The Five Compound Platonic Solids

Do check out my video with Adam Savage over on Tested: • Can Adam Savage and Ma...
Yes, I'm back in NYC for An Evening of Unnecessary Detail on Sunday April 14. www.eventbrite...
Pre-order my new book! mathsgear.co.u...
The excellent graphics of spinning dodecahedrons were made by Ben Sparks. You can see the GeoGebra file here: www.geogebra.o...
Some more details about space diagonals on Math Stack Exchange.
math.stackexch...
Huge thanks to my Patreon supporters. They keep my polyhedra intersecting. / standupmaths
CORRECTIONS
None yet, let me know if you spot anything!
Filming and editing by Alex Genn-Bash
Written and performed by Matt Parker
Laser cutting by Lisa Mather
Extra graphics by Ben Sparks
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
Training on using a bandsaw by Adam Savage
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinran...
UK book: mathsgear.co.u...

[@UnicornPowerzez]

the autotuned "but i couldnt be bothered" cracked me up, this is why matt is the best

[@qwfp]

4:29 ϕ×ϕ this is my new favourite emoticon!

[@JavSusLar]

Fun fact: by definition, φxφ=Φ+1

[@mySeaPrince_]

🤯

[@5ucur]

фхф

[@felicityc]

ϕwϕ classic cat eyes

[@onebronx]

¯\_(Φ×Φ)_/¯ - PARKER DIAGONAL IN SPACE!

[@gallium-gonzollium]

The “Diagonals in SPACE” interjection might be the best highlight for this channel in a while. And I’m glad to be a part of it when it becomes a happy meme. :)

[@falfires]

Matt still trying to make us forget about the Parker Square. But we will never forget. :D

[@richbuilds_com]

I was expecting a more Piiiiigs iiiiiiin Spaaaaaace vibe.

[@MasterHigure]

Clearly Science Asylum inspired, if you ask me. Not that I'm complaining.

[@orbitalshawn0625]

I love dodecahedrons but our relationship will always be platonic

[@Elesario]

Groan... but also cute.

[@frba9053]

Perfect pun

[@idontwantahandlethough]

nice

[@greanbeen2816]

What a shame, I thought things were just golden.

[@abstractapproach634]

That would break plato's heart, he thought the dodecahedron would always be your everything

[@LSA30]

D I A G O N A L S I N S P A C E

[@aiocafea]

sadly youtube bitrate compression messes with my full enjoyment of D I A G O N A L S I N S P A C E

[@Zi7ar21]

🛸👾

[@Yezpahr]

It is just a blatant theft from Science Asylum, but I'm not even mad. Well done by Parker-man.

[@victormunroe2418]

@@Yezpahr nah, clearly it's blatant theft from The Muppets

[@FLPhotoCatcher]

I'm sure Adam Savage is a man. I'm tired of the misuse of my language by an elite few, who are trying to spread the misuse.

[@zeotex2851]

The quiet echoey "space" at 3:45 killed me 😭😭😭💝💝💝

[@Elesario]

My condolences to your family.

[@zeotex2851]

@@Elesario than you, its so sick how you still have access to KZbin in the afterlife, didn't expect that 💝💝💝

[@aMessvv]

Was about to comment this hahaha great attention to detail

[@kbsanders]

1:31 Alex, do you want to give me a hand with this? Alex: Sure Caption: No

[@thomaskaldahl196]

made me think I was insane since I had to scroll so far to find this 😭

[@2ndfloorsongs]

Shamelessly stolen from Jean-Luc Godard.

[@robinsparrow1618]

@@2ndfloorsongs who?

[@2ndfloorsongs]

@@robinsparrow1618 Jean-Luc Godard was a famous French filmmaker. One of the many things he was noted for was having English subtitles that were frequently different from the spoken French soundtrack of his films. These were not slight differences in the translation, they contained different storylines, conversations, and descriptions of what was happening. They were frequently written by literary authors he'd invited and they were told just to view the movie and write their own script that went along with the visual film and not to worry about what the original French film was about. He was a legendary innovator and invented the "jump cut" film transition among many other things. I didn't mean to imply this was actually stolen, this was meant as a humorous joke.

[@robinsparrow1618]

@@2ndfloorsongs oh ok, this is actually really interesting and cool to know about. and it's a good joke with this context, thank you

[@HunterJE]

Great job on emulating the old educational film aesthetic for those insert animations, really sent me back...

[@gormster]

I think a reference to Look Around You

[@andreasbaus1554]

It reminded me of the animated sequences from the classic BBC Hitchhiker's Guide to the Galaxy TV series

[@nxpnsv]

Excellent. I especially liked the D I A G O N A L S I N S P A C E.

[@imaginaryangle]

If you want Steve's subscribers, you need to fill that thing with water. You were so close! 😄 I really dig how your personality and style come through even in the bits other people help you with these days. Been a fan of yours for many years, you always bring me smiles, quite a few belly laughs and a ton of inspiration

[@dummyaccount1706]

I see that VFX department got a raise recently

[@tsawy6]

Timing department getting their budgets slashed

[@stevewithaq]

4:42: maybe tropic would be a better word than equator, as there are two of them parallel and equidistant from the central plane.

[@jeffclarke3191]

This was so much fun to watch and in my opinion one of Matt’s best in terms of pure enjoyment and entertainment. Matt’s enthusiasm is totally infectious and a delight to watch. The brilliant choice of music only added a new dimension (!) and I cannot praise this video enough!

[@agrajyadav2951]

agreed

[@GoranNewsum]

Ben: Hey Matt! I've made a spinning dodecahedron in Geogebra! Matt (after this video): I don't need you anymore! I can make my own spinning polyhedra!

[@terdragontra8900]

The cube dodecahedron relationship is like, my favorite thing about 3d geometry, its so beautiful

[@needamuffin]

Mine is the three orthogonal golden rectangles forming the verticies of the icosahedron.

[@terdragontra8900]

@@needamuffin oh yes, in fact, that is also a result of same connection between the symmetry groups. (the icosahedron is duel to the dodecahedron, and three orthogonal planes have an associated cube)

[@estherstreet4582]

Every d12 I own (which is 2, I'm not a weird dice hoarder) has the cube shape drawn on in sharpie, it's so satisfying to look at. I also like how the pieces you'd have to "cut off" to make the dodecahedron into a cube are shaped like little rooftops.

[@HunterJE]

The smaller solids left behind by the shapes discussed are super satisfying in their proportions too, both the sort of flattened, obliquely truncated triangular prism you get from cutting along the square/cube and the frustrum of a pentagonal pyramid cut off by the near-equatorial pentagon...

[@jace.miller]

I like several of the integrated shapes. Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: kzbin.info/www/bejne/aKarl3mmZr12hsU

[@redoxxed]

I absolutely adore the visual representation of the see through dodecahedron with the tape to show the square, pentagons etc! it's just such a satisfying visual proof of the lengths of the space diagonals

[@dysphoricpeach]

13:34 the convex hull of the 5 octahedron compound is the icosidodecahedron. I know this video is about regular dodecahedrons, but I was a little sad when you brushed it off. It’s my favorite compound, my favorite stellation, and my favorite faceting. It also looks a whole lot like my one of my favorite polyhedra, the disdyakis triacontahedron!

[@galoomba5559]

The icosidodecahedron _is_ its convex hull. I don't know what Matt was talking about, maybe he meant that the convex hull is not regular.

[@columbus8myhw]

@@galoomba5559It really sounds like he accidentally skipped a word.

[@SineEyed]

It goes even deeper than that. As you may know, the octahedron is dual to the cube. As you mentioned, the convex hull described by the compound-5 octahedron is an icosadodecahedron. Well, the _interior space_ described by the intersection of the cubes comprising a compound-5 cube, is a rhombic tricontahedron, which is dual to the icosadodecahedron. Pretty neat--as above, so below..

[@dysphoricpeach]

@ VERY cool, thanks for sharing!!!

[@bizm]

Matt, you are honestly a master educator. I'm in my thirties and failed nearly every math class I ever took (and whatever I did manager to learn, I promptly forgot when I graduated high school). Every time I watch one of your videos I learn something and I'm able to truly understand and retain concepts that boggled my mind before.

[@PeterFreese]

I was not prepared for the joke at the end. Well done.

[@NoNeedForRandomNumbers]

Oh god the SFX budget went sky high for this video!

[@aikumaDK]

One might even say it is IN SPACE

[@taureon_]

12:00 a good excuse for drawing 12 pentagrams on a dodecahedron

[@rsyvbh]

Matt is summoning something in the exact center of the dodecahedron so that he can trap it

[@Bluesine_R]

Fun fact: Both the small stellated dodecahedron and the great stellated dodecahedron can be thought of as 3D versions of a pentagram. They are both very cool shapes.

[@nicholasvinen]

All hail Satan^12.

[@frojojo5717]

@@Bluesine_Rwell, duh! How else would you trap a demon in the centre?

[@blue2003fordwindstar]

the editing on this is impeccable

[@DaxSkrai]

Everyone taking about "diagonals in space" but 11:41 is the best voice sample for an EDM song.

[@wyattstevens8574]

And what about the "but I couldn't be bothered" from 7:47?

[@asheep7797]

@@wyattstevens8574they couldn't be bothered to mention it

[@ps.2]

Hot take: _Howard Carter's entire soundtrack_ for Matt's entire channel is, like, the only _good_ EDM I've ever heard.

[@walderlopes3372]

Oh, yeah! I have Steve's last video on the "watch later" list but I always forget that list. Thanks for reminding me, Matt!

[@arxaaron]

When I started learning 3D modeling and animation on the Amiga circa 1988, one of the bigger challenges I set for myself was modeling and animating regular pentagonal dodecahedron with a raised star on each face (similar to the Chrysler logo) -- thus the dodeca-deathstar was born. A couple years later, working in high end video post production, I used the mathematical precision of the amazing Ampex digital optics device with a precise pentagon matte to layer a spinning dodecahedron with different video on each face -- calculating exact angles and depth offsets with an HP-15c calculator was a wonderful challenge that grew my maths skills considerably. Sorry Matt, but the platonic dodecahedron is, and always will be, the BEST dodecahedron.

[@Zejgar]

The dodecahedron is slowly de-throning the icosahedron from being my favorite platonic solid, thanks to crazy fun stuff like this.

[@Zenzicubic]

I've always loved the regular compounds and their beautiful symmetry. When I built my first raytracer and figured out how to raytrace cylinders, the compound of 5 tetrahedra (which is my favorite) was one of the first things I made a render of. The regular compounds were the first things I printed when I first got my hands on a 3D printer. Great video as always!

[@MrDivinity22]

Once again, you're knocking it out of the Parker with these videos!

[@gnothisauton2116]

There is something SO satisfying about those taped models. Thank you.

[@andynicholson7944]

8:32 it tickles me no end to learn that Matt is a Look Around You fan

[@Nebula_ya]

7:43 It's the "Parker Fluorescent Embedded Cube", he's done it again!

[@qwertydragon8385]

Matt thanks for running the only math channel I've found that will always explain things in a way that makes sense and makes me laugh every time! I've been watching your videos for a long time and you've only gotten better with time!

[@andyb9124]

That's a lovely, easy to visualize, and excellent way to explain these conceps. Absoulytely a great example of how to teach a concept really well. Good job, Matt.

[@robinsparrow1618]

8:00 the rotation due to parallax and the actual rotation cancel out briefly. very cool to see

[@agrajyadav2951]

your videos are capable of pulling one out of depression and make them fall deeper in love with mathematics. Thanks a lot for your work, sir.

[@olgastec-mitura3890]

I love the over-the-top editing style.

[@Reprint001]

Just goes to show how you can't please everyone. I hate it.

[@Audey]

I almost audibly gasped when you taped that square on. This was a really cool way of showing everything, better even than a 3d animation or something I think.

[@superj1e2z6]

I couldn't be bothered marking the insides translates to "I gave it a go". Totally a parker cube.

[@davidioanhedges]

"Lots of ridiculous maths things" .... is possibly the best description of this channel I have heard ....

[@Artaxo]

Are you THE Matt Parker from the Parker Square? What an honor!

[@Schambes]

I love your visualization, it makes the whole thing insanely well understandable for me

[@THESP-rz3hg]

I aspire to enjoy my work as much as Matt

[@smanni01]

A masterpiece of maths and editing

[@ironpro7217]

8:24 matt's mental maths is on point

[@TheeAncientUrchin]

Loved the book! I love how you were able to invent *time traveling* with trig! Mark my words, This is going to be the best-selling book in history!

[@daniwalmsley611]

6:45 the coolest part of this was that this was wholely unsurprising thanks to your previous videos on the rhombic dodecahedron It's lovely when one maths investigation is helpful in understanding a completely unrelated one

[@paulzagieboylo7315]

4:25 This diagram is the longest space diagonal, not the medium-sized one Matt is talking about in this segment. But the length (phi^2 = phi + 1) is correct for the medium-sized one!

[@Peterwhy]

Thanks, I paused and looked for this comment.

[@馬陸]

phi^2 = phi + 1 Golden ratio quadratic equation.

[@dajac]

So good, Matt!

[@charlesmarshall7045]

Turning obscure math into real world objects, keep up the good work Matt!

[@deliciousrose]

7:40 this is next level editing XD

[@jajssblue]

3:30 I immediately know where this video is going and I love it!

[@milandavid7223]

I had to make a geometric solid out of paper as a highschool project. I chose a dodecahedron and it was pretty wild finding out that the whole net can be constructed (with straightedge and compass) using just a unit side and like 3 or 4 powers of the golden ratio. Imagine unfolding one half of the dodecahedron into a flower shape. That flower is bounded by a pentagon that's phi^2 larger than the faces.

[@mrautistic2580]

This is one of my favorite Stand-Up-Maths video!!!!!

[@BryndanMeyerholtTheRealDeal]

Legend says that he still says "Diagonals in Space"

[@ZedaZ80]

This was such a good visual demonstration!

[@scv4236]

The editing is genius

[@QuantenMagier]

I always was a fan of the Icosahedron, but this video made me appreciate the Dodecahedron.

[@mikeychrisanthus9948]

The subtle joke for diagonals in space about 4 minutes in was really good. I imagine you were thinking, this is a bit silly, no one’s gonna even care. I care. That caught me off guard.

[@degv364]

It hits better when you can visualize it in real life. Thanks Matt

[@ricdavid]

I love the ones where you can tell how much fun he had with it, and also where the concepts don't fly too far above my head. Also I can see myself making a shitty scaled down version of this in the future.

[@Chronicallywitty]

“Seemed clever at the start, I regretted it immediately”… that can basically be the theme of my life 😂

[@babilon6097]

One thing to do would be to also tape the insides, but wait, there's more... you could have taped each cube with a tape (or drew with a marker) that reacts to a different wavelength of UV. Then by switching different blacklights you could switch between the cubes instead of having them on all at the same time.

[@TheGreatAtario]

Are we sure there is such a product?

[@ciarangale4738]

@@TheGreatAtario product? unlikely. could it exist? definitely

[@unpythonic]

This is one of the most awesome things I've seen. So much better than CGI

[@heugvlinder]

What a lovely coincidence I'm building nested platonic figures in bamboo sticks (up to 3m) with my students at the moment and analyzing this video is their homework. Thanks, Matt.

[@HereticB]

the editing is amazing!!!

[@patrycjar1026]

You should be honest - "You might know me from Numberphile video with the Parker Square"

[@awebmate]

The first time Matt had a collab with Adam, he referred to him as "Adam Savage from Mythbusters". In return, Adam Savage referred to Matt as "Matt from Numberphile".

[@CBWP]

Adam was with mythbusters. Matt isn't with numberphile...

[@wierdalien1]

​@CBWP I mean he is, he has been doing videos since the start

[@CBWP]

@@wierdalien1 Numberphile is a collection of math. Was Matt in the first video? Are they friends? (those are rhetorical) Numberphile is a channel. Matt is a guest on their channel.

[@wierdalien1]

@@CBWP yes and yes and yes.

[@emperorbless120]

Matt Parker: "There are 5 regular polyhedra." Me, a jan Misali enjoyer: "there are 48 regular polyhedra"

[@joelcooper6441]

great vid, and looking forward to your book and, as a UK resident, can't wait for the 6th day of 20th month to get it

[@XplosivDS]

Good ol' small stellated dodecahedron and the great stellated dodecahedron

[@TheLastPhoen1x]

Novice sorcerer: Pentagram on the floor, demon flies away. Experienced Warlock: PENTAGRAM DODECAHEDRON!

[@itsEnyo]

man my workbook is getting full thank you for that note

[@kenmcfa]

Thanks, Matt. Thatt.

[@TheGeoffable]

Brilliant example of fairly simple geometry being done really, really beautifully, love the UV :)

[@ffggddss]

The compound polyhedron made of a pair of intersecting regular tetrahedra, is aka the "stella octangula." It was a favorite of Johannes Kepler, the guy who fiddled around with the 5 Platonic solids to try to explain the relative sizes of the planetary orbits, and the guy who formulated the famous "3 Laws of Planetary Motion" that bear his name. Anyway, the 8 vertices of the stella octangula are the vertices of a cube. Which also explains the 10 regular tetrahedra in the regular dodecahedron, once you've highlighted the 5 cubes in it. Fred PS. Also interesting to note, is that the main (longest) diagonal of an n-dimensional hypercube of unit edge, is √n.

[@ahsanuddin89]

Did not disappoint with the Steve Mould banter.

[@jonathanrobertson7059]

i will never look at a megaminx the same way after this

[@EliotChildress]

This video made me realize why I’m not a mathematician. I can fully understand why a square being a integral part of a dodecahedron is fascinating to some people, but i literally said out loud in a room by myself “oh, I don’t like that”. I find it supremely uncomfortable.

[@jace.miller]

Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: kzbin.info/www/bejne/aKarl3mmZr12hsU Let me know if a tool like that could aid in visualization. You could possibly do a follow-up on the hexagon within the hexahedron.

[@SineEyed]

Fun fact: if you take that hexagon found within a cube, and replicate it three times by rotating it 90 degrees each time along one of the three four-fold axes (through the center of opposing faces), you'll have four hexagons whose edges describe those of the cuboctahedron..

[@crawley6957]

@12:53 seven swans a-swimming, six geese a-laying, FIIIIIVE INTERSECTING CUUUUBES

[@volodyad195]

You are on fire, so many videos

[@jaytaffer9641]

Ten seconds in and my mind is already blown!!!

[@adityavardhanjain]

We want a t shirt of this with text "Diagonals IN SPACE"

[@GarryDumblowski]

I have to be honest, I really like the stella octangula (the compound of two tetrahedra) just because it has a simplicity that a lot of the other regular compounds don't have. You can take a single glance at it and instantly know how it's constructed.

[@vigilantcosmicpenguin8721]

I like how the editing of this video makes every joke feel like it glitched out and entered Matt's subconscious where he comes up with these gags.

[@GlizzyTrefoil]

A cube with roofs on the faces? So that the roof planes of one face continously match up with the roofplanes of the neighbour faces. Or the triangle part of one roof matches up with the trapezium part of another roof to make the pentagon without any kinks. LOVE IT!

[@Hamuel]

The long awaited cube asset video!!

[@Apes-With-Computers]

I am excited for your new book!

[@JosephParker_Nottheboxer]

I just want to say Tau vs Pi needs a revisit. Just to beat Steve again. That was a glorious fun video.

[@josh34578]

I like the way you've presented this. The group theory students might know that the rotational symmetry group of the dodecahedron is isomorphic to the alternating group A_5 and wondered what 5 things are being permuted. These 5 cubes! Or also the 5 tetrahedra as mentioned in the video. The tetrahedra come from joining opposite diagonals on the faces of the cubes. So technically you do get 10, but under the symmetry of the dodecahedron they split into two orbits of 5 and 5. Note also that the symmetry group of each tetrahedron is a subgroup of the symmetry group of the dodecahedron, but the symmetry group of the cube is not a subgroup of the dodecahedron. Note You can turn a cube a quarter turn around a face but doing so does not take the dodecahedron back to itself. For the lengths mentioned in the beginning, there's also a way to calculate the lengths that involves computing the eigenvalues of the adjacency matrix of the dodecahedron graph. From this one can derive the dot products between different vectors from the origin to the different vertices and then compute the lengths. This generalizes nicely to other graphs as well.

[@Howtheheckarehandleswit]

There are, in fact, more than 5 regular polyhedra! jan Misali has a great video on this, titled "There are 48 regular polyhedra" if I recall correctly

[@InhumanEntity]

Rollie Williams would be proud of the video's style I reckon

[@nadionmediagroup]

The title of the book is perfect.

[@vyreck]

Easily the best polyhedral based mini-rave I’ve had all week!

[@MegaMinerd]

I can solve a megaminx, so I've spent a long time looking at dodecahedra. I noticed long ago that at a certain orientation you can find a set of 6 edges that are parallel to one of the 3 axes and you can therefore make those edges all line up with a face of a circumscribed cube. It's really cool to see the numbers behind an inscribed cube. 14:00 I think I know how to create all of them. You can make the double tetrahedron with the medium space diagonal of the dodecahedron, aka the face diagonal of the cube. You might even be able to get the fifth using cube-octahadron duality. I believe this would be drawing diagonals between the edges of the dodecahedron, specifically the 6 edges mentioned in paragraph 1

[@jeremyjw]

another fun way to build a dodecahedron take a bunch of inflatable tubes (innertube , donut , torus) and lash them together i managed to build all of the solids except for the icosahedron it collapsed on itself you end up with some very large pool toys

[@the3nder1]

Imma need that track. 🎶"I couldnt be bothered."🎶

[@brandontylerburt]

Maybe it's because spring is coming, but I've suddenly begun noticing how adorable Matt Parker is. What a math hunk!

[@JTISD3LL]

Lovely stuff, such a clear way to see the lengths of the various space diagonals. It's fun, I think, to consider also which symmetries of the dodecahedron fix one of those cubes (and also which symmetries of the embedded cube are *not* symmetries of the dodecahedron). One pedantic correction (although, the mistake set alarms off in my head): At 13:30, you say the compound of five octahedra "doesn't even have a convex hull". I believe you meant that it (uniquely among the five) doesn't have a *regular* convex hull. Its convex hull is an Archimedean solid whose faces are (regular) triangles and pentagons and, if I'm not mistaken, its edges are all and only the space diagonals of a certain kind of that Archimedean solid (to bring the discussion back on theme)! I find the uniqueness curious. In particular, I don't think it's obvious that any of the regular compounds should have *regular* convex hulls and, moreover, it's maybe a bit surprising that duality does not preserve this property. I wonder if one can produce an example (in some high dimension or in hyperbolic 3-space perhaps) of a regular compound whose convex hull has even less regularity (say, having irregular polygons as faces or more than two types of faces).

[@galoomba5559]

In 4 dimensions, the convex hull of the regular compound of 2 5-cells is a polytope whose cells are 30 disphenoids (a type of non-regular tetrahedron). Similarly, the hull of the regular compound of 2 24-cells (as well as the inscribed compounds of 6 tesseracts and 6 16-cells) has 288 disphenoids as cells. It makes sense that duality doesn't preserve regularity of the convex hull. The dual of a convex hull is the convex _core_ of the dual.

[@JTISD3LL]

@@galoomba5559 You have inspired me to return to Coxeter's text. It's brilliant but dense. I wasn't aware of the regular compounds you mentioned, do you have a reference (I don't think Coxeter has one of two 5-cells or of two 24-cells but his list was not exhaustive)? [SEE EDIT BELOW] Also, I'm not familiar with the convex core concept, would you cite something about this idea? By the way, I'm not sure if you meant in your parenthetical that the six tesseracts with six 16-cells should count as another example or merely are as another (irregular) compound having the same convex hull as your other example but I would not count such a compound as **regular** as its components are not congruent. (Admittedly, what qualifies as a "regular polytope/compound" varies in different contexts but usually, one doesn't allow allow that the components of a regular compound are non-congruent polytopes.) EDIT: While the compounds you mentioned are quite nice, I don't believe they are regular compounds by any of the usual definitions, namely, neither has the vertex set of a regular polytope (because the compound of two 5-cells has ten **distinct** vertices and there is no regular 4-polytope with ten vertices and similarly for the compound of two 24-cells) so, in particular, it's clear that such compounds have irregular convex hulls and these don't provide the "more severe" examples I was originally inquiring about. Indeed, the sequence of compounds of two n-simplices, beginning with the six-pointed star {6/2} (or in compound notation {6}[2{3}]) in 2D, the stella octangula {4,3}[2{3,3}] in 3D, the compound of two 5-cells in 4D, and so on simply fails to be regular in dimension ≥ 4, yet another example of the elusiveness of regularity in higher dimensions.

[@galoomba5559]

@@JTISD3LL Seems like my reply got deleted, possibly because it contained a link. Great. I'll rewrite it. I think Coxeter used a different definition of "regular compound" which required the convex hull or convex core (or both? I don't remember exactly) to be regular. I'm not sure why he used that restriction, it seems quite arbitrary to me. The definitions I'm more familiar with simply generalise the definition of a regular polytope, namely a regular compound is either - vertex-, edge-, face-, etc. -transitive (the elements of each dimension are indistinguishable), or - flag-transitive (the flags are indistinguishable, a flag being a set with one element of each dimension that are all contained in one another, e.g. a flag of a polyhedron is a face plus an edge on that face plus a vertex on that edge); alternatively it consists of congruent regular polytopes and preserves all symmetries of its components. For connected polytopes, these two definitions are equivalent, but for compounds they are not. There are 5 regular compounds in 3D by the first definition, but only 1 by the second definition. I'll call the first definition "weakly regular" and the second "strongly regular". There are 7 strongly regular compounds in 4D: 2 5-cells, 2 24-cells, 3 tesseracts, 3 16-cells, 6 tesseracts, 6 16-cells, and 120 5-cells. The first two are simply the compounds of two dual 5-cells and 24-cells, their hulls are the duals of the bitruncated 5-cell and bitruncated 24-cell respectively. The next two consist of the 3 tesseracts and 3 16-cells inscribed in the 24-cell. The next two are the same plus a second copy inscribed in the dual 24-cell. The last is the 120 5-cells inscribed in a 120-cell. As far as I know there are many more weakly regular compounds in 4D, most using the index-10 subgroup of [5,3,3] and/or the index-2 subgroups of [4,3,3] and [3,4,3]. You asked about regular compounds with non-regular convex hulls, but then said regular compounds have to have regular convex hulls by definition. That doesn't really work. As for the convex core, it's the dual concept to the convex hull - the intersection of, for each facet, the half-space bounded by the plane of that facet and containing the origin. It's possible that this concept tends to go by a different name in the literature, I'm not a professional mathematician.

[@JTISD3LL]

@@galoomba5559 You're absolutely right, one must be very careful to distinguish the various regularity notions. To be clear, four of the five regular compound polyhedra in 3D are vertex-regular, meaning that they consist of copies of some regular polytope Q inscribed in a regular polytope P (i.e., vert Q ⊆ vert P) in such a way that some subgroup of the symmetry group of P acts transitively on the copies of Q and on the vertices of P. (Coxeter's notation for such a compound is mP[nQ] meaning that the n copies of Q cover vert P m times.) I think you're right that vertex regularity implies that the convex hull is P, hence regular, and so vertex regularity precludes the possibility I was asking for originally. The dual-notion on the other hand is facet-regularity, the compound denoted [nQ]kR is the dual of the vertex-regular compound kR'[nQ'] where R',Q' are the dual polytopes of R and Q, respectively. The fifth regular 3D compound [5{3,4}]2{3,5} of five octahedra is facet-regular but *not* vertex regular, and so has an irregular convex hull. Merely demanding d-cell transitivity, your "weak regularity", might be equivalent in 3D to the assertion that the compound is vertex-regular OR facet-regular (although I'm not sure) but it seems that the latter is a strictly stronger requirement in dimensions ≥ 4. (This issue is a subject of dispute for the "Regular compounds" sections of Wikipedia's "Polytope compound" entry.) In particular, the compound of two 5-cells is not vertex-regular, and being self-dual, also not facet-regular, but its symmetry group acts transitively on is vertices, edges, etc. In my opinion, these distinctions are relevant. In fact, I would go so far as to say that the three regular compounds of polyhedra (in 3D) which are BOTH vertex- and facet-regular are in a sense _more regular_ than the two (the dual pair of five cubes and of five octahedra) each having only one of these properties, and these in turn more regular than any compounds having neither property. With this in mind, a better formulation of my original question is "which _facet-regular_ compounds in 4D and higher have notably irregular convex hulls?" This probably isn't too difficult to answer but I am no expert in thinking about these convex hulls.

[@galoomba5559]

@@JTISD3LL There are compounds that are facet- and vertex-regular but not weakly regular. An example is the compound of 2 16-cells in a tesseract; its convex core is a 24-cell and its convex hull is a tesseract, but it's not cell-transitive (some cells are coplanar, some are not). As far as I know there are hundreds of facet- or vertex-regular compounds, while the number of weakly regular compounds is a lot smaller. Makes sense, because there are many subgroups of the symmetry groups of the regular 4-polytopes under which they remain vertex-transitive but not element-transitive for other dimensions of elements. As for facet-regular compounds that are not vertex-regular, there are probably many in 4D, for similar reasons. I found a compound of 50 16-cells in my files (not discovered by me) whose convex core is a 120-cell and whose convex hull is the grand antiprism.

[@kdgs2]

I've always thought that the dodecahedron is the best Platonic solid. This just strengthens my view!

[@Entroper]

This was mind blowing, definitely one of my favorite Matt videos. Almost cool enough to make me unsubscribe from Steve Mould.