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!!
“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 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 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 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?
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"
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
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."
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 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.
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.
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.
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.
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.
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.
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!
"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.
*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 would like to have it known that this video is responsible for one of my most “in character” moments of all time. My brand new girlfriend got in my car for the first time and said “Ooh! I get to find out what music you listen to.” All I could do was press play. At 23:30. This is not music. I was LISTENING to a video about Geometry while driving. I was listening to a video about DARK GEOMETRY while driving
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.
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
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.
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
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.
@@zzasdfwasno, 8/2 in this comment means two square stacked on each other, one of them is tilted 45 degrees. (so it looks like a kind of 8-star) and in this video, the "regular polygon" has to be a single connected shape (you can travel to a vertex to any other vertex by moving along the segments) so it's still true that the 8/2 should not be allowed in this video's context.
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.
@@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 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
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
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
@@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.
"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."
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!
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).
"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.
"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
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.
@@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.
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”
"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
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.
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.
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.
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.
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
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.
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!
This video literally reduced me to tears. First in laughter, and that slowly devolved into sobs. I think this is only half because of the sleep deprivation
@@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 :)
@@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.
Watch some of AntVenom's videos on the true structure of Minecraft's farlands. It varies by version and edition but generally the region that has normal minecraft world generation and physics is less than a trillionth of a trillionth or a trillionth of the area one can hypothetically visit. From what I recall of a fairly old version: most of it has no ground at all, only clouds, and normal motion is impossible because position is too discretised for you to move, so you can only teleport. Most of the remainder is corner farlands that have intangible ground. Most of the remainder is edge farlands that are similar. Most of the rest is corner farlands that are at least tangible. Most of the rest is edge farlands that are similar. Most of the rest is normal terrain with noticeably jerky movement. The tiny remaining part is the "normal" minecraft world.
@@SimonClarkstone I watched the first few seasons of KurtJMac's Far Lands or Bust when it was coming out weekly, friend. That stuff's just IEEE 754 double precision errors in perlin noise generators. This? This nonsense is what melts brains.
This entire video is amazing but one of my favourite parts is at the bottom of the iceberg, where one of the shapes is accompanied by “(DO NOT RESEARCH THIS)”, like it’s an SCP or something.
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
At a certain point these videos make me want to start crying, partly out of frustration/not understanding and partly out of wonder and sheer admiration for the world we live in.
This is really impressive. I'm a PhD student in mathematics and had never come across many of the things you mentioned. Extraordinary research! As for why "anyone would write a geometry paper without including any diagrams of the shapes they're talking about", I believe most mathematicians would consider the abstract interpretation of a geometric structure considerably easier to grasp and less complicated to "do mathematics with" than the actual shapes. For example, it's often easier to understand and prove properties about polytopes in terms of their isometry or reflection groups than by looking at their shapes (you can tell, for instance, what other regular polytopes can/cannot be immersed within a polytope by studying its isometry subgroups). The graph structure (and its homology) is similarly helpful. That said, intuition often arises from looking at something from a perspective we're not really familiar with, which may as well be a purely geometric one. I thought I was already subscribed, but in any case, subscribed again.
I have no idea about how i got here and dont understand how there can be so many people who understand what is going on and what is the real life use of all of this since so many people seem to study it, too advanced, help me
@Null Pointer Wow, are you one of the authors of that 1997 article? That's exciting! I couldn't really grasp everything in the paper, but found it very interesting nonetheless (despite the lack of "nice pictures to look at" :D).
...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
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.
I should probably get some sleep _janMisali uploads_ Oh cool, Numberphi-- oh. Lets go. Edit: you said this was gonna be a math video not a conlang video
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.
Being a mathematician-in-training, yes that is the 'general' introduction. The 'specific' introduction has a prerequisite of first year university mathematics.
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?
Me 5 mins in: Oh yes, this is reasonable Me 10 mins in: Wow, I'd never think about that. Nice. Me 15 mins in: ...Why would you do this?! Me 20 mins in: *Insanity*
Me 30 minutes after the video. Dazed. Then I discover that this is simple. Its just an extension of a quantum state! ... Meaning at one point in time I both DO and DO NOT believe I understand what I am watching, as I rewatch it for the 4th time. Meaning its a four dimensional quantum state of uncertainty across the axis of time?! (sarcasm lol)
i believe this may be one of my favorite jan Misali videos solely for its absolute disregard for what i consider a shape and my personal safe little bubble of shapes. thank you, Mitch, for giving me a new favorite polygon: the pentagram.
The pentagram? C'mon, there's the apeirogon of infinite sides meaning that the external angle of all of them is 180° so the polygon is actually a non-curved line segment but it can't be a line segment in 1D space since you need 2 coordinates to define a point in it yet it is.
@@user-pr6ed3ri2k Duo-cylinder Two circles made perpendicular to eachother in the fourth dimension and then connected Great Grand stellated hecatonicosachron A stellated, greatened, and grandified version of the 120-cell which is a 4d shape made up of 120 dodecahedra
22:01 I did not know it was possible to be jumpscared by the next step of a calm explanation of geometry. Now I do. I think I gasped aloud the first time I watched this and got to that part. Good stuff.
As a math soon-to-be major, I just can't resist the urge to engage with this kind of content! Surprisingly, this sort of geometric, classificatory, finite and not-very-abstract math is (unfortunately) not discussed in many circles I'm a part of. I guess "real" mathematicians like to spend their days solving infinite-dimensional equations or whatever. So, thanks! I also want to thank you on the amount of work and research you must have endured. Also, can we have a link for the polytope discord? I'd like to point that just because there are infinitely many polygons, doesn't mean it's boring; it's that it's too easy to classify them. You choose the number of vertices and it... just works, no strings attached. It's also simple to find the intersecting ones by number theory. That's the real interest with 3+ dimensions: it's much harder to produce regular solids than regular polygons. Directly answering your question about geometry papers, what matters about the polyhedra is the inherent symmetries it has, and also, shape alone can't distinguish between solids. Well, then we could simply equate the polyhedra with some of its properties, and discard the visual/positional necessity altogether. Then, we are dealing with an abstract object, defined not by its visuals but by its relations. All the information it contains can be described in that small set of numbers and words. Then there is no incentive to ever take the time and produce a visual representation, since none of the people engaging with it are expected to use a visual model. This is much more precise and easier to manipulate (with math tools) although admittedly much less intuitive. This leads me to my last point. Even with that fixed definition of regular polyhedra, how do you know that the list ends there? How can you be sure that an extra bendy, different line arrangement or something can't give rise to a new polyhedra? In other words, why is this list complete? (EDIT: after a quick look at the reference paper, this classification result is very similar to the one at part two, but instead of spacially combining polygons, you instead look at the symmetries themselves and just combine them until there are no more ways to do so)
I'm just upset that nobody is objecting to when he ventuered into pretrial "polyhedra," and said that there is nothing in the definition of polygon that restricts polygons to 2 dimensions. *Yes. There is.* It's one of the core defining elements. He might as well have said "there's nothing in the definition of polygon restricting the line segments to being straight, so here are some polygons with curved lines."
@@LeoStaley The definition for polygon used is: "a polygon is a shape made out of line segments(edges) where the defining endpoints(vertices) are each shared by exactly two line segments" None of this restricts the edges in question to a flat plane. The whole point of the video is to show all the places you can go if you don't also restrict the definition to "no self-intersections", "polygons must be 2D", "polyhedra must be enclosed" and probably another that I've missed. Those extra restrictions are often necessary. If you want to build a container that's a regular polyhedron, then the petrial mucube isn't going to be much use to you. But the point is these restrictions are imposed by us, and if we choose to remove them we can find new and interesting mathematical shapes that still hold to a formal definition of a polyhedron. Someone said it elsewhere in the comments, but is it not intriguing that even removing these assumptions, and relaxing the definition of regular polyhedra there is still a finite number of them?!
@@Minihood31770 that isn't the normal definition. That is much looser than the technical definition normaly used. The normal definition can be found on Wikipedia.
@@LeoStaley Let me try and give a bit of deeper intuition. The standard technical view of a regular polygon is a set of n vertices, all symmetrically equivalent, and a set of edges, all symmetrically equivalent. This definition agrees with the standard one so long as we restrict symmetries to mean rigid movements in the 2D plane. Now, when we pass to 3 dimensions, it's our interest to define this for polyhedra. Again, shape, position and scale shouldn't matter, so we look at the set of symmetries. But, if we insist that polygons remain flat, we have a problem. Because we now can perform symmetries in all of 3D space, to check that a thing is a polyhedra, we have to check that the symmetries of edges don't escape their plane, which is an unnatural condition and hard to verify. In other words: the natural algebric definition of a polyhedra is a good theoretical basis for the geometric polyhedra, but it does not need to contain geometric polygons. So, to ease the study of these objects, we can expand the definition of polygons. Or we can just ignore them; it's not like the fundamental structure of an object needs a name to exist. Fun observation: this algebric definition of polygons does cover curved edges. If all edges are symmetrical then the curve itself doesn't matter, and the symmetries are the same as an usual non-curved polygon. A Reuleaux triangle has the same set of symmetries as a regular triangle, and so it counts as the same thing (the same way two triangles with different size count as the same type of polygon, despite not sharing most of its points)
these are the kinds of shapes i spent late nights browsing wikipedia to find out about... thanks for the vid, i thought i knew about some weird polyhedra but this blew me away!
![Noob To Max With DRAGON REWORK In Blox Fruits [FULL MOVIE]](http://i.ytimg.com/vi/LnBMOinoOvA/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.
😂
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 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 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
Neat.
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.
“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.
And big flat is hiding 2 and 1-gons!
@NoName-sv2uz where is square
@@Radio_ink114 in *The Plane*
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
The moment you realise there are geometry Discord servers dealing in illegal polyhedra.
Oh shit
@@gameplaysuffering1620 *oh no*
Oh God
Oh zoinks
Oh My
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
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
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."
It truly feels like the magic professor doing class.
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'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.
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!
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.
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?
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 🤣🤣
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 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
@@soupisfornoobs4081 *but it's good to know and you should probably, and in a friendly manner, remind them of so.
@@polygontower yes thank you not every correction on the internet has to be hostile
"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.
*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
"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 would like to have it known that this video is responsible for one of my most “in character” moments of all time. My brand new girlfriend got in my car for the first time and said “Ooh! I get to find out what music you listen to.”
All I could do was press play. At 23:30.
This is not music. I was LISTENING to a video about Geometry while driving. I was listening to a video about DARK GEOMETRY while driving
🌿that is the best kind of video to be caught listening to
sounds fun tbh
are you still together
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.
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
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’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
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_ .
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 universe is extremely lucky that we have a linguist who loves shapes.
The fact that this video codifies the names for some of the polyhedra it describes is amazing.
This is how you get Thagomizers.
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.
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!
“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.
Thank u my guy
@@zzasdfwasno, 8/2 in this comment means two square stacked on each other, one of them is tilted 45 degrees. (so it looks like a kind of 8-star)
and in this video, the "regular polygon" has to be a single connected shape (you can travel to a vertex to any other vertex by moving along the segments) so it's still true that the 8/2 should not be allowed in this video's context.
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
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 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 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!
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 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
"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
thought you were gonna hit misali with it 😭
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
@@MegaDudeman21There's at least 50 American education systems
"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
"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
@@CastafioreOnRUclips Dear god...
SCP-478+23i
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!
This video felt like someone explaining to my how geometry is just an elaborate ARG, I love it
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).
"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!
"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?
i like that all of these videos become utterly incomprehensible in the second half
It's not incomprehensible?
@@trappedcosmosthe caveat is: for mere mortals like me and OP. if you get it, cg
Experiencing horror the way Lovecraft intended.
@@trappedcosmos they are to me
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.
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
"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'm not kidding, this is literally comfort media to me.
“Wow my brain is starting to go mushy”
“that’s the 15th polyhedra. And from here things are gonna get a lot weirder “
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.
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.
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?
"The technical name for this shape is a zig-zag"
Technically gonna have to give you this one, that's technically true
I've decided that this is a new form of torture. The fact that I still watched it and clicked on the like button changes nothing.
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.
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!
"The technical name for this is 'a zig zag'"
You know, I'm something of a mathematician myself.
lmao
Brilliant
I never thought I would procrastinate doing maths homework by watching more complicated maths
yep
yea :v
It's the circle of math
I never thought I would procrastinate on ART homework by watching math
Its better because you don't understand it
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
Just seeing the spinning truncated octahedron made my day. Truly my favorite shape
This video literally reduced me to tears. First in laughter, and that slowly devolved into sobs. I think this is only half because of the sleep deprivation
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!!
This is why golden retrievers shouldn’t be allowed to study math.
Racist
...
@@sineadthomas2024 Ok millenial
@@doommaker4000 Ok racist
Doom Maker Ok Boomer
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☺️
It's been a _very_ long time since mathematics has made me feel existential dread.
Well done.
Vsauce
Not since Calculus II *shudders*
Watch some of AntVenom's videos on the true structure of Minecraft's farlands. It varies by version and edition but generally the region that has normal minecraft world generation and physics is less than a trillionth of a trillionth or a trillionth of the area one can hypothetically visit. From what I recall of a fairly old version: most of it has no ground at all, only clouds, and normal motion is impossible because position is too discretised for you to move, so you can only teleport. Most of the remainder is corner farlands that have intangible ground. Most of the remainder is edge farlands that are similar. Most of the rest is corner farlands that are at least tangible. Most of the rest is edge farlands that are similar. Most of the rest is normal terrain with noticeably jerky movement. The tiny remaining part is the "normal" minecraft world.
@@SimonClarkstone I watched the first few seasons of KurtJMac's Far Lands or Bust when it was coming out weekly, friend. That stuff's just IEEE 754 double precision errors in perlin noise generators.
This? This nonsense is what melts brains.
You spend way too little time thinking about math then.
The geometry version of “But wait there’s more”
Say goodbye to the 69 likes
(puts 6 squares around a common vertex)
Everyone: wait, that's illegal
This man: nah mate that's valid
If it's not explicitly forbidden, then it is allowed. First and only law of thinking outside the box.
the fact that there is a polytope discord with someone named "compund of 48384 penaps" is hilarious and entirely unsurprising
hey, my boyfriend owns that polytope discord, this video made his discord grow alot and thats pretty epic
@Eric Lee yeah why wouldn't i be?
Are you homisexual?
@@metachirality well you're the founder so you still have more power than the owner
@@metachirality and its still technivally your server
@@metachirality thats not possible, you the discord server so no matter what rank you are you will always have more power than everyone
As a mathematician I didnt expect to be so surprised, floored, and awed at different ways to consider polygons. Stellar work as always!
This entire video is amazing but one of my favourite parts is at the bottom of the iceberg, where one of the shapes is accompanied by “(DO NOT RESEARCH THIS)”, like it’s an SCP or something.
I *think* it's a reference to the 1995 Mario 64 creepypasta?
all it is, is a seven-dimensional shape, not that scary
keterean geometry
@@somerandomgoblin2583 Yes
my favorite is "zigzag" being in the second lowest tier of the iceberg
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
At a certain point these videos make me want to start crying, partly out of frustration/not understanding and partly out of wonder and sheer admiration for the world we live in.
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 increasingly degenerative drawings of all of the polygons are fantastic.
lmao thats true
Yes
This is just mathematicians taking a break from whatever they were doing and going "you know what would be really cool..."
This is really impressive. I'm a PhD student in mathematics and had never come across many of the things you mentioned. Extraordinary research!
As for why "anyone would write a geometry paper without including any diagrams of the shapes they're talking about", I believe most mathematicians would consider the abstract interpretation of a geometric structure considerably easier to grasp and less complicated to "do mathematics with" than the actual shapes.
For example, it's often easier to understand and prove properties about polytopes in terms of their isometry or reflection groups than by looking at their shapes (you can tell, for instance, what other regular polytopes can/cannot be immersed within a polytope by studying its isometry subgroups). The graph structure (and its homology) is similarly helpful.
That said, intuition often arises from looking at something from a perspective we're not really familiar with, which may as well be a purely geometric one.
I thought I was already subscribed, but in any case, subscribed again.
I have no idea about how i got here and dont understand how there can be so many people who understand what is going on and what is the real life use of all of this since so many people seem to study it, too advanced, help me
On "Abstract interpretations vs diagrams", is there any potential reason against doing both?
@Null Pointer Wow, are you one of the authors of that 1997 article? That's exciting! I couldn't really grasp everything in the paper, but found it very interesting nonetheless (despite the lack of "nice pictures to look at" :D).
@@LowestofheDead I'm no geometer, but maybe not to bloat an otherwise elegant straightforward article or just because of the sheer work required.
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.
I should probably get some sleep
_janMisali uploads_
Oh cool, Numberphi--
oh.
Lets go.
Edit: you said this was gonna be a math video not a conlang video
lol
I actually just sorta started hearing noises more than words when he got to the recap.
@@barmacidic2257 i was feeling the beat of his voice and not hearing the actual words, lol
I kinda like to think that he went "oh I should upload this to the internet so I confuse some minds"
Lel
petridualofthemutetrahedronorasaskewdualofthedualofthepetrialhalvedmucube
I love the increasing asterisks at the beginning of the video just getting more and more specific. Math really do be like that sometimes.
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.
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.
Every jan Misali video has some tipping point in it where it begins to feel like a mathematical or linguistic (or both) Junji Ito story
Like Junji Ito, this video includes spirals that make my head hurt trying to understand them.
@@dappercuttlefish9557 oh god no, anything but UZUMAKI
this is unironically one of my favourite videos on youtube
nice pfp
“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?
Me 5 mins in: Oh yes, this is reasonable
Me 10 mins in: Wow, I'd never think about that. Nice.
Me 15 mins in: ...Why would you do this?!
Me 20 mins in: *Insanity*
Me 25 mins in: head explodes
Me 30 minutes after the video. Dazed. Then I discover that this is simple. Its just an extension of a quantum state!
...
Meaning at one point in time I both DO and DO NOT believe I understand what I am watching, as I rewatch it for the 4th time.
Meaning its a four dimensional quantum state of uncertainty across the axis of time?! (sarcasm lol)
@@pathwaystoadventure i feel like if you tried hard enough, you could publish that as an new field of quantum physicis
@@MatheusAmaral23 I should! It would continue the fine tradition of psychologists misinterpreting hard science!
i believe this may be one of my favorite jan Misali videos solely for its absolute disregard for what i consider a shape and my personal safe little bubble of shapes. thank you, Mitch, for giving me a new favorite polygon: the pentagram.
The pentagram? C'mon, there's the apeirogon of infinite sides meaning that the external angle of all of them is 180° so the polygon is actually a non-curved line segment but it can't be a line segment in 1D space since you need 2 coordinates to define a point in it yet it is.
@@Dexuz you have a very valid point but my reasoning is mostly that "the pentagram looks cool hee hee"
I'm partial to the duocylinder and the great grand stellated hecatonicosachoron
^ the person above me is saying real non nonsensical words ^
@@user-pr6ed3ri2k
Duo-cylinder
Two circles made perpendicular to eachother in the fourth dimension and then connected
Great Grand stellated hecatonicosachron
A stellated, greatened, and grandified version of the 120-cell which is a 4d shape made up of 120 dodecahedra
22:01 I did not know it was possible to be jumpscared by the next step of a calm explanation of geometry. Now I do. I think I gasped aloud the first time I watched this and got to that part. Good stuff.
As a math soon-to-be major, I just can't resist the urge to engage with this kind of content!
Surprisingly, this sort of geometric, classificatory, finite and not-very-abstract math is (unfortunately) not discussed in many circles I'm a part of. I guess "real" mathematicians like to spend their days solving infinite-dimensional equations or whatever. So, thanks! I also want to thank you on the amount of work and research you must have endured. Also, can we have a link for the polytope discord?
I'd like to point that just because there are infinitely many polygons, doesn't mean it's boring; it's that it's too easy to classify them. You choose the number of vertices and it... just works, no strings attached. It's also simple to find the intersecting ones by number theory. That's the real interest with 3+ dimensions: it's much harder to produce regular solids than regular polygons.
Directly answering your question about geometry papers, what matters about the polyhedra is the inherent symmetries it has, and also, shape alone can't distinguish between solids. Well, then we could simply equate the polyhedra with some of its properties, and discard the visual/positional necessity altogether. Then, we are dealing with an abstract object, defined not by its visuals but by its relations. All the information it contains can be described in that small set of numbers and words. Then there is no incentive to ever take the time and produce a visual representation, since none of the people engaging with it are expected to use a visual model. This is much more precise and easier to manipulate (with math tools) although admittedly much less intuitive.
This leads me to my last point. Even with that fixed definition of regular polyhedra, how do you know that the list ends there? How can you be sure that an extra bendy, different line arrangement or something can't give rise to a new polyhedra? In other words, why is this list complete? (EDIT: after a quick look at the reference paper, this classification result is very similar to the one at part two, but instead of spacially combining polygons, you instead look at the symmetries themselves and just combine them until there are no more ways to do so)
the polytope discord is a sacred place. you don't find the link to it, the link finds you
I'm just upset that nobody is objecting to when he ventuered into pretrial "polyhedra," and said that there is nothing in the definition of polygon that restricts polygons to 2 dimensions. *Yes. There is.* It's one of the core defining elements. He might as well have said "there's nothing in the definition of polygon restricting the line segments to being straight, so here are some polygons with curved lines."
@@LeoStaley The definition for polygon used is: "a polygon is a shape made out of line segments(edges) where the defining endpoints(vertices) are each shared by exactly two line segments"
None of this restricts the edges in question to a flat plane.
The whole point of the video is to show all the places you can go if you don't also restrict the definition to "no self-intersections", "polygons must be 2D", "polyhedra must be enclosed" and probably another that I've missed.
Those extra restrictions are often necessary. If you want to build a container that's a regular polyhedron, then the petrial mucube isn't going to be much use to you.
But the point is these restrictions are imposed by us, and if we choose to remove them we can find new and interesting mathematical shapes that still hold to a formal definition of a polyhedron.
Someone said it elsewhere in the comments, but is it not intriguing that even removing these assumptions, and relaxing the definition of regular polyhedra there is still a finite number of them?!
@@Minihood31770 that isn't the normal definition. That is much looser than the technical definition normaly used. The normal definition can be found on Wikipedia.
@@LeoStaley Let me try and give a bit of deeper intuition. The standard technical view of a regular polygon is a set of n vertices, all symmetrically equivalent, and a set of edges, all symmetrically equivalent.
This definition agrees with the standard one so long as we restrict symmetries to mean rigid movements in the 2D plane.
Now, when we pass to 3 dimensions, it's our interest to define this for polyhedra. Again, shape, position and scale shouldn't matter, so we look at the set of symmetries. But, if we insist that polygons remain flat, we have a problem. Because we now can perform symmetries in all of 3D space, to check that a thing is a polyhedra, we have to check that the symmetries of edges don't escape their plane, which is an unnatural condition and hard to verify.
In other words: the natural algebric definition of a polyhedra is a good theoretical basis for the geometric polyhedra, but it does not need to contain geometric polygons. So, to ease the study of these objects, we can expand the definition of polygons. Or we can just ignore them; it's not like the fundamental structure of an object needs a name to exist.
Fun observation: this algebric definition of polygons does cover curved edges. If all edges are symmetrical then the curve itself doesn't matter, and the symmetries are the same as an usual non-curved polygon. A Reuleaux triangle has the same set of symmetries as a regular triangle, and so it counts as the same thing (the same way two triangles with different size count as the same type of polygon, despite not sharing most of its points)
these are the kinds of shapes i spent late nights browsing wikipedia to find out about... thanks for the vid, i thought i knew about some weird polyhedra but this blew me away!
this has the same level of "woah holy shit" as that "turning a sphere inside out" video
This is incredibly true
Accurate!
That video was my childhood
@regibus361 ruclips.net/video/wO61D9x6lNY/видео.html
@regibus361 here
ruclips.net/video/sKqt6e7EcCs/видео.html