there are 48 regular polyhedra
ฝัง
- เผยแพร่เมื่อ 14 พ.ค. 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
![[Vowel]물고기는 물에서 살아야 해🐟🤣Fish have to live in the water #funny](http://i.ytimg.com/vi/3G9MH063Cx4/mqdefault.jpg)
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.
😂
“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
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
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 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?
Reeling from the ramifications of Big Shape hiding Dark Geometry from me.
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
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
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!
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
The fact that this video codifies the names for some of the polyhedra it describes is amazing.
This is how you get Thagomizers.
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
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.
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 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!
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 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 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
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
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.
*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
"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.
This video felt like someone explaining to my how geometry is just an elaborate ARG, I love it
“Dark geometry”… never knew I needed this in my life
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 🤣🤣
The moment you realise there are geometry Discord servers dealing in illegal polyhedra.
Oh shit
@@gameplaysuffering1620 *oh no*
Oh God
Oh zoinks
Oh My
i like that all of these videos become utterly incomprehensible in the second half
It's not incomprehensible?
Just seeing the spinning truncated octahedron made my day. Truly my favorite shape
"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
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?
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☺️
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
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.
The universe is extremely lucky that we have a linguist who loves shapes.
Some architects are gonna have the time of their lives designing like this.
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
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" 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
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.
this is unironically one of my favourite videos on youtube
nice pfp
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!
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
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
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_ .
"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 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
I love the increasing asterisks at the beginning of the video just getting more and more specific. Math really do be like that sometimes.
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
"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
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!!!
This is just mathematicians taking a break from whatever they were doing and going "you know what would be really cool..."
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.
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!
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 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.
"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!
“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.
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 💯
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!!!
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
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
one of the best geometry videos I've seen in a long while, thank you!
This is and very probably always will be my favorite video on the entire platform.
This is why we need the term "Platonic solids": So we don't have to keep saying "regular closed convex polyhedra up to Petrie duality."
why not just define "platonic polytopes" as being closed, finite and orientable and then have them be:
vertex-transitive edge-transitive face-transitive cell-transitive etc.
but more specifically, we can define an n-dimensional analogue of vertices/edges/faces/cells/etc recursively by only allowing "platonic polytopes" as counting, essentially meaning that a platonic polytope must have its vertices/edges/faces/cells/etc made of platonic polytopes in order to count as a platonic polytope.
then, **i think**, we get the intuitive notion of the generalisation of a platonic solid.
@@UnordEntertainment That's essentially what they already do. It's part of the definition of regularity. Note that even the abstract polyhedra mentioned in this video are composed entirely of regular polygons. Similarly, regular polychora are composed entirely of regular polyhedra. The general rule is that they have to have every possible symmetry. They have to be transitive on every flag (vertex, edge, face, facet, etc.). If we further require them to be closed (thus finite) and convex (thus not self-intersecting), we get the usual list (up to Petrie duality).
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
I've watched this so many times. I enjoy your content a ton dude!
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.
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
The geometry version of “But wait there’s more”
Say goodbye to the 69 likes
I appreciate your knowledge of the difference between number and amount as well as the difference between fewer and less.
I love watching the video and knowing what’s going on and slowly fading into madness as he explains tiling
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.
"The Petrial mutetrahedron can either be derived either as the Petri dual of the mutetrahedron or as the skew dual of the dual of the Petrial halved mucube" what did i just watch
Idk man I need to learn those stuffs
Nice rap verse
Reading this exactly when he said it spooked me
I read your post out loud and by bed started floating please help
@@memeulous4ft247 no one can help you now, sorry
I come back to this video so often!
I really hope a good math channel or even this channel would make a follow-up video!
And yes, I also don't understand why people write about such shapes without having pictures of them 😅
This is now my comfort video essay. I watch it at least once a month
1:33 "We can plot any two points in space and connect them to form a line segment"
7:04 "... but there's nothing in the rulebook that says a golden retriever can't construct a self-intersecting non-convex regular polygon"
That just went from 0 to 100 real quick!
"But there's nothing in the rulebook that says a golden retriever can't.." I've watched this video about eight times and just now understood the air bud joke. Quality content
Literally same I only just got this joke on this viewing thanks to Vsauce XD.
Never saw that, but got it from context, and knowledge of goldens. 🙂
@@lvlupproductions2480 how vsauce ?
@@adithyan9263 He references that line in Air Bud at one point
what is the joke?
Thank you very much. Closing one's eyes and visualising these 48 polyhedra one by one may lead to some form of enlightenment. 🙃
you actually introduced me to video essays with this, so thanks
“Wow my brain is starting to go mushy”
“that’s the 15th polyhedra. And from here things are gonna get a lot weirder “
"there's nothing in the rulebook that says a golden retriever can't construct a self-intersecting non-convex regular polygon" is maybe the most jan misali sentence that's ever been jan misali'd
wow ! i love shapes!
back four months later, i still love shapes!!
Thank you for this video. I cried in distress multiple times when watching it. Truly horrifying and very informational.
That moment when you stay in the wrong class first day of school because you’ve been there so long it would be rude to leave
I’m fascinated but horrified
Happened to me once xD School gave the wrong schedule and I ended in a class I shouldn't be.
And yet somehow it makes perfect sense to you, but you know it will evaporate out your brain when the class stops...
"The technical name for this is 'a zig zag'"
You know, I'm something of a mathematician myself.
lmao
Brilliant
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.
Mind blown. I LOVE this stuff! Note: dragged here by my daughter after I was watching Adam Savage's Rhombic Dodecahedron One Day Build. I have now found a new TH-cam obsession.
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.
Virgin tetrahedron: well known, invented and defined centuries ago, known by children
Chad stellated dodecahedron: barely known, curiosity of geometry nerds and professors
THAD dual of petrial halved mucube: consumes infinite 3d reality to simply exist, still only known by a few researchers, impossible for mere humans to comprehend or visualize
@Eric Lee Honestly that felt like what this video was for me, as a dude with a MSc in Psychology who never had any sort of geometry in college other than my own personal curiosity since age 13 in high school lol. Structural model equations in statistics is the closest I've done to anything geometry related.
I'm ABSOLUTELY using this shiz in my next D&D session.
I'll be honest, I got entirely lost around the 20 minute mark.
Even still, this was a fantastic video that very clearly explained the more complex natures of fuckin shapes, and I loved every moment of it!
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.
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.
watching this felt like physically sinking into the lovecraftian void of my calc textbook. i geniunely believed i could have no further hatred for a branch of mathematics in my life. i think i burned a few brain cells watching this. thank you.
I’m currently taking a discrete math 2 course and we just got to the graph theory section where we learned about Schläfli symbols. I thought I kinda understood that, but this video blew my mind lol 🤯.. in a good way 😊
this is one of my top favorite videos on all of youtube
At 10:00, when you first showed the numbers as representing shapes, it *immediately* clicked that we’d be using stars as vertice numbers and I audibly groaned “oh goooood”
oh good or oh god?
if hes groaning then its probably oh god
@@NoName-rd6et Or he’s being sarcastic.
@@mariafe7050 rrrrrrrrr
@@AshtonSnapp
Internet thread go brrrrr
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.
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!