there are 48 regular polyhedra 

plato: a regular polyhedron has equal edges and equal vertex angles diogenes: *holds up infinite square tiling* behold, a regular polyhedron

For the people who read the comments first: A cube is made up of 4 hexagons.

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"

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.

17:02 "There's nothing in the definition that restricts polygons to two dimensions" *Dear God*

Thanks for being brave enough to stand up to Big Shape.

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

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!

*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*

The moment you realise there are geometry Discord servers dealing in illegal polyhedra.

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.

The fact that this video codifies the names for some of the polyhedra it describes is amazing.

Bart: There are 48 regular polyhedra. Homer: There are 48 regular polyhedra so far.

“I’m making this for general audiences” *15 minutes later* : D A R K G E O M E T R Y

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

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

Jan misali: *big smart words* Me: cool shapes go spinny

"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.

This video felt like someone explaining to my how geometry is just an elaborate ARG, I love it

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.

Reeling from the ramifications of Big Shape hiding Dark Geometry from me.

“Roll the 50 polyhedra” “All we have is 48 polyhedra and 2 marbles” “Close enough”

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.

“Dark geometry”… never knew I needed this in my life

I never thought I would hear the words “dark geometry”

I'm actually astonished that this incredibly loose definition of a polyhedron does not lead to an infinite number of regular polyhedra.

i like that all of these videos become utterly incomprehensible in the second half

Just seeing the spinning truncated octahedron made my day. Truly my favorite shape

This is why golden retrievers shouldn’t be allowed to study math.

“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.

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

새로운 정다면체의 정의와 이걸 기존에는 정다면체로서 이야기 못했다는점과 이 혼돈의 카오스 스크립트를 전부 번역했단게 전부 놀랍다.... 특히 번역하신분 ㄹㅇ..

They make sense as soon as you rip the skin off geometry and start reorganizing the algebraic bones in otherwise impossible shapes.

"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?

I'm not kidding, this is literally comfort media to me.

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 must be that crazy "crystal math" stuff I've heard about on the news.

This feels like a video that years from now will be the equivalent of what the "Turning a sphere inside-out" video became.

I love the increasing asterisks at the beginning of the video just getting more and more specific. Math really do be like that sometimes.

Some architects are gonna have the time of their lives designing like this.

"The dark side of the geometry is a pathway to many shapes some consider to be... unnatural..." -Grünbaum, probably

Making a shirt with a petrial cube and the caption "This is not a cube" to feel superior to my unenlighted peers.

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.

This is just mathematicians taking a break from whatever they were doing and going "you know what would be really cool..."

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

Him: It has to be in _Euclidean_ 3-space Me: NOOOO Not my Order-4 Dodecahedral Honeycomb!

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 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.

"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."

"dark geometry" is the most intimidating phrase I've heard all year

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

this is unironically one of my favourite videos on youtube

The geometry version of “But wait there’s more”

The further this went the more it felt like the insane ramblings of a math thatcher gone off the deep end

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!!!

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 💯

The universe is extremely lucky that we have a linguist who loves shapes.

The “Big Shape” I’m figuratively dying

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!!!

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.

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.

What exactly IS a polygon? A miserable pile of vertexes.

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."

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.

"there's nothing restricting polygons to 2 dimensions" oh yeah? then why am i standing here with a hammer? get back in 2d

Before watching: I can't believe general education channels ignored such an important fact! After watching: oh.

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.

one of the best geometry videos I've seen in a long while, thank you!

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"

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.

This is and very probably always will be my favorite video on the entire platform.

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

At this point, we should just redefine a regular polyhedron as also having a defined (or definable) volume, to stop mathematicians from going mad.

alternative title: man bullies shapes for 28 minutes straight

I've watched this so many times. I enjoy your content a ton dude!

wow ! i love shapes! back four months later, i still love shapes!!

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.

new genre: Lovecraftian geometry

the fact that there is a polytope discord with someone named "compund of 48384 penaps" is hilarious and entirely unsurprising

I love watching the video and knowing what’s going on and slowly fading into madness as he explains tiling

"The technical name for this is 'a zig zag'" You know, I'm something of a mathematician myself.

"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

I appreciate your knowledge of the difference between number and amount as well as the difference between fewer and less.

This is now my comfort video essay. I watch it at least once a month

that's the second air bud joke in the edutainment sphere this week

"The technical name for this shape is a zig-zag" Technically gonna have to give you this one, that's technically true

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 😊

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 😅

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

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

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

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

“Wow my brain is starting to go mushy” “that’s the 15th polyhedra. And from here things are gonna get a lot weirder “

"what even is this spiky thing?" *KIKI*

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!

“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

This is the RU-vid equivalent of hard liquor.

this has the same level of "woah holy shit" as that "turning a sphere inside out" video

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.