The ALMOST Platonic Solids

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

[@TheWolfboy180]

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.

[@terdragontra8900]

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.

[@Omicron23-sj4wu]

its also funny to say "Snub Disphenoid"

[@Buriaku]

Yeah! I once tried designing a Rubik's-cube-like twisty puzzle with the snub disphenoid. It bent my brain.

[@soleildj1572]

I like the snub disphenoid, partly because the name is silly and partly because Vsauce mentioned it, mostly because I think it's pretty.

[@marcomoreno6748]

​@@Buriaku"... you must realize the truth." "And what is that?" "It is not the snub disphenoid that bends, it is you."

[@craz2580]

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"

[@TheCreatorIsHere]

Nah you should have named him "Disdyakis Triacontahedron"

[@NikiTricky2]

Omg platonic solids

[@Kona120]

Why did I read this in the “omg I love chipotle” voice??

[@timpunny]

​@@Kona120platonic is my liiiiiiife

[@vaclavtrpisovsky]

> platonic solids But wait! There's more!

[@user-sn6gt6rz1z]

Almost

[@JGM.86]

😑

[@DissonantSynth]

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

[@jan_Eten]

same

[@KinuTheDragon]

I came here to mention that video, lol.

[@jan_Eten]

@@KinuTheDragon same

[@choco_jack7016]

same

[@malkistdev]

Same

[@kayleighlehrman9566]

Platonic solids Familial solids Romantic solids

[@onlykflow]

the kepler-poinsot polyhedra are sexual solids

[@asafesouza2015]

Dude WTF 💀

[@asafesouza2015]

Okay then sorry

[@alexterra2626]

Sexual solids- **gets shot**

[@KaesoARhombil]

Alterous solids

[@chaotickreg7024]

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.

[@brandtyee6257]

d10 and percentile dice are pentagonal trapezohedrons

[@estherstreet4582]

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.

[@emilyrln]

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.

[@Green24152]

can't wait for when we figure out a way to make dice in the shape of the star polyhedra

[@AkamiChannel]

I can describe your panic: trivial

[@terdragontra8900]

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

[@nnanob3694]

It's even better when you realize it can tile 3d space! That's something most Platonic solids can't even do

[@terdragontra8900]

@@nnanob3694 hey, this guy gets it! :)

[@malkistdev]

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! 😁

[@KakoriGames]

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.

[@Kuvina]

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.

[@CananaMan]

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

[@redpepper74]

Omg I would totally buy that

[@crazygamingoscar7325]

Someones gotta make that, that'd be so cool!

[@TaranVaranYT]

@@crazygamingoscar7325maybe i can

[@someknave]

For dice, face transitivity is much more important than corner transitivity, so Catalan solids are much more useful.

[@zactron1997]

This is an excellent followup for Jan Miseli's video on a similar topic! Thanks for making this!

[@chaotickreg7024]

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.

[@erikhaag4250]

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

[@LeoStaley]

You'll probably enjoy this puzzle by Oskar can Deventer. kzbin.info/www/bejne/Z4OoqYt7rdCCqMk. The peices are almost rhombuses

[@user-qd9sk8ih4h]

There's a rhombic hexecontahedron? I thought it's always a dodecahedron or triacontahedron.

[@erikhaag4250]

@@user-qd9sk8ih4h There is, It's also the logo for wolfram alpha. en.wikipedia.org/wiki/Rhombic_hexecontahedron

[@MichaelDolenzTheMathWizard]

What's a rhombic hexecontahedron?

[@erikhaag4250]

​ @MichaelDolenzTheMathWizard en.wikipedia.org/wiki/Rhombic_hexecontahedron

[@RonuPlays]

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

[@dysphoricpeach]

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

[@Kuvina]

Thank you so much! This is one of the most in depth comments of praise I've received and it's very encouraging :)

[@valentine6162]

Me watching this at 2 am, half asleep: “I like your funny words magic person”

[@0ans4ar-mu]

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

[@AlphaFX-kv4ud]

So basically it's a 3d version of the hexagon

[@Currywurst-zo8oo]

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.

[@0ans4ar-mu]

youe could well be right, im no polygon-zoologist @@Currywurst-zo8oo

[@Drachenbauer]

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.

[@HesterClapp]

I've watched this once, twice opposite, twice non-opposite and three times and I still don't really understand all of them

[@binauraltreatments6178]

Vastly Underrated Comment

[@Pixelarity64]

15:21 It must be my birthday! Look at that beautiful little chartreuse gremlin spin! Oh, how my heart radiates with joy!

[@jkershenbaum]

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!

[@JoseSanchezLopez-yf3lo]

this is by far the best video I've seen on the topic! it's incredibly well explained

[@Zekiraeth]

I don't know why, but polyhedra like these are inherently appealing to me. I just really love me some shapes.

[@MrBrain4]

This is an incredible video. Fantastic job, and thank you!

[@user-bu2mj2tk9q]

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!!

[@robkb4559]

Great video - I've been fascinated by polyhedra for decades and I learned some new things here. Well done!

[@davecgriffith]

Had to pause to comment - this video is excellent. Great job. Interesting topic, good visuals, good narration. Kudos!

[@ToadJimmy]

Beautiful very well done and well paced video! I love it and thanks!

[@leannviolet]

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.

[@node_deer]

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

[@stickmcskunky4345]

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. 🎉

[@Kuvina]

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)

[@user-dt8fr4up6j]

You are the literal personification of underrated

[@user-nm7gb3rw9c]

why tf would you need to normalise this tq+ bullshit literally in a math video smh

[@Shauryousee]

Highly appreciate the compilation ❣️

[@NHGMitchell]

Fascinating video, thanks for posting. Some years ago I assembled some of the Johnson Solids using Polydron (plastic panels that clip together)

[@realmless4193]

I've been looking for a good video about this exact topic for ages. So glad there finally is one.

[@lexinwonderland5741]

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 ;)

[@phobosdiscord5183]

You deserve way more than 4k subs, this a brilliant video

[@clarise-lyrasmith3]

I have been trying to find a good explanation of Johnson Solids for YEARS and this one finally satisfies me. Thank you :D

[@SunroseStudios]

these shapes are really cool, we enjoy how ridiculous the names get lol

[@louiesumrall358]

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

[@euanmccabe4962]

Excellent video! thank you so much

[@oliverstack7055]

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.

[@aidanmaniaMusic]

These are incredibly interesting, like platonic solids but stranger and there are way more. Love it!

[@RoxanneClimber]

Loved the video!

[@greggregoryst7126]

Wow thats one great video. To go through so many cases It must've taken a long time to make, good stuff

[@bennyloodts5497]

Solid work, my compliments!

[@colettekerr279]

Gonna be printing some of these. A+ infodump. Super well done

[@silas6446]

this channel is so underrated love your videos!!!!

[@zackf13]

First time seeing any video of yours, already my favorite enby math teacher

[@TheMDCXVII]

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

[@codatheseus5060]

Awesome! Good work!

[@mrbananahead2005]

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

[@davidsiriani9586]

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!

[@clockworkkirlia7475]

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!

[@samueldeandrade8535]

My Euler! This channel is a gem!!!

[@mrbenjiboy9527]

I will now use this information in life. Thank you so much.

[@matheuscastello6554]

fantastic video!

[@nono-xm8yl]

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.

[@Kuvina]

Thank you! I put a lot of thought into the colors so I'm really happy that it goes appreciated!

[@Harmonikdiskorde]

This was so chilling and exciting. And also as an origami person, I was basically thinking of how to construct each one!

[@dorianjack2240]

I absolutely love your videos

[@DissonantSynth]

The shapes are all so beautifully presented; could you please share the software you used? Or is it a code library, perhaps?

[@Kuvina]

I used blender! You can download all the STLs from wikimedia commons, and they're automatically public domain since they're simple geometry!

[@DissonantSynth]

@@Kuvina awesome; many thanks!

[@vaclavtrpisovsky]

​@@KuvinaI didn't know Wikimedia hosts 3D files. Thanks!

[@soleildj1572]

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.

[@muuubiee]

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.

[@Farzriyaz]

You: "This is a truncated icosahedron." Football: Am I a joke to you ?

[@michaellyga4726]

This KZbin 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.

[@inheritedwheel2900]

I'm thankful another person has commented on the incredible quality of this video. I agree!

[@NickenChicken]

Now I wish I had hundreds of magnet shapes, so that I could make these in real life. They look so collectible.

[@a-love-supreme]

i really liked all the solids constructed with lunes! my favourite has to be the bilunabirotunda, it's just so pretty

[@kennyearthling7965]

I loved this, especially the explanation on why there are only 13 Archimedian solids, great work!

[@rickyardo2944]

Master video presentation!! very very well done! and thank you.

[@-NGC-6302-]

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.

[@epikoof]

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

[@noone-ld7pt]

sensational video! Loved the term honorary platonic solids, definitely stealing that one! My personal favourite is the rhombic dodecahedron! :)

[@LeWolfYT]

so in other words tetrahedrons can create everything

[@LeoStaley]

I was so happy when you included those 4 honorary platonic solids!

[@thebigcheese10]

congrats on 6k subscribers

[@weillio1993]

your channel is so cool omg

[@jonahwolfe3396]

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.

[@IcosaMarty]

mine too!

[@antoniolewis1016]

I like these shapes - shapes are cool!

[@BunchaWords]

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!

[@millerwhite6915]

The blender is incredible! I love the little introductory twirl tytytytyty

[@cs127]

great video! once, twice opposite, twice not opposite, or three times

[@CoolyanEmoji]

Best vid ever, i rewatched like 5 times

[@Scraebler]

Very useful video, thank you.

[@ramonhamm3885]

This is a most excellent video! As a 3d puzzle designer and laser polyhedra sculptor, this helps show the relations between the shapes. ⭐

[@PretzelBS]

I have no idea how you make everything feel so concise and ordered. If I wanted to research this it would be so messy

[@jimiwills]

I love your videos ❤

[@TomatoBulb]

These shapes made my braid happy

[@aer0a]

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

[@WizardOfDocs]

Thank you for making a version of jan Misali's 48 Regular Polyhedra that respects its audience. I needed that.

[@atmatey]

Awesome video, thank you. It was really cool to see the systematic way of finding the Archimedean, Catalan and Johnson solids. Maybe you could talk about stellated polyhedra as well? I have looked at some stellations of the dodecahedron and the icosahedron and they are really beautiful but it is a bit unclear to me how to find them all and what are counted as proper stellations. I think they're so pretty that we even 3D-printed and assembled a small stellated dodecahedron as a prop for our university play.

[@jjchouinard2327]

Just wow! Knowledge dense, but not confusing.

[@mekkler]

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.

[@TaranVaranYT]

same with the icosidodecahedron (which is pretty much if the two fused together dragon ball z style)

[@SineEyed]

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..

[@Mr-__-Sy]

Glad you managed to show us all the crystal simetry cases with one video except for the pyritohedron

[@lord0fthecubes]

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!

[@funwithtommyandmore]

Lol there is 2 Dodecs

[@timnewsham1]

Your mathematical curiosity is beautiful and scary. Thank you.

[@aykarain]

i really like thinking about 3D shapes and stuff like that, so this is nice :D

[@kingwolf3044]

Very interesting. I love shapes

[@Team98]

Hell yeah! New kuvina video

[@ezdispenser]

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

[@Patricia_Taxxon]

rooting for this channel

[@perplexed_hapax_144]

I am happy that this video exists