I think my favorite Johnson solid has to be the Snub Disphenoid. The idea that a "digon" (line) has a use case at all as a polygon, despite being degenerate, is just so funny to me.
yes! i get a weird sense of joy using degenerate cases in math, such as for example, 0! = 1actually being intuitive if you think about it, there really is exactly one way to arrange 0 items in a line on your desk after all.
Son: "dad, why is Daisy called like that?" Dad: "because you mother really loves daisys" Son: "i love you dad" Dad: "i love you too Great Rhombicosidodecahedeon III"
Dad, why is Daisy called like that? Because when she was young a daisy fell on her head. And how did you come up with my name? No further questions whilst I'm reading, brick.
I can't describe my panic at the Dungeons & Dragons table looking at my dice and realizing that there were so few regular platonic solids. I bothered my DM about it for weeks. And then finally I saw in a video showed there are very many regular platonic solids as long as you don't care what space looks like, and that put my mind at ease. A good collection of *almost* regular objects is going to seriously put my mind at ease. I should make plush versions of these solids to throw around during other hair pulling math moments. Yeah this is really giving context to the wikipedia deep dive I tried to do. Lots of pretty pictures but they didn't make sense until you showed the animations.
If you want more dice, the catalan solids all make nice fair dice. The disdyakis tricontrahedron makes a particularly great dice, with 120 sides you can replicate any "standard" single dice roll by just dividing the result, since 4,6,8,10,12,20 are all factors of 120.
Plush solids would be so cute! Might want to use mid- to heavy-weight interfacing on the faces so they don't all turn into puffy balls when stuffed with polyfill… although that could be cute, too, especially if you marked the edges somehow, e.g. by sewing on some contrasting ribbon or cord (you could ignore this step or use different colors for the adjacent faces). Now I want to make some 😂 I sewed some plushie ice cream cones recently and have been itching to make more cute things.
rhombic dodecahedron is my favorite among all these guys. i like how unfamiliar it looks even though it has cubic symmetry. and its 4d analogue, the 24 cell, is completely regular! i wish i could look at it, its beautiful
I just started watching this channel and I love how you can visualize and explain all this information in a way that is easy to understand. Great video! 😁
Incredible video, great work on it all! A lot of new names for solids I never knew before A giant grid of all of the solids as a flowchart of different operations to get to them would be a hella cool poster tbh
I had a weird math panic attack when I learned there weren't more platonic solids and that Jan Miseli video really put my mind at ease, and then went even farther and blew my mind a few times. Great video. And his stuff on constructed languages has taught me so much about linguistics that just keeps coming up in my regular language study, it's awesome. Love that guy.
if you take the deltoidal hexecontahedron. and force the kite faces to be rhombi, you get a concave solid called the rhombic hexecontahedron, and it is my favorite polyhedron
this is fast becoming my favorite video on youtube. i'm so happy to see that there are other people out there who care this much about polyhedra. the disdyakis triacontahedron is also my favorite, it's like a highly composite solid! just as 120 is highly composite! this is closely followed by the rhombic dodecahedron (because it's like the hexagon of solids!) and then the rhombic triacontahedron. this video has taught me so much, like how snubs work, and the beautiful relationship between the archimedean and catalan solids. not to mention half triakis (i had always wondered how someone could think up something as complex as the pentagonal hexacontahedron.) and johnson solids! i hadn't even heard of them before this video! thanks for educating, entertaining, and inspiring me! i'm so glad i stumbled across this. 120/12, would recommend
my favourite solid has always been the truncated octahedron because it evenly tiles space with itself, and it has the highest volume-to-surface-area ratio of any single shape that does so. its the best single space filling polyhedra! if you were to pack spheres as efficiently as possible in 3d space, and then inflate them evenly to fill in the gaps, you get the truncated octahedron
I dont think thats quiet true. The shape you get when inflating spheres is a rhombic dodecahedron. You can see this by looking at the number of faces. The truncated octahedron has 14 faces but a sphere only has 12 neighboring spheres.
@@feelshowdy It's not 100% accurate, because not all of the Bejeweled gems are platonic or almost platonic solids of course, but I wanted to include all of them in the comment since they're all so equal to each other.
A few years ago I was very intrigued about a very similar thing, but with tetrominoes, aka tetris pieces. It's well know that there's only 5 ways to connect 4 squares on a plane, with 2 of them being chiral, hence the 7 tetris pieces we all know, but once you start to dig deeper you start to have so many questions. What about 5 squares? 6 squares? 7? What about other shapes, like triangles? Or maybe cubes in 3D, aka tetracubes? What if you keep only squares, but allow them to go in 3 dimensions (they are called Polyominoids)? Turns out there's lots of ways one could extend the idea of tetrominos, by either using different shapes, getting into higher dimensions or simply changing the rules of how shapes are allowed to connect.
I've been interested in that also! Not counting reflections, there are 12 pentominoes, and it's a classic puzzle to arrange them into a rectangle. You can actually make 4 different types of rectangle, 3x20, 4x15, 5x12, and 6x10.
Watching this for the 17th time. Thank you for getting this all this down into one video. I can tell you worked really hard to put all the faces together for this one. 🎉
I saw descriptions about these solids at high school, and couldn't grasp many concepts yet getting really intrigued. Your explanation was excellent. Thank you sooooo much!!
Really fantastic video! You did a beautiful job with the visuals and in organizing the explanation. I have shown it to a wide range of viewers - from a 7 year old to a guy with a phd in math. Everyone loved it and had the same basic reaction - it was entrancing!
pentagonal hexecontahedron is clearly my favorite with it's "petal" sides if you consider 5 faces connected on their smallest angle, or heart shaped sides, if you only consider 2 faces
Thanks! Great video. Have you ever looked at the geometric net of these kinds of solid. I know the cube has 11 possible nets. I would like to see a video that dives into the possible nets of some of the other shapes as well.
Thank you so much! I do have some degree of experience with the nets of the catalan and archimedean solids after making them all out of paper. Some of them I even modified to fit better on 1 piece of paper!
The hebesphenorotunds (last one explained 27:03) looks really similar a gem-cut. Think about the side with the 3 pentagon down into the socket and the hexagon outside and visible.
I watched this whole video and found at least five of my new favorite solids. They will never beat my favorite shape, the snub disphenoid! Also, please make a video on some of the near miss johnson solids.
I am a particular fan of the disdyakis triacontahedron because it is the largest roughly spherical face-transitive polyhedron, so it's the largest fair die that can be made (ignoring bipyramids and trapezohedrons)
after watching jan Misali's platonic solids video and vsauce's strictly convex deltahedra video, seeing some concepts i got from there return here was nice and cool, like a callback from across my brain :3
I was expecting this to be like a reduced version of Jan Misali's video about the 48 regular polyhedra... what a fantastic surprise! I love geometry, those were some great explanations.
I've been looking into these solids for years, but had no idea what the process of discovering them was. Half-truncation is one hell of a leap, especially for someone born a few thousand years too early for computers. It's amazing he found them all
I love this video! I'm glad that I found your videos. I have a love for mathematics and geometry, and it's cool someone made a video about platonic-y solids! I liked the video "there are 48 regular polyhedra" by jan Misali and this is the type of stuff I like. I think you would like that video, too.
Wow, haven't seen so clean, concentrated and convenient explanation, without unnecessary effects it's even easier to understand. Your format is my favorite among others since I went in for geometry 11 years ago. My suggestion for next topic is "3D Honeycombs" because it's logical continuation of solids. There are "regular" ones which consist of the same solids you were talking about in this video. The particularly brilliant thing is there were found some irregular (!) 3D honeycombs. Most of them are of similar polyhedra, both convex and not. The only irregularity in them were the colors which cube faces had or something like this. But maybe there are some of them I missed which look like 3D version of Penrose tiling. Edit: Pentakis Dodecahedron is my favourite solid (the second one is Icosahedron) because it's one of the roundest solids which consists of equal polygons.
My favourite catalan solid is the pentagonal hexacontahedron. I find it very pretty how the flower patterns with 5 petals interlock to make chiral corners at the boundary.
Your color choices for each polyhedron are lovely. This whole video tickles my brain wonderfully. I want a bunch of foam Catalan solids to just turn over in my hands.
Let's face it most underrated youtuber I have ever come across (is you)! Well done and Thank You, you are a wonderful edgeucator c: who always gets even very complicated points across, not to mention the volume of information in each video is enormous!
Amazing video!!! Very in depth and yet easy to follow, I really enjoyed some of the smaller details like sphericity!! i look forward to your future uploads!!! -from another friend of Blahaj ;)
Just discovered your channel and am loving it. You are covering all my favourite topics. I personally find the Catalan solids more beautiful than the Archimedean ones.
The Dice Lab is a company that makes some unusual ones. Their large set has a truncated tetrahedron, truncated octahedton, rhombic dodecahedron, deltoidal icosahedron, disdyakis dodecahedron, deltoidal hexecontahedron, and disdyakis triacontahedron.
This channel is going onto the list. Hopefully once this nightmare of a degree (math) is done I'll have time to get through these interesting videos/topics.
i'm honestly surprised that you've explained it this well, i was able to keep up pretty much the whole time,, i was so shocked that i could understand what was happening i want to commend you for the use of color coding for things like rotundas and cupolas, you've done an amazing job at making this more digestible and it was very helpful excellent job on the video, kuvina
My favorite Catalan solid is the 30-sided rhombic polyhedron based on the Golden Ratio because I figured out how to make it in Sketchup. It is closely related to the icosahedron and dodecahedron.
If you're into Sketchup and geometry then you might find a few videos I've done on my channel to be interesting. Also, you guys know the Sketchup team does a livestream every Friday? Fun times..
I wondered if there are solids where instead of relaxing the properties 2: all faces being the same 3: all corners being the same we relaxed: 1: faces don't have to be regular polyhedra. These solids do exist! But it's a single class of solids. The first thing we can note is that all the angles that are "supplied by the faces" have to be "consumed by the corners". Or in other words, if a face has angles a,a,b,c then a corner has to use up the same amount, or a multiple. That means that each corner could have 3 3-sided faces 4 4-sided faces 5 5-sided faces ... meet. But 5 5-sided faces would make a hyperbolic surface, and 4 4-sided faces just make a distorted square grid. Therefore 3 3-sided faces is the only type of these that can exist (see below). You could also have 3 6-sided faces or 6 3-sided faces meet. But for similar reasons, they'd be distorted planar grids. And combining multiplicities 4 and 8 or 3 and 9 (or above) doesn't work. 2-sided faces don't exist, but we _could_ have 2 4-sided faces meet at each corner. Except that that would just be 2 rectangles back to back with zero volume enclosed. *Thus a distorted tetrahedron is the only type of "fully transitive solids",* as I would call them, that could exist. Or in other words, "cursed d4 dice". And all that remains is to prove that it isn't an impossible construction. (And that the construction from a given set of faces doesn't allow for more than 1 type of solid.) The only problem that distorting a tetrahedron could cause is that making a triangle with 3 angles that aren't all the same is that the edges will have different lengths too. But luckily, any two congruent triangles always share a side of common length, along which we can join them. Let's call that side length "a" and the angles on its ends "beta" and "gamma". You can't join two "beta" or two "gamma" angles together in the same vertex, or you won't get identical corners. (Each vertex has to use one of each angles.) That means we can only join these two triangles with sides "a" against each other and angle "beta" touching "gamma" and vice versa. This shows that the solid can be completed, and that it can only be constructed in a single way. (The two remaining faces will have their edges "a" joined together in the same way. And then edges "b" and "c" can only be joined to edges of the same length. This leaves two possibilities, of which one is just two sets of coplanar triangles - which form a parallelogram - joined back to back, with zero volume.)
twitter.com/kuvina_4 instagram.com/kuvina_4 *Correction* : At the beginning I mislabeled the icosahedron as dodecahedron. (copied textbox but forgot to change text)
I LOVED this video!! I am a huge geometry nerd and learning about polyhedral families and the construction methods to generate new ones makes them all feel so intertwined and uniform. If I may request, please do a video on higher dimensional projections into the third dimension like fun cross sections of polytopes through various polyhedra. TYSM
There is another category of almost platonic solids where you only use property 1 and 2 and don't care about the verticies being identical. These are the triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism and gyroelongated square bipyramid, otherwise known as the irregular deltahedra.
Thank you for such an interesting video. A lot of these I was hearing about for the first time and I found great joy in hearing you pronounce the name, getting surprised that this one is longer than the last one, and then laughing as I struggled to pronounce the name myself. My favorite was either the “Snub Dodecahedron” or the “Pentagonal Hexacontahedron”. The Snub Dodecahedron looks so satisfying having a thick border of triangles around the pentagon, but there was something about that Pentagonal Hexacontahedron that I found really pretty. I think it’s because of the rotational symmetry. Again, thank you for taking the time to make such interesting and engaging videos. I look forward to watching another one.
Even as someone who knew where most of this was going in the first half, I didn't realize why you were delaying explaining the relationship between the cube, octahedron, and cuboctahedron until you started talking about duals.
4:37 You can also make a rhombicuboctahedron by expanding a cube, which is done by moving the faces away from the centre and then connecting them with rectangles on the edges and whichever polygon is needed on the corners. The same can be done but by rotating each face and connecting them with triangles instead of rectangles to make a snub cube
And the real fun (and actual research) starts when you go to higher dimensions. The higher dimensional Archimedean solids are called uniform polytopes, and noone so far has been able to classify them. Same for edge-transitive polytopes. There are lists that are conjectured to be complete, but no one knows. Conway found a new uniform polytope in dimension 4 (the grand antiprism) which had to be added to the list, so no one knows whether there is not something else we have missed so far.
This is the type of video I hope gets preserved after the internet gets destroyed or restricted or some great data loss happens within TH-cam’s servers
This TH-cam video has earned a spot in my all-time top 100, and definitely on the upper end of that 100. I’ve been watching YT since 2007. You’re seriously underrated, so if it helps, you’ve earned a new subscriber.
I hate to be that guy but 15 seconds in, the icosahedron is labeled as a dodecahedron. That's the only thing I could think of that was wrong with this video. Amazing work!
8:41 "and anything more than 360° requires hyperbolic geometry" This made me pause the video and go "No, you could just fold some of the folds inwards, so this proof can't hold unless there are more than those 2 criterias for what is an Archimedean solid!" I just checked wikipedia. Both Platonic and Archimedean solids *also* have the criteria that they have to be convex. That said, I have no idea if it would be possible to generate other Archimedean solids if concave ones were also allowed. The type of solid I'm thinking of isn't the en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron solids, as they all have intersecting planes. What I'm thinking of would be a non-convex solid made up by only regular polygon and every corner is identical, yet doesn't have any intersecting sides. But it's very possible the reason the type of solid I'm thinking of simply can't exist, so that's why I can't find them on wikipedia. I invite you or anyone else reading this to get nerd sniped by this, so that *I* don't have to go down the rabbit hole of disproving/finding them. Or if anyone knows that they exist and have a name, then I'd really appreciate a reply with their name.
I think my favorite Johnson solid has to be the Snub Disphenoid. The idea that a "digon" (line) has a use case at all as a polygon, despite being degenerate, is just so funny to me.
yes! i get a weird sense of joy using degenerate cases in math, such as for example, 0! = 1actually being intuitive if you think about it, there really is exactly one way to arrange 0 items in a line on your desk after all.
its also funny to say "Snub Disphenoid"
Yeah! I once tried designing a Rubik's-cube-like twisty puzzle with the snub disphenoid. It bent my brain.
I like the snub disphenoid, partly because the name is silly and partly because Vsauce mentioned it, mostly because I think it's pretty.
@@Buriaku"... you must realize the truth."
"And what is that?"
"It is not the snub disphenoid that bends, it is you."
Son: "dad, why is Daisy called like that?"
Dad: "because you mother really loves daisys"
Son: "i love you dad"
Dad: "i love you too Great Rhombicosidodecahedeon III"
Nah you should have named him "Disdyakis Triacontahedron"
Dad, why is Daisy called like that?
Because when she was young a daisy fell on her head.
And how did you come up with my name?
No further questions whilst I'm reading, brick.
Isn't the last johnson solid the shape of a diamond.
@@taxing4490Oh no
@@TheCreator-178 Should have called it gyroelongated pentagonal birotunda
Spectacular video!
I also enjoyed Jan Misali's video about "48 regular polyhedra" which talks about some of the ones you excluded at the beginning
same
I came here to mention that video, lol.
@@KinuTheDragon same
same
Same
I can't describe my panic at the Dungeons & Dragons table looking at my dice and realizing that there were so few regular platonic solids. I bothered my DM about it for weeks. And then finally I saw in a video showed there are very many regular platonic solids as long as you don't care what space looks like, and that put my mind at ease. A good collection of *almost* regular objects is going to seriously put my mind at ease. I should make plush versions of these solids to throw around during other hair pulling math moments.
Yeah this is really giving context to the wikipedia deep dive I tried to do. Lots of pretty pictures but they didn't make sense until you showed the animations.
d10 and percentile dice are pentagonal trapezohedrons
If you want more dice, the catalan solids all make nice fair dice. The disdyakis tricontrahedron makes a particularly great dice, with 120 sides you can replicate any "standard" single dice roll by just dividing the result, since 4,6,8,10,12,20 are all factors of 120.
Plush solids would be so cute! Might want to use mid- to heavy-weight interfacing on the faces so they don't all turn into puffy balls when stuffed with polyfill… although that could be cute, too, especially if you marked the edges somehow, e.g. by sewing on some contrasting ribbon or cord (you could ignore this step or use different colors for the adjacent faces).
Now I want to make some 😂 I sewed some plushie ice cream cones recently and have been itching to make more cute things.
can't wait for when we figure out a way to make dice in the shape of the star polyhedra
I can describe your panic:
trivial
For dice, face transitivity is much more important than corner transitivity, so Catalan solids are much more useful.
magic man*
rhombic dodecahedron is my favorite among all these guys. i like how unfamiliar it looks even though it has cubic symmetry. and its 4d analogue, the 24 cell, is completely regular! i wish i could look at it, its beautiful
It's even better when you realize it can tile 3d space! That's something most Platonic solids can't even do
@@nnanob3694 hey, this guy gets it! :)
I just started watching this channel and I love how you can visualize and explain all this information in a way that is easy to understand. Great video! 😁
pixel land guy
I've watched this once, twice opposite, twice non-opposite and three times and I still don't really understand all of them
Vastly Underrated Comment
understandable
"twice non-opposite"
What? It’s sight readable.
Omg platonic solids
Why did I read this in the “omg I love chipotle” voice??
@@Kona120platonic is my liiiiiiife
> platonic solids
But wait! There's more!
Almost
😑
The most important thing I noticed in this video is a new way to get to irrational numbers and ratios via geometry
This was so chilling and exciting.
And also as an origami person, I was basically thinking of how to construct each one!
Incredible video, great work on it all! A lot of new names for solids I never knew before
A giant grid of all of the solids as a flowchart of different operations to get to them would be a hella cool poster tbh
Omg I would totally buy that
Someones gotta make that, that'd be so cool!
@@crazygamingoscar7325maybe i can
15:21
It must be my birthday!
Look at that beautiful little chartreuse gremlin spin! Oh, how my heart radiates with joy!
never before have i ever thought "damn i wish i had a collection of archimedean solids in my house" and then i saw 1:11 and spontaneously melted
I want one too but they cost like 80$ per shape lol
@@funwithtommyandmore they look like paper though, i'm sure an exacto knife and strong enough glue should be enough to recreate them
@@Yvelluap looks like weeks of work I'm not willing to put into some shapes lol
🥜 : cube
🧠 : square prism
🌀 : triangular trapezohedron
🤓: inverted truncated triangular trapezoidhedronakaliod
Supertriakis tetrahedron.
pirax
Me watching this at 2 am, half asleep: “I like your funny words magic person”
with the music buildup at the end i was hoping for a scrolling lineup of all of the polyhedra lol. amazing explanation and 3d work btw
This is an excellent followup for Jan Miseli's video on a similar topic! Thanks for making this!
I had a weird math panic attack when I learned there weren't more platonic solids and that Jan Miseli video really put my mind at ease, and then went even farther and blew my mind a few times. Great video. And his stuff on constructed languages has taught me so much about linguistics that just keeps coming up in my regular language study, it's awesome. Love that guy.
Platonic solids
Familial solids
Romantic solids
the kepler-poinsot polyhedra are sexual solids
Dude WTF 💀
Okay then sorry
Sexual solids- **gets shot**
Alterous solids
if you take the deltoidal hexecontahedron. and force the kite faces to be rhombi, you get a concave solid called the rhombic hexecontahedron, and it is my favorite polyhedron
You'll probably enjoy this puzzle by Oskar can Deventer. th-cam.com/video/1RExXExkOrg/w-d-xo.html. The peices are almost rhombuses
There's a rhombic hexecontahedron? I thought it's always a dodecahedron or triacontahedron.
@@FranklinWilliamWelker There is, It's also the logo for wolfram alpha. en.wikipedia.org/wiki/Rhombic_hexecontahedron
What's a rhombic hexecontahedron?
@MichaelDolenzTheMathWizard
en.wikipedia.org/wiki/Rhombic_hexecontahedron
I don't know why, but polyhedra like these are inherently appealing to me. I just really love me some shapes.
this is fast becoming my favorite video on youtube. i'm so happy to see that there are other people out there who care this much about polyhedra. the disdyakis triacontahedron is also my favorite, it's like a highly composite solid! just as 120 is highly composite! this is closely followed by the rhombic dodecahedron (because it's like the hexagon of solids!) and then the rhombic triacontahedron. this video has taught me so much, like how snubs work, and the beautiful relationship between the archimedean and catalan solids. not to mention half triakis (i had always wondered how someone could think up something as complex as the pentagonal hexacontahedron.) and johnson solids! i hadn't even heard of them before this video! thanks for educating, entertaining, and inspiring me! i'm so glad i stumbled across this. 120/12, would recommend
Thank you so much! This is one of the most in depth comments of praise I've received and it's very encouraging :)
3:18 is that my channel
my favourite solid has always been the truncated octahedron because it evenly tiles space with itself, and it has the highest volume-to-surface-area ratio of any single shape that does so. its the best single space filling polyhedra! if you were to pack spheres as efficiently as possible in 3d space, and then inflate them evenly to fill in the gaps, you get the truncated octahedron
So basically it's a 3d version of the hexagon
I dont think thats quiet true. The shape you get when inflating spheres is a rhombic dodecahedron. You can see this by looking at the number of faces. The truncated octahedron has 14 faces but a sphere only has 12 neighboring spheres.
youe could well be right, im no polygon-zoologist @@Currywurst-zo8oo
Bejeweled gems timestamps:
0:06 Amethyst Agate (Tetrahedron), Amber Citrine (Icosahedron), kinda Topaz Jade (Octahedron)
2:38 Ruby Garnet (Truncated Cube)
2:46 Quartz Pearl (Truncated Icosahedron/"Football" shape)
16:12 Emerald Peridot (Deltoidal Icositetrahedron)
20:11 kinda Sapphire Diamond (Halved Octahedron)
OMG thank you for this comment, I was wondering about this!
@@feelshowdy It's not 100% accurate, because not all of the Bejeweled gems are platonic or almost platonic solids of course, but I wanted to include all of them in the comment since they're all so equal to each other.
Why are you calling this ⚽ a football that's obviously a soccer ball there's a giant difference
A few years ago I was very intrigued about a very similar thing, but with tetrominoes, aka tetris pieces. It's well know that there's only 5 ways to connect 4 squares on a plane, with 2 of them being chiral, hence the 7 tetris pieces we all know, but once you start to dig deeper you start to have so many questions. What about 5 squares? 6 squares? 7? What about other shapes, like triangles? Or maybe cubes in 3D, aka tetracubes? What if you keep only squares, but allow them to go in 3 dimensions (they are called Polyominoids)? Turns out there's lots of ways one could extend the idea of tetrominos, by either using different shapes, getting into higher dimensions or simply changing the rules of how shapes are allowed to connect.
I've been interested in that also! Not counting reflections, there are 12 pentominoes, and it's a classic puzzle to arrange them into a rectangle. You can actually make 4 different types of rectangle, 3x20, 4x15, 5x12, and 6x10.
Watching this for the 17th time. Thank you for getting this all this down into one video. I can tell you worked really hard to put all the faces together for this one. 🎉
Great now I need a hystericaly elaborate polyhedra family tree diagram >:(
I need a bucket of blocks with solids from each family to play with
this is by far the best video I've seen on the topic! it's incredibly well explained
I saw descriptions about these solids at high school, and couldn't grasp many concepts yet getting really intrigued. Your explanation was excellent. Thank you sooooo much!!
This is a most excellent video! As a 3d puzzle designer and laser polyhedra sculptor, this helps show the relations between the shapes. ⭐
Gonna be printing some of these. A+ infodump. Super well done
Really fantastic video! You did a beautiful job with the visuals and in organizing the explanation. I have shown it to a wide range of viewers - from a 7 year old to a guy with a phd in math. Everyone loved it and had the same basic reaction - it was entrancing!
Truncated Icosahedrons = soccer ball pattern
Yes! I was wondering when someone would notice! 😄
Us: How many 3-d solids you want?
Kuvina Saydaki: yes
i really liked all the solids constructed with lunes! my favourite has to be the bilunabirotunda, it's just so pretty
My Euler! This channel is a gem!!!
I have been trying to find a good explanation of Johnson Solids for YEARS and this one finally satisfies me. Thank you :D
pentagonal hexecontahedron is clearly my favorite with it's "petal" sides if you consider 5 faces connected on their smallest angle, or heart shaped sides, if you only consider 2 faces
these shapes are really cool, we enjoy how ridiculous the names get lol
Thanks! Great video. Have you ever looked at the geometric net of these kinds of solid. I know the cube has 11 possible nets. I would like to see a video that dives into the possible nets of some of the other shapes as well.
Thank you so much! I do have some degree of experience with the nets of the catalan and archimedean solids after making them all out of paper. Some of them I even modified to fit better on 1 piece of paper!
The hebesphenorotunds (last one explained 27:03) looks really similar a gem-cut.
Think about the side with the 3 pentagon down into the socket and the hexagon outside and visible.
I watched this whole video and found at least five of my new favorite solids. They will never beat my favorite shape, the snub disphenoid!
Also, please make a video on some of the near miss johnson solids.
I am a particular fan of the disdyakis triacontahedron because it is the largest roughly spherical face-transitive polyhedron, so it's the largest fair die that can be made (ignoring bipyramids and trapezohedrons)
I want a toy set that's just all of these solids,
not sure what i'd do with them,
but it seems cool...
I LOVE WATCHING EDUCATIONAL GEOMETRY VIDEOS MADE BY NON BINARY PEOPLE ‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️
Now I wish I had hundreds of magnet shapes, so that I could make these in real life. They look so collectible.
after watching jan Misali's platonic solids video and vsauce's strictly convex deltahedra video, seeing some concepts i got from there return here was nice and cool, like a callback from across my brain :3
I was expecting this to be like a reduced version of Jan Misali's video about the 48 regular polyhedra... what a fantastic surprise! I love geometry, those were some great explanations.
Beautiful very well done and well paced video! I love it and thanks!
Seriously the best use of visual examples in explaining these, I am sure there will never be a better explanation as long as I live.
DEGREES • FACES • EDGES • VERTICES
Triangle:
* Degrees: 180
* Faces: 1 (triangle)
* Edges: 3
* Vertices: 3
Square:
* Degrees: 360
* Faces: 1 (square)
* Edges: 4
* Vertices: 4
Pentagon:
* Degrees: 540
* Faces: 1 (pentagon)
* Edges: 5
* Vertices: 5
Hexagon:
* Degrees: 720
* Faces: 1 (hexagon)
* Edges: 6
* Vertices: 6
Tetrahedron:
* Degrees: 720
* Faces: 4 (equilateral triangles)
* Edges: 6
* Vertices: 4
Octagon:
* Degrees: 1080
* Faces: 1 (octagon)
* Edges: 8
* Vertices: 8
Pentagonal Pyramid
* Degrees: 1440
* Faces: 6 (5 triangles, 1 pentagon)
* Edges: 10
* Vertices: 6
Octahedron:
* Degrees: 1440
* Faces: 8 (equilateral triangles)
* Edges: 12
* Vertices: 6
Stellated octahedron:
* Degrees: 1440
* Faces: 8 (equilateral triangles)
* Edges: 12
* Vertices: 6
Pentagonal Bipyramid
* degrees: 1800
* Faces: 10 (10 triangles)
* Edges: 15
* Vertices: 7
Hexahedron (Cube):
* Degrees: 2160
* Faces: 6 (squares)
* Edges: 12
* Vertices: 8
Triaugmented Triangular Prism:
* Degrees: 2520
* Faces: 10 (6 triangles, 4 squares)
* Edges: 20
* Vertices: 14
Octadecagon (18-sided polygon):
* Degrees: 2880
* Faces: 1 (octadecagon)
* Edges: 18
* Vertices: 18
Icosagon (20-sided polygon):
* Degrees: 3240
* Faces: 1 (icosagon)
* Edges: 20
* Vertices: 20
Truncated Tetrahedron
* Degrees: 3600
* Faces: 8 (4 triangles, 4 hexagons)
* Edges: 18
* Vertices: 12
Icosahedron:
* Degrees: 3600
* Faces: 20 (equilateral triangles)
* Edges: 30
* Vertices: 12
Cuboctahedron or VECTOR EQUILIBRIUM
* Degrees: 3600
* Faces: 14 (8 triangles, 6 squares)
* Edges: 24
* Vertices: 12
3,960 DEGREES
88 x 45 = 3,960
44 x 90 = 3,960
22 x 180 = 3,960
11 x 360 = 3,960
Rhombic Dodecahedron
* Degrees: 4,320
* Faces: 12 (all rhombuses)
* Edges: 24
* Vertices: 14
* Duel is Cuboctahedron or vector equilibrium
Tetrakis Hexahedron:
* Degrees: 4320
* Faces: 24 (isosceles triangles)
* Edges: 36
* Vertices: 14
Icosikaioctagon (28-sided polygon):
* Degrees: 4680
* Faces: 1 (icosikaioctagon)
* Edges: 28
* Vertices: 28
5040 DEGREES
5400 DEGREES
5,760 degrees
6,120 degrees
Dodecahedron:
* Degrees: 6480
* Faces: 12 (pentagons)
* Edges: 30
* Vertices: 20
7560 DEGREES
6840 DEGREES
7,200 DEGREES
7560 DEGREES
Truncated Cuboctahedron
* Degrees: 7920
* Faces: 26 (8 triangles, 18 squares)
* Edges: 72
* Vertices: 48
Rhombicuboctahedron:
* Degrees: 7920
* Faces: 26 (8 triangles, 18 squares)
* Edges: 48
* Vertices: 24
Snub Cube:
* Degrees: 7920
* Faces: 38 (6 squares, 32 triangles)
* Edges: 60
* Vertices: 24
Trakis Icosahedron:
* Degrees: 7920
* Faces: 32 (20 triangles, 12 kites)
* Edges: 90
* Vertices: 60
8,280 DEGREES
8640 DEGREES
9000 DEGREES
9,360 degrees
9,720 degrees
Icosidodecahedron:
* Degrees: 10080
* Faces: 30 (12 pentagons, 20 triangles)
* Edges: 60
* Vertices: 30
? 10,440 degrees
Rhombic Triacontahedron:
* Degrees: 10,800
* Faces: 30 (rhombuses)
* Edges: 60
* Vertices: 32
11160 DEGREES
11,520 DEGREES
11,880 DEGREES
12,240 DEGREES
12,600 DEGREES
12960 DEGREES
END OF POLAR GRID
Small Ditrigonal Icosidodecahedron:
* Degrees: 16,560
* Faces: 50 (12 pentagons, 20 triangles, 18 squares)
* Edges: 120
* Vertices: 60
Small Rhombicosidodecahedron
* Degrees: 20,880
* Faces: 62 (20 triangles, 30 squares, 12 pentagons)
* Edges: 120
* Vertices: 60
Rhombicosidodecahedron
* Degrees: 20,880
* Faces: 62 (30 squares, 20 triangles, 12 pentagons)
* Edges: 120
* Vertices: 60
Truncated Icosahedron:
* Degrees: 20,880
* Faces: 32 (12 pentagons, 20 hexagons)
* Edges: 90
* Vertices: 60
Disdyakis Triacontahedron:
* Degrees: 21600
* Faces: 120 (scalene triangles)
* Edges: 180
* Vertices: 62
Deltoidal Hexecontahedron
* Degrees: 21,600
* Faces: 60 (kites)
* Edges: 120
* Vertices: 62
Ditrigonal Dodecadodecahedron:
* Degrees: 24480
* Faces: 52 (12 pentagons, 20 hexagons, 20 triangles)
* Edges: 150
* Vertices: 60
Great Rhombicosidodecahedron
* Degrees: 31,680
* Faces: 62 (12 pentagons, 20 hexagons, 30 squares)
* Edges: 120
* Vertices: 60
Small Rhombihexacontahedron:
* Degrees: 31,680
* Faces: 60 (12 pentagons, 30 squares, 20 hexagons)
* Edges: 120
* Vertices: 60
Pentagonal Hexecontahedron:
* Degrees: 32,400
* Faces: 60 (pentagons)
* Edges: 120
* Vertices: 62
I've been looking into these solids for years, but had no idea what the process of discovering them was. Half-truncation is one hell of a leap, especially for someone born a few thousand years too early for computers. It's amazing he found them all
I love this video! I'm glad that I found your videos. I have a love for mathematics and geometry, and it's cool someone made a video about platonic-y solids! I liked the video "there are 48 regular polyhedra" by jan Misali and this is the type of stuff I like. I think you would like that video, too.
Wow, haven't seen so clean, concentrated and convenient explanation, without unnecessary effects it's even easier to understand. Your format is my favorite among others since I went in for geometry 11 years ago. My suggestion for next topic is "3D Honeycombs" because it's logical continuation of solids. There are "regular" ones which consist of the same solids you were talking about in this video. The particularly brilliant thing is there were found some irregular (!) 3D honeycombs. Most of them are of similar polyhedra, both convex and not. The only irregularity in them were the colors which cube faces had or something like this. But maybe there are some of them I missed which look like 3D version of Penrose tiling.
Edit: Pentakis Dodecahedron is my favourite solid (the second one is Icosahedron) because it's one of the roundest solids which consists of equal polygons.
My favourite catalan solid is the pentagonal hexacontahedron. I find it very pretty how the flower patterns with 5 petals interlock to make chiral corners at the boundary.
Your color choices for each polyhedron are lovely. This whole video tickles my brain wonderfully. I want a bunch of foam Catalan solids to just turn over in my hands.
Thank you! I put a lot of thought into the colors so I'm really happy that it goes appreciated!
Great video - I've been fascinated by polyhedra for decades and I learned some new things here. Well done!
Let's face it most underrated youtuber I have ever come across (is you)! Well done and Thank You, you are a wonderful edgeucator c: who always gets even very complicated points across, not to mention the volume of information in each video is enormous!
I'm trying to get a pun in here but your comment fills so much of the available space that I'm pretty sure it's a tileable solid!
i like the cupolas
also i admire how you were able to say so many syllables so confidently lol- it probably took a few takes
Amazing video!!! Very in depth and yet easy to follow, I really enjoyed some of the smaller details like sphericity!! i look forward to your future uploads!!!
-from another friend of Blahaj ;)
Just discovered your channel and am loving it. You are covering all my favourite topics. I personally find the Catalan solids more beautiful than the Archimedean ones.
Imagine having dice of every single one of these
The Dice Lab is a company that makes some unusual ones. Their large set has a truncated tetrahedron, truncated octahedton, rhombic dodecahedron, deltoidal icosahedron, disdyakis dodecahedron, deltoidal hexecontahedron, and disdyakis triacontahedron.
ENBY DETECTED!!
LOVE, AFFECTION, AND SUPPORT MODE ACTIVATED!!
This channel is going onto the list.
Hopefully once this nightmare of a degree (math) is done I'll have time to get through these interesting videos/topics.
I was so happy when you included those 4 honorary platonic solids!
sensational video! Loved the term honorary platonic solids, definitely stealing that one!
My personal favourite is the rhombic dodecahedron! :)
i'm honestly surprised that you've explained it this well, i was able to keep up pretty much the whole time,, i was so shocked that i could understand what was happening
i want to commend you for the use of color coding for things like rotundas and cupolas, you've done an amazing job at making this more digestible and it was very helpful
excellent job on the video, kuvina
Had to pause to comment - this video is excellent. Great job. Interesting topic, good visuals, good narration.
Kudos!
My favorite Catalan solid is the 30-sided rhombic polyhedron based on the Golden Ratio because I figured out how to make it in Sketchup. It is closely related to the icosahedron and dodecahedron.
same with the icosidodecahedron (which is pretty much if the two fused together dragon ball z style)
If you're into Sketchup and geometry then you might find a few videos I've done on my channel to be interesting.
Also, you guys know the Sketchup team does a livestream every Friday? Fun times..
I wondered if there are solids where instead of relaxing the properties
2: all faces being the same
3: all corners being the same
we relaxed: 1: faces don't have to be regular polyhedra. These solids do exist! But it's a single class of solids.
The first thing we can note is that all the angles that are "supplied by the faces" have to be "consumed by the corners". Or in other words, if a face has angles a,a,b,c then a corner has to use up the same amount, or a multiple.
That means that each corner could have
3 3-sided faces
4 4-sided faces
5 5-sided faces
... meet. But 5 5-sided faces would make a hyperbolic surface, and 4 4-sided faces just make a distorted square grid. Therefore 3 3-sided faces is the only type of these that can exist (see below).
You could also have
3 6-sided faces or
6 3-sided faces
meet. But for similar reasons, they'd be distorted planar grids.
And combining multiplicities 4 and 8 or 3 and 9 (or above) doesn't work.
2-sided faces don't exist, but we _could_ have 2 4-sided faces meet at each corner. Except that that would just be 2 rectangles back to back with zero volume enclosed.
*Thus a distorted tetrahedron is the only type of "fully transitive solids",* as I would call them, that could exist. Or in other words, "cursed d4 dice". And all that remains is to prove that it isn't an impossible construction. (And that the construction from a given set of faces doesn't allow for more than 1 type of solid.)
The only problem that distorting a tetrahedron could cause is that making a triangle with 3 angles that aren't all the same is that the edges will have different lengths too. But luckily, any two congruent triangles always share a side of common length, along which we can join them. Let's call that side length "a" and the angles on its ends "beta" and "gamma". You can't join two "beta" or two "gamma" angles together in the same vertex, or you won't get identical corners. (Each vertex has to use one of each angles.) That means we can only join these two triangles with sides "a" against each other and angle "beta" touching "gamma" and vice versa.
This shows that the solid can be completed, and that it can only be constructed in a single way. (The two remaining faces will have their edges "a" joined together in the same way. And then edges "b" and "c" can only be joined to edges of the same length. This leaves two possibilities, of which one is just two sets of coplanar triangles - which form a parallelogram - joined back to back, with zero volume.)
Thank you for making a version of jan Misali's 48 Regular Polyhedra that respects its audience. I needed that.
twitter.com/kuvina_4
instagram.com/kuvina_4
*Correction* : At the beginning I mislabeled the icosahedron as dodecahedron. (copied textbox but forgot to change text)
You are the literal personification of underrated
why tf would you need to normalise this tq+ bullshit literally in a math video smh
Fascinating video, thanks for posting. Some years ago I assembled some of the Johnson Solids using Polydron (plastic panels that clip together)
I LOVED this video!! I am a huge geometry nerd and learning about polyhedral families and the construction methods to generate new ones makes them all feel so intertwined and uniform. If I may request, please do a video on higher dimensional projections into the third dimension like fun cross sections of polytopes through various polyhedra. TYSM
There is another category of almost platonic solids where you only use property 1 and 2 and don't care about the verticies being identical. These are the triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism and gyroelongated square bipyramid, otherwise known as the irregular deltahedra.
You should make a video about tilings and hyperbolic tilings.
20:35
Johnson Solids
J01 Square Pyramid
J02 Pentagon Pyramid
J03 Triangular Cupola
J04 Square Cupola
J05 Pentagonal Cupola
J06 Pentagonal Rotunda
J07 Elongated Tetrahedron
J08 Elongated Square Pyramid
J09 Elongated Pentagonal Pyramid
J10 Gyroelongated Square Pyramid
J11 Gyroelongated Pentagonal Pyramid
J12 Triangular Bipyramid
J13 Pentagonal Bipyramid
J14 Elongated Triangular Bipyramid
J15 Elongated Square Bipyramid
J16 Elongated Pentagonal Bipyramid
J17 Gyroelongated Square Bipyramid
J18 Elongated Triangular Cupola
J19 Elongated Square Cupola
J20 Elongated Pentagonal Cupola
J21 Elongated Pentagonal Rotunda
J22 Gyroelongated Triangular Cupola
J23 Gyroelongated Square Cupola
J24 Gyroelongated Pentagonal Cupola
J25 Gyroelongated Pentagonal Rotunda
J26 Twisted Triangular Biprism
J27 Twisted Triangular Bicupola
J28 Twisted Square Bicupola
J29 Square Bicupola
J30 Twisted Pentagonal Bicupola
J31 Pentagonal Bicupola
J32 Pentagonal Cupola-Rotunda
J33 Twisted Pentagonal Cupola-Rotunda
J34 Twisted Pentagonal Birotunda
J35 Elongated Triangular Bicupola
J36 Twisted Elongated Triangular Bicupola
J37 Pseudo Rhombicuboctahedron
J38 Elongated Pentagonal Bicupola
J39 Twisted Elongated Pentagonal Bicupola
J40 Elongated Pentagonal Cupola-Rotunda
J41 Twisted Elongated Pentagonal Cupola-Rotunda
J42 Elongated Pentagonal Birotunda
J43 Twisted Elongated Pentagonal Birotunda
J44 Gyroelongated Triangular Bicupola #
J45 Gyroelongated Square Bicupola #
J46 Gyroelongated Pentagonal Bicupola #
J47 Gyroelongated Pentagonal Cupola-Rotunda #
J48 Gyroelongated Pentagonal Birotunda #
J49 Square Pyramid on Triangular Prism
J50 Two Square Pyramids on Triangular Prism
J51 Three Square Pyramids on Triangular Prism
J52 Square Pyramid on Pentagonal Prism
J53 Two Square Pyramid on Pentagonal Prism
J54 Square Pyramid on Hexagonal Prism
J55 Two Opposite Square Pyramids on Hexagonal Prism
J56 Two Non-Opposite Square Pyramids on Hexagonal Prism
J57 Three Square Pyramids on Hexagonal Prism
J58 Pentagonal Pyramid on Dodecahedron
J59 Two Opposite Pentagon Pyramids on Dodecahedron
J60 Two Non-Opposite Pentagon Pyramids on Dodecahedron
J61 Three Pentagonal Pyramids on Dodecahedron
J62 Two Cuts From an Icosahedron
J63 Three Cuts From an Icosahedron
J64 Three Cuts From an Icosahedron + Tetrahedron
J65 Triangular Cupola on a Truncated Tetrahedron
J66 Square Cupola on a Truncated Cube
J67 Two Square Cupolas on a Truncated Cube
J68 Pentagonal Cupola on a Truncated Dodecahedron
J69 Two Pentagonal Cupolas on Truncated Dodecahedron 1
J70 Two Pentagonal Cupolas on Truncated Dodecahedron 2
J71 Three Pentagonal Cupolas on a Truncated Dodecahedron
J72 Twisted Rhombicosidodecahedron
J73 Double Opposite Twisted Rhombicosidodecahedron
J74 Double Non-Opposite Twisted Rhombicosidodecahedron
J75 Triple Twisted Rhombicosidodecahedron
J76 Cut Rhombicosidodecahedron
J77 Opposite Cut and Twist Rhombicosidodecahedron
J78 Non-Opposite Cut and Twist Rhombicosidodecahedron
J79 Cut and Double Twist Rhombicosidodecahedron
J80 Opposite Double Cut Rhombicosidodecahedron
J81 Non-Opposite Double Cut Rhombicosidodecahedron
J82 Double Cut and Twist Rhombicosidodecahedron
J83 Triple Cut Rhombicosidodecahedron
J84 Snub Disphenoid
J85 Snub Square Antiprism
J86 Dilunic Octahedron
J87 Dilunic Octaherdon + Square Pyramid
J88 Dilunic Icosahedron
J89 Trilunic Icosahedron
J90 Gyroelongated Elongated Octahedron
J91 Dilunic Rotunda
J92 Hexagonal Dilunic Rotunda
The End.
This video fulfilled a craving I’ve had for years. Thank you.
I would love to see a video looking at the stellated versions of some of these and how the math works out for self-intersecting planes in these shapes
Thank you for such an interesting video. A lot of these I was hearing about for the first time and I found great joy in hearing you pronounce the name, getting surprised that this one is longer than the last one, and then laughing as I struggled to pronounce the name myself.
My favorite was either the “Snub Dodecahedron” or the “Pentagonal Hexacontahedron”. The Snub Dodecahedron looks so satisfying having a thick border of triangles around the pentagon, but there was something about that Pentagonal Hexacontahedron that I found really pretty. I think it’s because of the rotational symmetry.
Again, thank you for taking the time to make such interesting and engaging videos. I look forward to watching another one.
mine too!
Even as someone who knew where most of this was going in the first half, I didn't realize why you were delaying explaining the relationship between the cube, octahedron, and cuboctahedron until you started talking about duals.
These are incredibly interesting, like platonic solids but stranger and there are way more. Love it!
4:37 You can also make a rhombicuboctahedron by expanding a cube, which is done by moving the faces away from the centre and then connecting them with rectangles on the edges and whichever polygon is needed on the corners. The same can be done but by rotating each face and connecting them with triangles instead of rectangles to make a snub cube
I have no idea how you make everything feel so concise and ordered. If I wanted to research this it would be so messy
This is an incredible video. Fantastic job, and thank you!
First time seeing any video of yours, already my favorite enby math teacher
You deserve way more than 4k subs, this a brilliant video
And the real fun (and actual research) starts when you go to higher dimensions. The higher dimensional Archimedean solids are called uniform polytopes, and noone so far has been able to classify them. Same for edge-transitive polytopes. There are lists that are conjectured to be complete, but no one knows. Conway found a new uniform polytope in dimension 4 (the grand antiprism) which had to be added to the list, so no one knows whether there is not something else we have missed so far.
This is the type of video I hope gets preserved after the internet gets destroyed or restricted or some great data loss happens within TH-cam’s servers
This is the first time I've ever heard of a disdyakis triacontahedron, but upon discovering what it is, I now want one.
I loved this, especially the explanation on why there are only 13 Archimedian solids, great work!
I’ve always LOVED the Catalan solids, definitely more than the Archimedean solids, …maybe more than the Platonic solids.
this channel is so underrated love your videos!!!!
This TH-cam video has earned a spot in my all-time top 100, and definitely on the upper end of that 100. I’ve been watching YT since 2007. You’re seriously underrated, so if it helps, you’ve earned a new subscriber.
I'm thankful another person has commented on the incredible quality of this video. I agree!
Came for the 3d shapes
Stayed for the enby explaining the 3d shapes
I hate to be that guy but 15 seconds in, the icosahedron is labeled as a dodecahedron. That's the only thing I could think of that was wrong with this video. Amazing work!
Lol there is 2 Dodecs
8:41 "and anything more than 360° requires hyperbolic geometry"
This made me pause the video and go "No, you could just fold some of the folds inwards, so this proof can't hold unless there are more than those 2 criterias for what is an Archimedean solid!"
I just checked wikipedia. Both Platonic and Archimedean solids *also* have the criteria that they have to be convex.
That said, I have no idea if it would be possible to generate other Archimedean solids if concave ones were also allowed. The type of solid I'm thinking of isn't the en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron solids, as they all have intersecting planes. What I'm thinking of would be a non-convex solid made up by only regular polygon and every corner is identical, yet doesn't have any intersecting sides. But it's very possible the reason the type of solid I'm thinking of simply can't exist, so that's why I can't find them on wikipedia.
I invite you or anyone else reading this to get nerd sniped by this, so that *I* don't have to go down the rabbit hole of disproving/finding them. Or if anyone knows that they exist and have a name, then I'd really appreciate a reply with their name.
11:50 i like the pacman reference