a regular polygon (...as in a "flat shape that has equal angles/side-lengths and no internal angles"...) join it up with other (identical) polygons to create a regular 3D shape. The Platonic solids are the full set of these that aren't: a) flat, an infinite plane, b) angled both inwards and outwards, c) self-crossing its surfaces, d) leaves gaps, e) have different-looking corners, ....etc.

It can be shown that (given the above restrictions): a square can be used only to make a cube; a triangle can make a tetrahedron (triangular-based pyramid), an octahedron (two square-based pyramids stuck base-to-base) or an icosahedron (a twenty-sided solid); a pentagon can make a dodecahedron (a twelve-sided solid); no other combinations are possible. Hexagons will tile infinitly (or create a flat 'double-sided' free-floating hexagonal plate) heptagons and more won't even tile. There are thus five of these 'basic' regular solids, described by/ascribed to Plato, as seen (discounting the jorb).

Further 'regular' shapes exist if you undo some of the restrictions (infinite sheets, alternating in-and-out angles, using regular self-crossing shapes and/or allowing the shapes to cross each other whilst forming the solid, etc... e.g. for which someone else recently added links to the additional set of 'stellated' regular polyhedra). Or abandon Euclidean space/expand beyond three dimensions (both of which make Platonic solids no longer fully valid, at the same time). And if you consider "regular convex polyhedra" beyond you, at this point where you've already seen links to a potential wikiwalk to fill in gaps in your knowledge about what these terms mean, then I'm not sure what more I can say. Either here or in the main Explanation. Not without hand-holding you through points that I learnt in maths lessons when I was no more than 11 (and probably already knew, at least in parts). Can't account for current practice, or the localised curriculum that you were taught, of course, but then a 'refresher' or post-education remedial 'filler' lesson isn't going to be possible just through a paragraph or two of blind monologue. Check the links, and ask more specific questions than just not understanding any of it. 172.71.178.66 10:24, 23 January 2024 (UTC)