there are 48 regular polyhedra
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- Опубліковано 22 тра 2024
- a comprehensive list of all 48 regular polyhedra in 3D Euclidean space
primary source: link.springer.com/article/10....
bgm: queerduckrecords.bandcamp.com...
visualization tool for the shapes in this video: cpjsmith.uk/regularpolyhedra
/ hbmmaster
conlangcritic.bandcamp.com
seximal.net
/ hbmmaster
/ janmisali
0:00 - introduction
1:06 - part one: what?
4:06 - part two: the platonic solids
6:21 - part three: the Kepler solids
9:00 - part four: the Kepler-Poinsot polyhedra
11:26 - part five: the regular tilings
13:15 - part six: the Petrie-Coxeter polyhedra
16:51 - part seven: the Petrials
21:08 - part eight: the blended apeirohedra
22:39 - part nine: the pure Grünbaum-Dress polyhedra
25:03 - part ten: summary
plato: a regular polyhedron has equal edges and equal vertex angles
diogenes: *holds up infinite square tiling* behold, a regular polyhedron
Okay, that's perfect.
Underrated comment
Perhaps the Nerdiest joke I've ever understood
@@iamdigory ...so far.
😂
For the people who read the comments first:
A cube is made up of 4 hexagons.
I hate this
I'm sorry to say, but you are truly evil.
This is the funniest comment I’ve ever read
Psicologist: The Petrial cube isn't real, it can't hurt you.
The Petrial cube: {6,3}v4
The more I think about it, the more it oddly makes sense.
my dad had the opposite reaction: i told him about the video and he said "why only 48?'
i then told him the euclidean space restriction and he went "oh ok"
Yeah, once you go off into non-euclidean symbols you're likely to summon something.....
@@johnmccartney3819 i knew it, i knew this video contained eldritch knowledge
@@somedragonbastard It summons a 4D hound or something
@@samuilzaychev9636oh no , get rid of all the angles
@@have_a_cup_of_water_08biblically accurate angles
I found the paper "Regular Polyhedra - Old And New" by Branko Grünbaum in 1977, which list all 47 regular polyhedra. The one that was found by Andreas Dress is the Skew Muoctahedron
Cool, good to know!
pog
Link pls?
search the paper name in google with quotes around it so only results containing the exact name show up
@@axehead45
17:02 "There's nothing in the definition that restricts polygons to two dimensions"
*Dear God*
There's more
@@boldCactuslad No!
Saint Scott!!
Would that mean that there is nothing restricting polyhedra to 3 dimensions?
@@ondrej2871 by his definition, there was, but he left it open to explore removing that restriction.
Bart: There are 48 regular polyhedra.
Homer: There are 48 regular polyhedra so far.
I'd watch that episode
@@Asger1703 that line is from the movie.
Wasn't Homer an author though?
@@hyliandragon5918 everyone knows, it is a joke
Halfway I was laughing from the joy of discovery.
By part 8 I was crying from the horror of discovery. By then, I felt like I was diving into an eldritch horror.
Same here, man. This video has so much emotion hidden inside it. It's a masterpiece of drama.
This is all still Euclidean though, which Eldritch horror is clearly described as not being.
Allowing for non-Euclidean curved space would presumably pretty easily allow for infinite regular polyhedra, stuff like angles adding up to 360 degrees doesn't apply anymore so you could have a septagon sided shape etc.
@@xTheUnderscorex HP Lovecraft was naive. Non-Euclidean geometry doesn't have to be eldritch (just look at flight plans for aircraft, which take place entirely in spherical geometry, or really anything based on the surface of the Earth), meanwhile this video showed that it's more than possible to find Eldritch horrors entirely within Euclidean geometry.
I love how this is packed with easy-to-digest info distilled into half an hour but at the same time you can _feel_ how deep Jan had to stare into the abyss to do that. Like, well done bro, you truly suffered for your art here!
'jan' just means person/people in tokipona. If you want to refer to them by name, you should call them 'Misali'.
@@Sapien_6 (they don't mind and you don't have to correct people on it)
@@soupisfornoobs4081they also go by he
"There's nothing in the rulebook that says a golden retriever can't construct a self intersecting non-convex regular polygon."
Never change jan Misali, never change.
I read this right before he said it lol
It's the sheer confidence with which he says it that just catches you off guard and leaves you wheezing.
I loved that line too! Especially since the last Vsauce episode referenced that part of Air Bud too. Still fresh in mind.
Thanks for being brave enough to stand up to Big Shape.
you're welcome petrial halved mucube
IS THAT A... nevermind
You're welcome (look up 120 sided polyhedron(
" to square up"
Yeah down with Cube!
I’m in college learning more advanced math and computer science now, but I still come back to this video on occasion to keep myself humble.
>username: uwufemboy
>"computer science"
Ah ok that makes sense
The fact that this video codifies the names for some of the polyhedra it describes is amazing.
This is how you get Thagomizers.
*Plato:* "Nooo, you can't just call filthy abstractions of reality a platonic solid!"
*Haha blended Petrial hexagonal tiling go }{{⁶{}}⁶{{{}⁶}}}}⁶}{{{}⁶*
I'm don't understand, but I like it
platonic solids are convex regular polyhedra and have surface area
They're not really platonic aren't they... They're just... Regular.
Everybody gangsta until the brackets italicize themselves
May the touhou fan base rise up
What exactly IS a polygon? A miserable pile of vertexes.
*BUT ENOUGH TALK, HAVE AT YOU!*
Thanks
Vertices >:(
this is legitimately hilarious. underrated comment
Oh boy, yes Vertices.... I got my BS in Animation (2D&3D) & wen we model for animation we map our polygons, sometimes for repeatable textures- they do breakdown to triangles, but usually use 4 sided faces to make nice mappable squares/quads. 5 is a no no because of artifact/shading probs and such when animated. But holy heck- if you're using polygons & make a mistake early you're in for it. (Rudimentary comment don't come @ me w/aCtuAlLty ... I'm echoing the struggle for perky noobies.)
I want to comment on how most of this video is actually very easy to comprehend even though I know nothing beyond high school maths. Very well made explanation
Yes, agreed. I'm in high school currently taking Calculus, and I am a math nerd, but this kind of iceberg territory is usually incomprehensible, yet I somehow understand what a Petrial is now :D
wait, nullfoo? *the* nullfoo? in my jan Misali comments section?
@@dangerousglasses7995 it's more likely than you think!
The one thing that im frustrated with is this: In school, i was taught with the assumption that my questions where irrelevant or inappropriate. Yet this shows my questions had in the past been accurate. Thank you for all the effort you gave this video. Much appreciated
what the heck kind of school did you go to?
@@MegaDudeman21a bunch of schools are just stupid and bad
@@MegaDudeman21An American one. Most US schools are staffed by people who don't care about the subject they teach, and sometimes they don't even understand the subject themselves.
@@nikkiofthevalley that was never the case for me when I was in school
The moment you realise there are geometry Discord servers dealing in illegal polyhedra.
Oh shit
@@gameplaysuffering1620 *oh no*
Oh God
Oh zoinks
Oh My
“Roll the 50 polyhedra”
“All we have is 48 polyhedra and 2 marbles”
“Close enough”
you need to define rolling before you do that
@@_vicary ROLL THE PETRIAL SQUARE TILING
@@_vicary shake it about with gravity
How tf do you roll any tiling?
Actually spherical tilings are valid regular polyhedra.
to explain 5/2:
1. imagine you have five dots in a circle
2. connect those dots via lines to make a shape
3. make note of how many dots you move around the perimeter each time you connect a dot (Make sure these are equal)
4a. if you move 1 dot per line, you end up making a pentagon, therefore it would be 5/1, but you dont have to write the 1, as it is understood by default.
4b. if you move 2 dots per line, you end up making a pentagram (5 pointed star), therefore it would be 5/2
4c. if you move 3 dots per ling, you still end up making the same pentagram, just the other way around, so it would still be 5/2
another more complicated example:
There are multiple ways to make an 8 pointed star, and the schlaffle symbol allows us to distinguish between them.
1.have 8 dots in a circle
2.connect those dots in the same manner as the 5 dots
3. notice that now you have more choices on how many spaces you can go and make different polygrams (stars)
4a. 1 dot gives you an octogon, 8
4b. 2 dots give you a square octogram (an 8 pointed star made by stacking squares), 8/2
4c. 3 dots give you a different octogram (this one can be drawn withut lifting your pen), 8/3
4d. 4 dots give you an 8 pointed asterisk (the * symbol but with 8 points instead of 5), 8/4
4e. 5 dots makes 8/3 in the other direction.
now hopefully, you understand a little more about schlaffle symbols.
Thank you very much about this comment. I believe there was a vihart video I watched that made it easier to understand this comment. She didn’t use any notation but she was creating every type of stars including 5/1 (that is a pentagon I don’t remember whether she called it a star in the video or not), 7/2 or 6/3 or 6/2
Thank you very much. Really appreciate your explanation 😊
So 8/2 results in pairs of edges that completely overlap. Jan Misali was explicitly not allowing overlapping edges or faces or vertices, but if you did allow them, it would surely give infinite regular polyhedra.
This video felt like someone explaining to my how geometry is just an elaborate ARG, I love it
“This video is supposed to be for a general audience”
Are you really sure about that?
Well, his general audience. The kind that watches conlang reviews and very deep dives into hangman and the letter w.
Being a mathematician-in-training, yes that is the 'general' introduction. The 'specific' introduction has a prerequisite of first year university mathematics.
No, it's a video for an audience of generals.
thats why he defined them 😹😹
as a regular human, I can confirm that this video was very informative and entertaining. I'm not sure how much I actually understood, but that's not always the most important part, ight?
"there's nothing restricting polygons to 2 dimensions" oh yeah? then why am i standing here with a hammer? get back in 2d
2D or not 2D, that is the question!
@@simonmultiverse6349Highly underrated comment
I mean this as positively as possible, I have watched this video like 5 times, I have never made it to the end, I am genuinely interested in what you’re talking about but dear lord this video is like a sleep spell to me. I only watch it when I can’t fall asleep and nothing else works, 10 minutes in and I’m GONE. This is a blessing. Thank you.
And thus, the regular polyhedra brought peace to clown town...
_(I like your username)_
@@dantesdiscoinfernolol thank you :) I like yours too! Our usernames are like, same spectrum but opposite ends
Tip from me, If you need more, Just Pick a weird niche science topic, search a Uni class on it, choose Like the 5 class, and boom, ITS Just Professors saying words that dont mean anything and Its super nice
@Clown From Clown Town have you finally completed your quest to watch it?
How many times have you watched it by now?
i like that all of these videos become utterly incomprehensible in the second half
It's not incomprehensible?
“I’m making this for general audiences”
*15 minutes later* : D A R K G E O M E T R Y
See, THIS is what my conservative Catholic mother warned me about! That darn Pentagram leads to the path of Dark Geometry if you twist it with evil dark math!!
That was about the point I started feeling like one of my Call of Cthulhu characters.
Let's be honest anyone who watched until the dark geometry bit are definitely not part of the general audience.
;)
“I’m making this for general audiences”
“Look again, what your actually looking at is a infinite spiral pattern of squares spiraling into the 3 r d d i m e n s i o n “
Not the best example but still
I'm actually astonished that this incredibly loose definition of a polyhedron does not lead to an infinite number of regular polyhedra.
if it didn't have the extra rules Jan added, there probably would
I'm not sure it's been proved that these are the only ones, these are just the ones he found.
Nah, he deliberately set the definitions to exclude an infinite number of regular polyhedra. In the spesific definitions he set, he (probably) found all of em.
@@gustavjacobsson3332 That's also true. Just not an infinite set of polyhedra *classes.*
@@gustavjacobsson3332 Well, I should've specified, stricktly adhering to the definitions set here, an infinite amount of classes of regular polyhedra is impossible. Technically speaking it might be possible to construct more than jan Misali showed here, since that hasn't been disproven yet as far as I'm aware. But there probably isn't a way to create infinitely many classes of *regular* polyhedra that are unique.
“Dark geometry”… never knew I needed this in my life
새로운 정다면체의 정의와 이걸 기존에는 정다면체로서 이야기 못했다는점과 이 혼돈의 카오스 스크립트를 전부 번역했단게 전부 놀랍다.... 특히 번역하신분 ㄹㅇ..
The translator was probably on some strong drugs...
@@orbitalvagabond especially korean words are good for making new words about new "definition". but this is another problem that the words for anomaly(?) polygons are even hard to understand in english and also not in dictionary for evidences either. (i tried to find)
then it means the translator did kind of translating NEW abnormal mathematics into pretty reasonable korean words for make korean ppl understanding it well
maybe translator had a high grade of "MATH".
or "math".
or both of them :)
무서워요
진짜 공포
@@qkqk111 Translator here, and yeah, mucubes and Petrials were around the edge of previously available Korean translations and I had to invent some words from that point. Thankfully I only had to invent some; say, "Petrial halved mucube dual" needs four words "Petrial" (a proper noun), "halved" (translated), "mucube" (mu- invented) and "dual" (existing) but only one word has to be invented and reused.
And no, the only thing I have is a master's degree in computer science, which has a crossover with discrete mathematics but that's about all. An ability to parse academic papers did help, though. See also my older comment that links to detailed glossaries and references.
@@lifthras11r 관련은 얼마 없어도 컴공 석사는 진짜 아무나 할 수 있는 게 아닌 것 같습니다,,,😵💫 대단한!
자막 켜고 끝까지 잘(??) 봤습니다 ㅎ☺️
"There's nothing in the rulebook that says a golden retriever can't construct a self-intersecting non-convex regular polygon."
This is just like 8 minutes in... This will be a wild ride, won't it?
By the end of this you will realize we don’t need a fourth dimension to black magic/sci-fi things into existence because three dimensions are complex enough.
@@ravensquote7206 the what
@@engineerxero7767 the j
But what about staplers?
777th like! I'll make a wish!
Making a shirt with a petrial cube and the caption "This is not a cube" to feel superior to my unenlighted peers.
Bonus points: You also get to look like an Art snob at the same time!
@@An_Amazing_Login5036 SIGN ME UP! :D
Ce n'est pas un cube.
I would personally add parentheses around the not for an anime twist.
I would also really like this shirt
Just seeing the spinning truncated octahedron made my day. Truly my favorite shape
I love weird geometry stuff like this, but at the same time it's kind of scary. It's always kind of scary to learn something that contradicts what you always thought you knew. It's like learning that Uranus and Neptune are actually ice giants. I always thought they were made of gases and some liquids, with the only solid part of them being the relatively small rocky and metallic core. That's still true, but the "ice" in "ice giant" actually refers to substances heavier than hydrogen and helium such as water, methane, ammonia, elemental carbon (in the form of planet-wide liquid diamond oceans, to boot), neon, and carbon dioxide, among others, regardless of what state of matter they're in, and that they're called "ices" because they were probably solid when the planets first formed even though they aren't now. The truth can be confusing and you can end up feeling like everything you know is a lie even though you just had the confusing parts explained to you.
Galaxy Man pfp spotted
Jan misali: *big smart words*
Me: cool shapes go spinny
all I can think about now are those 5 monkeys spinning around with mario music
That me
Same
It me
Cool shapes go whrrrrrrrrr
“I don’t understand why anyone would write a geometry paper without including any diagrams of the shapes they’re talking about”
Oof that must have been rough.
Making pictures was a lot harder back then
Think about how satisfying those were to model though
@@undeniablySomeGuy or frustrating
@@jercki72 probably frustrating. i can't even think about it about programming them. _MATH MATH MATH MATH AAAAAAAAAAAA_
I looked at some of those articles and it's ridiculous. You spent 12 pages talking about polyhedra and did not make a single drawing? What's the point?
One of the restrictions you chose to include was that two points connected by line segments doesn't count as a polygon. That's a sensible exclusion, but that is actually my favorite shape, the digon. It's not very interesting in a plane by itself so explicitly excluding it for this video is a good idea, but on a sphere it's a really important shape called a lune, think of it as the boundary on a sphere of an orange wedge. But way more importantly, a digonal antiprism is a tetrahedron! it's so cool! a totally different way of constructing a tetrahedron. A tetrahedron is two line segments, degenerate digons, rotated 90° and connected vertex to vertex. If you allow the digon there's also at least 1 new regular polyhedron, The Apeirogonal Hosohedron, basically a tiling of the plane by infinitely long rectangles, or stripes.
This is my favorite video of your channel and it singlehandedly reignited my interest in geometry and topology.
Some architects are gonna have the time of their lives designing like this.
They make sense as soon as you rip the skin off geometry and start reorganizing the algebraic bones in otherwise impossible shapes.
That sounds metal as hell
that's a horrible way to put that, thank you
Best way to look at geometry: *Remove its skin*.
@@cyberneticsquid skin it and rearrange its skeleton
i don't understand
This feels like a video that years from now will be the equivalent of what the "Turning a sphere inside-out" video became.
thats precisely how i got here
hmmm what if instead of turning it inside-out, you view the sphere from the inside instead of from the outside
literally came here from that video
@@GhGh-ci8ld SAME
That was the video right after this one 🤣🤣
I'm not kidding, this is literally comfort media to me.
I would love for someone to 3D print the regular polyhedra that are possible, the solid, finite ones preferably. I would totally buy them. Cast them as well in some metal perhaps.
You mean... dice that you can buy in any store that sells board games/tabletop RPGs?
@@aralornwolf3140who makes stellated dice lmao
@@VectorJW9260,
People sell metal dice... so....
Give me a mucube but not infinite please
"The dark side of the geometry is a pathway to many shapes some consider to be... unnatural..." -Grünbaum, probably
Is it possible to learn that power…?
-not with a compass and a straightedge
AHAHAHAH
This is one of the best applications of this quote I hav ever seen lol!
Have you heard the tragedy of Darth Non-platonic solid the regular? I thought not, it's not a mathematical principal the Ancients would tell you
This is fricking gold
I never thought I would hear the words “dark geometry”
Dark geometry show me the forbidden polytopes
Greg Egan wrote a story, "The Dark Integers" but the definition of what they were was disappointing and not related to the story, even though the name was evocative of the story.
Queue dramatic striking sound
The Dark Side of geometry is a pathway to many shapes some consider to be... unnatural.
The Dark Arts of Mathematics!
This is just mathematicians taking a break from whatever they were doing and going "you know what would be really cool..."
i could kind of comprehend this video, but i love how, despite a hexagonal polyhedron being impossible, it all kept coming back to hexagons
i guess hexagons truly are the bestagons
This is why golden retrievers shouldn’t be allowed to study math.
Racist
...
@@sineadthomas2024 Ok millenial
@@doommaker4000 Ok racist
Doom Maker Ok Boomer
alternative title:
man bullies shapes for 28 minutes straight
Man bullies his viewers with shapes for 28 minutes straight
@Eric LeeIt’s*
Lmao
@Eric Lee It is, did you not read my correction?
@Eric Lee Don’t say such derogatory things!!
I love the increasing asterisks at the beginning of the video just getting more and more specific. Math really do be like that sometimes.
Honestly, Jan, your videos are the only ones that can genuinely rewatch 100 times, I seriously have seen bith this and caramelldansen more time than I can count, and they always perk up my mood, so thanks
Reeling from the ramifications of Big Shape hiding Dark Geometry from me.
Jan, I wanted to congratulate you. Fool that I was, I thought that after besting graduate-level dynamical system analysis, no topic in mathematics could make me irrationally angry upon learning it, yet you've proven me wrong.
I am simultaneously both thoroughly impressed by the ideas contained in this video, and utterly disgusted with them for having the gall to exist and ruin something I thought I previously understood.
Thanks for that.
thanks for your comment DickEnchilada
Very inciteful, DickEnchilada
@@franky2192 @Adrien Calin These comments will be really confusing if DickEnchilada changes their username.
@@franky2192 insightful.
mm, yes a very wise statement, DickEnchilada
this is unironically one of my favourite videos on youtube
nice pfp
정말 좋은 영상입니다.
특히 정사각형으로 이루어진 정육면체를 그리다보면 뒷부분의 모서리들을 점선으로 그려야하는데, 그 점선들이 한점에 모이게 되는 시점에서 정육각형이 보이게되는 것은 당연하다는 점에서 감동받았습니다.
나만 그렇게 느끼는 줄 알았습니다.
The “Big Shape” I’m figuratively dying
Thanks for not saying "literally dying"
You _are_ literally dying. We all are
@@blue_leader_5756 Assuming you're not a vampire or a lobster, you are literally dying as you read this.
@alper kaderli so you're like, getting hit by a bus while trying to escape an axe murderer?
@alper kaderli was the bus part of your escape route? that would be pretty ironic.
This must be that crazy "crystal math" stuff I've heard about on the news.
@Liyana Alam literally
i am both very angry and absolute thrilled that this made me laugh
this comment has layers.
I like how no matter what vocal you replace the a with in the word math it will still be a word (except u)
Math
Meth
Mith
Moth
@@CoingamerFL Be thankful you've never encountered the horrifying _Crystal Muth_ .
One of my favorite sentences ever
"The Petrial mutetrahedron can be derived either as the Petrie dual of the mutetrahedron or as a skew-dual of the dual of the Petrial halved mucube."
the fact that there is a polytope discord with someone named "compund of 48384 penaps" is hilarious and entirely unsurprising
this video perfectly captures how it feels to be enchanted into reading an eldritch tome, experiencing a type of madness that is coherent in the moment and that you are mentally and physically incapable of sharing the knowledge you've obtained
... u wot m8??...
@@valinorean4816 go try to tell your mom what a mucube is without showing her a picture or this video
"remember how as a child you were taught there was 1 god? there's actually 48"
Esoteric knowledge
*psychedelics
At this point, we should just redefine a regular polyhedron as also having a defined (or definable) volume, to stop mathematicians from going mad.
that's not gonna stop them and we all know it
@@literallyafishhook u right and i hate it
complex numbers count as "defined", right?
@@strangeWaters holy shit
Technically platonic solids do not have volume, they're surfaces curved into 3D space, just as how polygons are line segments curved into 2D space.
Everytime I watch this video, the summary makes my heart race. I understand all the lead up, and the final conclusions, but yowza, having the whole of it condensed into a few short minutes makes me excited!!!! Like, imagining space, and defining it, and being able to explain that definition is sooooooooo....!!!! So, like, fascinating!! Thank you!!!
Man I found you first through this one random one off video, then left and never thought of it again, until I found you again a year later when i got into linguistics. it's a really weird thing. Good video
"I've been Jan Misali, and I don't understand why anyone would write a geometry paper without including any diagrams of the shapes they're talking about."
You haven't met mathematicians enough
Me, a mathematician: Oh, like the Kepler-Poinsot polyhedron? (Also I saw the Petrie-Coxeter ones once but forgot about them.)
Jan Misali, a hobbyist: I'm about to ruin this man's whole day.
CuK
the virgin mathematician vs the chad petrial halved mucube
Jan? His name is Mitch
@@palatasikuntheyoutubecomme2046 I know that now, but only after seeing like all of his videos. I thought for the longest time his name was "Jan", like a Polish friend of mine.
"look at all those long names, c'mon guys, we can just call it Bob or something" - my friend, watching this video
When you watch Dexter's Laboratory and you pay a little too much attention into understanding all the scientific jargon Dexter talks to himself with
Me learning about Kepler solids: Ah! Technically correct! My favourite kind of correct.
Me learning about Petrials and infinite towers of triangles: This is witchcraft and it's making me anxious and honestly I don't think it should exist.
That's just a sign that we are going the right way and we need to go deeper.
This is one of the areas where using VR for study actually makes a lot of sense. I'd assume seeing all these shapes "in person" makes it much more simple and understandable.
Exactly
@@sdrawkcabmiay I might need to model some of these and bring them into VR.
I have a feeling that these would act like the dreaded "brown note", except instead of making you go mad from looking at them, you'd just be left extremely confused and would get a headache.
So an animation of some sort would be handy as well.
After seeing all of these in VR all of reality starts to look wrong and incomplete...
@@Alorand where did you get them?
I could watch this on repeat for the rest of my life and still not get it, but I can appreciate that you went through all that research to be able to present this almost unpresentable idea. I want more.
So curious how many people actually watched to the end like I did... this was an AMAZING video dude. I truly appreciate all of the research and effort you put into making this video great!!!
Full list:
- Platonic Solids
- - Tetrahedron {3, 3}
- - Cube {4, 3}
- - Octahedron {3, 4}
- - Dodecahedron {5, 3}
- - Icosahedron {3, 5}
- Star Polyhedra / Kepler-Poinsot Polyhedra
- - Small Stellated Dodecahedron {5/2, 5}
- - Great Stellated Dodecahedron {5/2, 3}
- - Great Dodecahedron {3, 5/3}
- - Great Icosahedron {5, 5/2}
- Flat Tilings / Apeirohedra
- - Triangle Tiling {3, 6}
- - Square Tiling {4, 4}
- - Hexagon Tiling {6, 3}
- Regular skew apeirohedra / Petrie-Coxeter polyhedra
- - Mucube {4, 6|4}
- - Muoctahedron {6, 4|4}
- - Mutetrahedron {6, 6|3}
Petrial Duals of all of the above
Unnamed
- Blended Square Tiling {∞,4}_4 # { }
- Blended Triangle Tiling {∞,6}_3 # { }
- Blended Hexagonal Tiling {∞,3}_6 # { }
- Helical Square Tiling {∞,4}_4 # {∞}
- Helical Triangle Tiling {∞,6}_3 # {∞}
- Helical Hexagonal Tiling {∞,3}_6 # {∞}
- Petrial Duals of all the above
- Halved Mucube {6, 6}_4 (and it's petrial dual {4, 6}_6}
- Dual of the Halved Mucube {6, 4}_6
- Trihelical Square Tiling {∞, 3} (the first one)
- Tetrahelical Triangle Tiling {∞, 3} (the other one)
- Skew Muoctahedron {God knows}
"God knows"
no.. God does not. dark geometry is beyond any divine influence
{GOD KNOWS}
doing God's work, my guy
Basshedron {69, 420}
@@wormius51 lmao
"dark geometry" is the most intimidating phrase I've heard all year
Now I want to open a bar named that. Complete with neon fixtures with these Edritchian polyhedra.
Reminds me of Lovecraft...
I raise you: Umbral Calculus
@@castafiorept7309 Dear god...
SCP-478+23i
I remember watching this video when it first came out. Don’t know or care anything about the topic, but I always get reminded by my YT recommended by how I intriguing and entertaining these are (specifically this video too). Anyway, long story short you can make something distasteful and seemingly simple into something pretty fascinating. Props to you 💯
This is and very probably always will be my favorite video on the entire platform.
that's the second air bud joke in the edutainment sphere this week
Where was the one in this video?
@@anselmschueler 7:00
Now imagine me watching those two videos in a row. I was like “??? Is it Air Bud appreciation week??”
Not only that but they were both referencing the same moment in Air Bud
Who was the other one? I remember watching the vid, but forgot who
"what even is this spiky thing?"
*KIKI*
Bouba
@@YitzharVered no
@@Xnoob545 bouba
@@ATBZ no
@@Xnoob545 bouba
I appreciate your knowledge of the difference between number and amount as well as the difference between fewer and less.
one of the best geometry videos I've seen in a long while, thank you!
this shit literally had me laughing the entire time, sure you could talk slower so i could understand more but everytime you pulled a new concept on me i was like "oh fUCK" and then a giant ass shape with a stupidly long name appeared and it was like the punchline to the funniest joke ever like unironically never stop making these
Oh man I keep coming back to this comment every once in a while because it makes me so unreasonably happy. Imagining you laughing at this anything-but-funny video makes me do a massive :) for whatever reason. Thank you.
The names in the video are short compared to stuff like the small dispinosnub snub prismatosnub pentishecatonicosatetrishexacosichoron.
@@danielsebald5639 dont say that ever again D:
the spinning mucube is making me lose my shit
the jokes just kept on coming
Before watching: I can't believe general education channels ignored such an important fact!
After watching: oh.
Lol. Simple minded.
I mean, the spiky pentagram ones are pretty simple and cool and shouldn't be left out as often as they are.
The rest, though, yeah, those can stay in the depths.
@@walugusgrudenburg3068 its probably because a lot of school curriculums leave out stars from being regular polygons/polyhedra (for no real good reason other than simplicity, i guess). if those educational channels want to help people with schoolwork they might leave out something a bit more complicated
100th like
Yeah but it would be reasonable to limit it to finite ones, constructed with flat polygons.
This would include the star polyhedra, but exclude:
the petrials (cause those ain't flat polygon faces)
the tilings (they're infinite)
and the petrie coxeter polyhedra (which are both infinite and don't have flat polygonal faces)
The restriction removed from the platonic solids is just that edges are now allowed to intersect.
I've watched this so many times. I enjoy your content a ton dude!
I was almost expecting to see a reference to origami, especially crease patterns, tesselations and 3D modulars by the time you were talking about "blended apeirohedra" in 3D.
As a mathematician, I can not thank you enough for doing something like this. I'm no expert on geometry, but regular polyhedron and polychora for 4d are some of the things I find the most interesting. Have not finished it yet but just the act of making it is wonderful.
Edit #1: Not done but when you introduce stellated dodecahedrons, you say they are called "stellated" because they are made from stars but this is technically inaccurate. Something being stellated is weirder than that and I am not an expert on the subject but look at en.wikipedia.org/wiki/Stellation.
Edit #2: It is immediatly noted that another way of thinking about it is the formal Stellation thing but so nvm I guess.
I always assumed that stellation referred to the fact they looked like stars; a pentagram looks like a pentagon with spikes instead of edges - similarly the faces of a dodecahedron or icosahedron were replaced with pyramids. Each face being uniformly augmented to a point.
For that reason i assumed they weren't regular, but i suppose being thinly defined as stars for faces caught me off guard.
They are however "Stellated" because they look like stars - a pentagram is technically a stellated pentagram
I'm just upset that nobody else is objecting to his use of skew polygons here, which are not actual polygons. Polygons are in fact defined as being 2 dimensional. I had other objections, but that's where I started shouting at my screen.
OmG Are YOu a REaL MatHeMATicIaN?
Theoretically, if you define a regular polygon as any polygon with edges of uniform length which share the property of edge and vertex transitivity where each vertex connects to two edges and each edge to two vertexes (a moderately restrictive definition, but definitely not what we think of as regular polygons) then by all means, skew polygons are entirely valid.
I appreciate the fact that Petrials still have uniform, transitive faces, edges, and vertices, and are rather simple if you understand them
@@signisot5264 but the technical definition of the polygon, in Euclidean space, states that it is a two dimensional figure. You can't have a polygon which extends into a 3rd dimension any more than you could have a polygon with a curved edge, or a square with 120 degree interior angles.
Me: "Don't you have to define that lines in regular polygons can't cross each other?"
Misali: "That's a surprise tool that will help us later"
Mickey Mouse Clubhouse?
@@AdityaKrishnan17293621_Osaka bahaha!
This is now my comfort video essay. I watch it at least once a month
플라톤 입체 이후부터 '하지만 정의에 이런 제한을 걸진 않았죠' 라면서 온갖 괴상한 것들을 들고 정다면체라며 소개하고 어떻게 정다면체인지 설명하는게...
악마는 디테일에 있다는 말이 떠오르고, 수학자들은 모두 악마 같다.
Precisely!
new genre: Lovecraftian geometry
...and the sky hast ruptured, and the f'rty eight harbing'rs of nightmare hast spill'd f'rth from the wound, each bearing the majestic f'rm of one of the regular polyhedrons, devouring space and timeth in their waketh, boiling m'rtal minds with their hideous beauty...
Lovecraft’s geometry is quite distinct from what is covered in this video... he actually described warped space in his books, but those violate the “3D _euclidean_ space” rule
@@gusbates-haus3209 i t s a j o k e
Given how poorly Lovecraft understood geometry in general because he had "too delicate a constitution for math," I am, in fact, truly horrified at the idea of living in a world with a geometry of that man's making.
an intelligent Jewish man discovered Special Relativity (space fucks with time: time dilates and lengths contract as you speed up, etc) and it both personally and philosophically horrified Lovecraft.
The best thing about this video is the increasingly scuffed drawing of all the polyhedra at the end of each part
EDIT: Also I don't know why but seeing and hearing 'part one: what?' made me laugh way too much
And eventually he just gives up on trying to visualize the creations of a geometry PhD with an aversion to diagrams.
Also the golden retriever
Welcome to the jan Misali style of humor.
I love the word scuffed, first encountered it in a speedrun video, it's just a fun word
I love watching the video and knowing what’s going on and slowly fading into madness as he explains tiling
This video. This video has an amazing plot. It's a descent into pure madness. Having finished this video, I feel like I've seen things God never meant a human to see or understand. This video is in the genre of cosmic horror.
We're lured into the video like naïve children during part two. The simple, easy to understand platonic solids are comfortable. Mom is happy.
During part three and four, we start to break "new" ground. We can understand these new shapes, even if they're a bit strange. This makes us feel smart and accomplished, since we've learned about new, pretty shapes. Mom doesn't understand, but she's happy we're learning.
Parts five, six, and seven are a slow, deliberate push into less and less sensible things. We get a glimpse back at where we started with the frequent connections to flat shapes like triangles, squares, and hexagons. It's like trying to think during a fever dream; a repetition of stranger and stranger things, all rooted to something simple. We're starting to go too far. Mom is scared.
Part eight. The diagram beginning this whole thing is so absurd. It's logically put together, but it's just so obtuse. We might as well be looking at some sort of occult magic diagram intended to summon Cthulhu.
Part nine is the climax of the insanity. Not even the creator of these diagrams, the maker of the video, can make sense of the last part. Not even God himself can't make sense of what man has wrought. Mom is dead.
Part ten.
The viewer's life flashes before his eyes. But something is wrong. All the familiar things are corrupted, entangled in the unknowable terror. The unknowable things have so thoroughly corrupted the video that even the past itself can't resist it. It almost sounds like the narrator is speaking another language. A language not ever meant for man's lips to speak.
Mom is dead, and she has always been dead.
Excuse me, i would like to know why you didn't render the entirety of the infinite shapes, it made it a lot harder for me to understand how they truly worked and i would like if next time you rendered the entirety of the infinite shapes. Thanks!
If they are infinite the computer cannot generate it. It could create larger sequences but never infinite.
@@vincenzorutigliano5435 you missed the joke
@@vincenzorutigliano5435
The joke -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Aw shit the entirety of the joke's infinitely long trajectory doesn't fit into this reply. Never mind then.
Dammit, Vincenzo! It was YOUR JOB to end that joke’s infinitely long trajectory!
⭐️
“Wow my brain is starting to go mushy”
“that’s the 15th polyhedra. And from here things are gonna get a lot weirder “
“If the five Platonic solids are fire, water, air, earth, and aether, what are the other 43 regular polyhedra?”
-
Tetrahedron-fire
Cube-earth
Octahedron-air
Dodecahedron-aether/heaven
Icosahedron-water
Small stellated dodecahedron-lightning
Great dodecahedron-wood
Great stellated dodecahedron-ice
Great icosahedron-metal
Petrial tetrahedron-heart
Petrial cube-soul
petrial octahedron-spirit
Petrial dodecahedron-chakra
Petrial icosahedron-dreams
Petrial small stellated dodecahedron-storm
Petrial great dodecahedron-poison
Petrial great stellated dodecahedron-snow
Petrial great icosahedron-rust
Square tiling-mountain
Triangular tiling-volcano
Hexagonal tiling-plasma
Petrial square tiling-desert
Petrial triangular tiling-glass
Petrial hexagonal tiling-bose-Einstein condensate
Blended Square tiling-waterfall
Blended triangular tiling-laser
Blended hexagonal tiling-steam
Petrial blended square tiling-shadow
Petrial blended triangular tiling-light
Petrial blended hexagonal tiling-ghost
Square tiling blended w/ an apeirogon-cannabis
Triangle tiling blended w/ an apeirogon-mind
Hexagonal tiling blended w/ an apeirogon-hellfire
Petrial square tiling blended w/ an apeirogon-explosions
Petrial triangular tiling blended w/ an apeirogon-disease
Petrial hexagonal tiling blended w/ an apeirogon-death
Mucube-space
Muoctahedron-sound
Mutetrahedron-time
Petrial mucube-gravity
Petrial muoctahedron-plant
Petrial mutetrahedron-vacuum energy
Halved mucube-chaos
Petrial halved mucube-oblivion
Skewed Petrial muoctahedron-cheese
Skewed muoctahedron-love
Facetted halved mucube-god
Petrial facetted halved mucube-existence
oh gosh
What abt bug type
wow ! i love shapes!
back four months later, i still love shapes!!
Him: It has to be in _Euclidean_ 3-space
Me: NOOOO Not my Order-4 Dodecahedral Honeycomb!
:(
That's a polychoron, no?
@@anselmschueler No, it's a hyperbolic honeycomb
You are both correct.
@@metachirality If you count a hyperbolic honeycomb as a polychoron, then you have to count the 2D hyperbolic tilings (Such as the heptagonal tiling) as polyhedra.
It's just good manners!
The further this went the more it felt like the insane ramblings of a math thatcher gone off the deep end
Thatcher!
gender-neutral bathroom but with math
There is no such thing as polyhedra. There are only individual edges and vertices, and there are faces.
a thatcher is just a British manufactured bathroom
@@slimsh8dy specifically a gender neutral british manufactured bathroom
Hey Misali, im rewatching this video, because im kinda in my math mood rn and i really love your content, if you were to remake this id recommend you look up 'minimal surfaces'... that would be a meaninful way to fill in 2d surfaces with vertecies in 3d space
a board game for geometrists where the entire path is just the schläfli map of all 48 regular polyhedra, and in order to move to the next square, you have to be able to name the shape you're on
I wish I could back in time and tell HP Lovecraft that we didn't even need to leave Euclidean space to have terrifying geometry
funny
I wish I could go back in time and tell him that he's a racist prick.
@@bored_person beat me to it
@@bored_person Both? Yeah let's do both.
I do think it's important to note that a majority of these polyhedra are abstract algebra constructs that cannot meaningfully exist in a physical space.
The geometry version of “But wait there’s more”
Say goodbye to the 69 likes
Mitch, I hate to point out an omission in this masterpiece of educational content, but using your definition, there is a fourth regular tiling, which would add at least one, but probably more polyhedra to your list. I am talking about the regular tiling of hexagrams. And to be clear - a hexagram is a *fundamentally* different shape than the compound of two equilateral triangles. If you disagree, I would love to persuade you. Anyways, this is one of my top 5 favourite videos on youtube, thank you so much for making it :D
The hexagram isn't fully connected.