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"

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.

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

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

Omg platonic solids

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

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.

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

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

Platonic solids Familial solids Romantic solids

The most important thing I noticed in this video is a new way to get to irrational numbers and ratios via geometry

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 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

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

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

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)

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

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

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

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

🥜 : cube 🧠 : square prism 🌀 : triangular trapezohedron

Us: How many 3-d solids you want? Kuvina Saydaki: yes

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

Truncated Icosahedrons = soccer ball pattern

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

Great now I need a hystericaly elaborate polyhedra family tree diagram >:(

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

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

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

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

This is the type of video I hope gets preserved after the internet gets destroyed or restricted or some great data loss happens within KZfaq’s servers

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

My Euler! This channel is a gem!!!

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

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. 🎉

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.

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!

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

i swear geometry is both infinite in possibilities AND beautifulness

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

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.

Hang on, aren't soccer balls truncated isocosahedrons?

This is the first time I've ever heard of a disdyakis triacontahedron, but upon discovering what it is, I now want one.

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!

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 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!!

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

I’ve always LOVED the Catalan solids, definitely more than the Archimedean solids, …maybe more than the Platonic solids.

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

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.

Fun fact, Many of these have several different ways of construction, The Rhombicuboctahedron, For example, Can be formed by rectifying a cube or octahedron, And I _presume_ it can also be acquired via truncation of a Rhombic dodecahedron, As that's where the name comes from, Although I can't actually confirm that.

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 my third time trying to watch this video, and my mind was finally ready.

Why do the shapes look delicious

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.

The Pseudo Rhombicuboctahedron is called "elongated square gyrobicupola". I love this video, could watch it over and over again. Thanks!

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

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.

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

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

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 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.

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

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.

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

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.

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

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

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

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

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 ;)

because of this video, i wanted to make all platonic, archimedean, and catalan solids and now my geometry teacher has an entire drawer full of paper polyhedra

This KZfaq 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.

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

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

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.

Fun fact: the platonic solids, catalan solids, and four infinite families make up all mathematically fair dice. You can actually buy most of them, and a metal D120 (your favorite) weighs half a pound.

if jan misali's video of "there are 48 regular polyhedra" is the science in making a nuclear bomb this video is the science in making a nuclear reactor no hate on jan misali's video and i love that video too but that video *feels* like a kid making some crazily creative ideas but lacks structure but this video pushes my button on "building with what you already have" and i love every second of this well done Mr Kuv! ❤

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

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

I LOVE WATCHING EDUCATIONAL GEOMETRY VIDEOS MADE BY NON BINARY PEOPLE ‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️‼️

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

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

These shapes made my braid happy

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 :)

19:42 I didn't expect this formula, I thought it would be simpler

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

Just wow! Knowledge dense, but not confusing.

I absolutely love your videos

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.

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

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

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!

Platonic solids: the OGs Archimedian solids: a bit generic but still cool Catalan solids: the “I’m not like other solids” gang (absolutely adore them btw) Johnson solids: the weird ones

You missed the fact that the rhombic dodecahedron can also make a honeycomb. Honeycombs are to 3D space as tiling is to 2D space. One thing even more special about the rhombic dodecahedron is the fact that its honeycomb is the most efficient for sphere packing.