The ALMOST Platonic Solids
ฝัง
- เผยแพร่เมื่อ 30 พ.ค. 2024
- This is my entry in #SoME3 . This video covers the Archimedean solids, Catalan solids, and Johnson solids. Geometry is one of the most beautiful parts of math, and polyhedra are one of my favorite parts of that. If you love geometry, make sure to check out my video on map projections!
Chapters:
0:00 Intro
1:17 Archimedean Solids
7:22 Proving there are 13
12:13 Catalan Solids
18:28 Johnson Solids
27:11 Outro
#math #geometry - วิทยาศาสตร์และเทคโนโลยี
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
😑
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.
Platonic solids
Familial solids
Romantic solids
the kepler-poinsot polyhedra are sexual solids
Dude WTF 💀
Okay then sorry
Sexual solids- **gets shot**
Alterous solids
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
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! :)
For dice, face transitivity is much more important than corner transitivity, so Catalan solids are much more useful.
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
Omg I would totally buy that
Someones gotta make that, that'd be so cool!
@@crazygamingoscar7325maybe i can
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.
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.
@@user-qd9sk8ih4h 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
Me watching this at 2 am, half asleep: “I like your funny words magic person”
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.
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
15:21
It must be my birthday!
Look at that beautiful little chartreuse gremlin spin! Oh, how my heart radiates with joy!
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)
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
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
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.
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 :)
I don't know why, but polyhedra like these are inherently appealing to me. I just really love me some shapes.
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.
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
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. 🎉
These are incredibly interesting, like platonic solids but stranger and there are way more. Love it!
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!
You: "This is a truncated icosahedron."
Football: Am I a joke to you ?
This was so chilling and exciting.
And also as an origami person, I was basically thinking of how to construct each one!
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
This is an incredible video. Fantastic job, and thank you!
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!!
The shapes are all so beautifully presented; could you please share the software you used? Or is it a code library, perhaps?
I used blender! You can download all the STLs from wikimedia commons, and they're automatically public domain since they're simple geometry!
@@Kuvina awesome; many thanks!
@@KuvinaI didn't know Wikimedia hosts 3D files. Thanks!
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.
I loved this, especially the explanation on why there are only 13 Archimedian solids, great work!
The most important thing I noticed in this video is a new way to get to irrational numbers and ratios via geometry
these shapes are really cool, we enjoy how ridiculous the names get lol
this is by far the best video I've seen on the topic! it's incredibly well explained
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
Hang on, aren't soccer balls truncated isocosahedrons?
they are !
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..
things i learned from this:
the geometrical name of a soccerball [2:49]
how to make my favorite shape even outside of archimedians (basically my favorite polyhedra) [4:44] from squares only
basically nothing else but
here is the info requested
great rhombicosidodecahedron
triakis icosahedron
hebesphenorotunda
I have been trying to find a good explanation of Johnson Solids for YEARS and this one finally satisfies me. Thank you :D
Great video - I've been fascinated by polyhedra for decades and I learned some new things here. Well done!
Now I wish I had hundreds of magnet shapes, so that I could make these in real life. They look so collectible.
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.
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!
this video was really good I enjoyed it a lot. good explanation of each in a way that was easy for me to understand and cool visuals. you earned yourself a sub from this. I really loved this video
i really liked all the solids constructed with lunes! my favourite has to be the bilunabirotunda, it's just so pretty
This is a most excellent video! As a 3d puzzle designer and laser polyhedra sculptor, this helps show the relations between the shapes. ⭐
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!
🥜 : cube
🧠 : square prism
🌀 : triangular trapezohedron
🤓: inverted truncated triangular trapezoidhedronakaliod
Supertriakis tetrahedron.
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
Beautiful very well done and well paced video! I love it and thanks!
The Pseudo Rhombicuboctahedron is called "elongated square gyrobicupola". I love this video, could watch it over and over again. Thanks!
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've been looking for a good video about this exact topic for ages. So glad there finally is one.
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
Excellent video! thank you so much
Fascinating video, thanks for posting. Some years ago I assembled some of the Johnson Solids using Polydron (plastic panels that clip together)
Your mathematical curiosity is beautiful and scary. Thank you.
i gotta say i appreciate your choice of favorite catalan solid, but in my case i just really enjoy the rhombic triacontahedron. the chiral deltoidal ones are tough runners up though. for my favorite johnson solid i was pleasantly surprised to see the snub disphenoid be a thing (i completely forgot it existed), which i think is just more interesting to look at than any of the "take a prism and put a rotunda/cupola on its face, or don't". my favorite archimedean solid is probably the snub dodecahedron. as you might be able to tell, i like snubs :)
What's your favorite Johnson solid? Mine is the gyrobifastigium, which also has the best name, which you didn't even mention! You just labeled it j26! I like it because it just feels so symmetrical, like it should almost count as an archimedean.
What's the gyrobifastigium?
@@user-qd9sk8ih4h youtube keeps deleting my comment. there's a Wikipedia article on it, and if you Google the term alongside the word dmccooey you'll find a site that let's you rotate it and look at different angles. There's I even a puzzle someone made on TH-cam out of the shape.
I need a bucket of blocks with solids from each family to play with
Had to pause to comment - this video is excellent. Great job. Interesting topic, good visuals, good narration.
Kudos!
You deserve way more than 4k subs, this a brilliant video
Highly appreciate the compilation ❣️
Wow thats one great video. To go through so many cases It must've taken a long time to make, good stuff
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!
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!
Here are a few names of certain Platonic & Archimedean Solids:
1. Octahedron: Triangular Antiprism/Square Dipyramid
2. Icosahedron: Gyroelongated Pentagonal Dipyramid
3. Cuboctahedron: Triangular Gyrobicupola
4. Rhombicuboctahedron: Elongated Square Orthobicupola
5. Icosidodecahedron: Pentagonal Gyrobirotunda
6. Rhombicosidodecahedron: Elongated Pentagonal Orthobicupola
BONUS: The pseudorhombicuboctahedron is called a elongated square gyrobicupola.
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.
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!
Platonic solids nah we got romantic liquids
solids that got drunk and screwed at the office christmas party last year
Us: How many 3-d solids you want?
Kuvina Saydaki: yes
First time seeing any video of yours, already my favorite enby math teacher
My Euler! This channel is a gem!!!
Loved the video!
i love the pentagonal hexecontahedron, the great rhombicosidodecahedron, and the snub squar antiprism :3
Why do the shapes look delicious
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
so in other words tetrahedrons can create everything
I have no idea how you make everything feel so concise and ordered. If I wanted to research this it would be so messy
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
Gonna be printing some of these. A+ infodump. Super well done
Solid work, my compliments!
My Faves: Snub Dodecahedron Pentagonal Hexcontahedron, and either J75 or J48*. I really like chiral polyhedra in general, but my favorite of these definitely is the pentagonal hexcontrahedron, it reminds me a bit of the "Einstein" aperiodic monotiling
Also, while the pentagonal hexcontrahedron (V3.3.3.3.5) is my favorite Catalan solid, I also *really* enjoy the tiling V3.3.3.3.6 and the compact hyperbolic V3.3.3.3.7
Great now I need a hystericaly elaborate polyhedra family tree diagram >:(
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 ;)
I absolutely love your videos
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)
My favorites of each category are: the Icosahedron (platonic), the Snub Cube (Archimedean), the Pentakis Dodecahedron (Catalan) and J71, which is the Truncated Dodecahedron with 3 Pentagonal Cupolas, mainly because of how egg-shaped it looks
Do you mean the triaugmented truncated dodecahedron
@@penguinandfishy4858 Yes
I enjoy seeing these kind of videos about 3D solids, because it gives me a chance to try and build some of the shapes irl. I hadn't heard of the snub square antiprism before, that was my project to make during this video. I ended up making a poor paper one. I tried to make one with magnetic shapes, but the structure wasn't ever stable enough for me to properly connect it up. Still had a great time, tho! Solid video, thanks for introducing me to some new shapes!
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.
Have you tried Openscad ?
In this software, you code your desired shape in a C based language (similar to Arduino), and when you execute the code, the shape is displayed. You just need to set the correct dimensions, or to calculate on the go with simple arithmetic and trigonometric operations, in your program.
The cube (and the parallellepipdes), the "sphere" and the cylinder (cones, and truncated cones and cylinders also), keep in mind that the program can't display an infinite number of edges.
For prisms, the extrude function is handy, and for more complex polyhedra, you either use tge Boolean operations (union, intersection, difference) or you hard code (with appropriate functions to do the job for you) the vertexes positions and the faces and gives that to the polyhedron function, I coded with it a helicoïd solids generator to make springs, screws,...
That's so cool! I will keep it in mind
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.)
I will now use this information in life. Thank you so much.
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