The Mysterious Hyperdice Sequence

In this class, we'll meet some mysterious sequences of numbers and shapes, use art to prove geometric truths, use the "four elements" to learn about history, and more! This is the longest and craziest episode I've made yet. Thanks for watching! Let me know in a comment which shape is your personal favorite.
Disclaimer: Do not copy any actions you see in this video. This is for educational purposes.
Some topics discussed in this video include: number sequences, dice, regular polygons (triangles, squares, pentagons, and beyond), the platonic solids (cube, tetrahedron, octahedron, dodecahedron, icosahedron), Plato's opinion of the universe's elements, convex vs. star polytopes, dual shapes, the Euler characteristic of faces/edges/vertices of polyhedra and planar graphs, the strange 4D shape known as the hyper-diamond or 24-cell, and what perfect polytopes can exist in the 5th dimension and beyond. We'll return to the 4D dice-mention in a later episode...
Combo Class, taught by Domotro, is an unconventional learning experience where anybody (whether they're a fan of normal school or not) can become excited to learn rare things about math, science, language, and more. Also check out the shorter videos on the Combo Class Bonus channel. Thanks for coming to Combo Class!

[@xbolt90]

Always was jealous how the fourth dimension gets one more platonic solid than we do. 4D D&D must be pretty fun!

[@sapphiresuicune6787]

😂😂😂

[@stevenjacobs2750]

More dice doesn't equal more fun haha. Single dice systems ftw

[@TheBasikShow]

Would they call it D&D&D?

[@publiconions6313]

Lol

[@cashkurtz5780]

The rhombic dodecahedron is the 3D equivalent to the last 4D polytope.

[@bonecanoe86]

I bet if they showed your videos in school a lot more kids would be interested in math!

[@iamdigory]

Do-mo-tro the math guy!

[@gabrielbarros493]

Some would be disencouraged

[@danieloconnor4638]

Super vid! Jan Misali has a video called "there are 48 regular polyhedra" that explores more concave polyhedra and other weird shenanigans like the "spiky" polyhedra that Domotro talked about.

[@SunroseStudios]

was gonna comment this! very good video

[@Hamboarding]

Me too, I should have looked for the comments first!

[@clutchbridge]

So was I lol. There was also a small part of me that just expected to see him randomly comment on the video himself

[@mesplin3]

Right... I was a little surprised that he didn't mention why the hexagon tilling of the plane was not considered.

[@vii-ka]

Here is the link: kzbin.info/www/bejne/lZnNg6mQjtB0naM I just realised people will think I am a bot, but no this is the video mentioned.

[@ghastlyGrenadine]

the only guy to simultaneously use both the singular "verticie" and the plural "vertexes"

[@gary.h.turner]

Perhaps he also thinks the plural of "spouse" is "spice"?

[@RyanSnead]

Squirrels like the one appearing in the video at 12:48 are natural experts a gliding through the air which I noticed is the Platonic element being discussed at 12:48. And 12 which is the number of minutes elapsed in the video can be multiplied by 4 to make 48 which is the number of seconds in the video which makes that particular neighborhood squirrel a very special guest in this video.

[@Wyrmfell]

I've known about the platonic solids but never knew *why* they were the only ones, really good job explaining!

[@sketchditty1054]

I really appreciate you bringing a fresh and interesting way of teaching math. Not just your topics but you as an entertainer is what makes this channel so special. Keep up the awesome work!

[@ww4830]

I love how loving and silly your videos are. Makes all of this complicated geometry seem so simple!

[@darkstar342]

Dude, you're making awesome stuff, glad I found your channel.

[@1234567zeek]

A "normal" person would require about half a year to produce this video. I catch myself laughing when I'm mentally saturated. Thank you for the excellent videos.

[@connorhart2793]

Just love this guy, the perfect mix of entertainment, enthusiasm and education.

[@Lugmillord]

The only channel where the clean up is the most time-consuming part of video production.

[@lexinwonderland5741]

This guy is fucking crazy. I love him. He's so excited to share and obviously the chaos aesthetic does him well. I feel so understood by him, this is about how it goes when I show friends or students my corner of academia. Subscribed!

[@stickmcskunky4345]

The Fourth Dice-mention. So. Very. Excited.

[@wesleydeng71]

12:47 A squirrel visits your backyard!

[@Metaller1982]

dude you're like a weird mix of explosions and fire and numberphile i absolutely love it!!

[@thatssokwekwe]

This channel should be required viewing for any mathlete

[@sirludicrous7823]

Awesome content, awesome video But is nobody going to talk about the scream at 2:20? xD

[@DecayedPony]

This channel is by far the craziest most entertaining mathematics related thing I've ever seen. Amazing

[@chinmayjoharle9065]

This is one of the most interesting channels I've seen. Keep up the quality videos man 👍

[@evancherpeski1876]

This is my favorite new channel; every video is very good education and entertainment!

[@qmoyer]

Hi Domotro! I've always been fond of the dodecahedron and the truncated icosahedron. Stellar video today!

[@jacobbates7766]

A beautiful sequence and a brilliant explanation! This is some of the most awesome educational content I've seen and it's really inspiring, can't wait for the fourth dimension!

[@coltenh581]

I’ve maybe never understood the “spherical x in a vacuum” joke more than I did when you had to put up the info card with all the caveats and precise language 5:30

[@tjeepert9782]

I literally can't express how much I love this video and your entire channel. KZbin needs more of your style.

[@Kreypossukr]

I LOVE your channel and your videos, keep up the good work !

[@azimuth4850]

Loving this series. Learn something I didn't know each time.

[@rhodesmusicofficial]

Your style of presentation is super refreshing! Really great stuff man

[@user-we9mj5yo1w]

i love the quality of your lessons and the amount of jokes. Your channel is amazing! hope you get some more subscribers soon 💜

[@BaldRocket]

Your stuff is incredible! I’m glad I came across your channel

[@HomieSeal]

Glad to see another video from you :D I’ve only seen a tiny bit so far, higher dimension shapes have always been very cool so this will be interesting

[@natalie5947]

I've seen about 6 combo class videos by this point and I'm still not acclimated to the chaotic energy demotro brings.

[@garlicbread5214]

Very cool to see more videos on the regular polyhedra. This also explained duals much better than Jan Misali's video that went into like all 40 or whatever many shapes there actually are without the restrictions you mentioned at the beginning.

[@ThomasHaberkorn]

Your style is quite unique, love it

[@zeshan144]

I do love your channel, I have recently took an interest into maths and enjoy geometry and number theory which I see you do alot of. Thanks for the help 😊

[@thejaywalker8922]

I love your videos man!

[@publiconions6313]

So glad YT put this dude in my feed - wonderful!.. good thing about math channels is there no shortage of fascinating content.

[@peppermann]

You have a truly gifted way of explaining both simple and complex maths topic Dimitri, I’ve taught about the Platonic solids for years and you’ve out-taught me without even trying.

[@vanevarnel7131]

Thanks great video. I love the lab coat and the desk. Good job filming the video out in the fresh air.

[@marveloussoftware1417]

Very cool!!

[@iranzijlion7674]

Thanks man, your videos are helpful. Keep delivering 👏

[@jesnoggle13]

Funny and informative and really dorky. I’m hooked!

[@james10492]

This content is so good... you sir - kudos.

[@themetal]

I can see why Plato believed the gods my have used 12 faced figures to create the universe; coincidentally, or not, 12 is a very important concept that regularly appears throughout almost everything.

[@frimi8593]

God these videos look so fun to shoot. I can’t wait for this channel to blow up

[@FirstLast-oe2jm]

incredibly fun video as always

[@aliasjon8320]

I thought pop culture desensitized me to mad scientists, but this guy is a mad mathematician

[@chri-k]

A mad mathematician might be worse than a mad scientist

[@sillypilly1234]

Great video! lots of interesting stuff and entertaining

[@Dracopol]

12:47 A squirrel scurried by!

[@epicdaniel508]

This is amazing

[@infinitesimalperinfinitum]

I don't know how you're doing it, but please continue spying on my watch history and releasing videos that explain the things I'm unsure of. It's unnervingly helpful.

[@brandonstokes5927]

great video as usual

[@vincentv.3992]

Thank you so much for your awesome videos! I would love to see the net of a 4D-Hyperoctahedron!:-) From the 4D-Hypercube, one can easily find many representations, but not from the other ones.

[@sthubbar]

Thank you.

[@charlygutierrez1338]

Gosh, as i said before, i'll watch every video until i get it, bc the way you're skills teaching are amazing!! Greetings from México lol,

[@GODDAMNLETMEJOIN]

The 24 cell is both the cuboctahedron and rhombic dodecahedron of the fourth dimension at the same time.

[@cec4]

this is top quality stuff

[@jmm1233]

Nice cameo for the squirrel

[@steelegagnon5273]

can’t wait for 4d 😝😝

[@DIPHENHYDRAMINEMAN]

I love this guy :)

[@flockofwingeddoors]

Just discovered you the other day, love what you're doing with your videos! The chaotic energy really compliments the math. Very interesting at 1.5x speed haha Also, isn't there an argument for infinite regular polytopes in 1D since you can make your like of any length you want? Alternatively, it should be zero since it doesn't enclose an area?

[@ComboClass]

Thanks! About your questions: different sizes of line segment wouldn’t count as different types of polytope, similar to how different sizes of cube all count as the same type. The specific length doesn’t matter, what matters is how the edges/sides/vertices go together, and if you considered different sizes as different types then any of these dimensions would have an infinite amount). And the generalized “hypervolume” of a shape’s interior in 1d is length (then in 2D it’s area, in 3D it’s volume, etc)

[@calebclark9114]

Nice video! Also did you compose the music in the intro?

[@ComboClass]

Yeah I made all the music in this episode (and many of the episodes)

[@calebclark9114]

@@ComboClass 😎

[@TheMayzeChannel]

combo class ost mixtape when domotro?

[@ComboClass]

@@TheMayzeChannel I'll definitely release more musical projects when the time is right :)

[@TheMayzeChannel]

@@ComboClass sweet

[@jesnoggle13]

The tetrahedron is also special because it hurts the most when you step on it barefoot.

[@november666]

Domotro is such a cool name. It sounds like someone from a comic book

[@filthysewerrat3490]

I like to imagine 4D shapes by visualizing 3D shapes stacked to fill a space folding outward in a regular pattern through a dimension that I can't really see

[@paulfoss5385]

The d10 is the dual to the pentagonal antiprism (two opposite facing pentagons on parallel planes joined by ten alternating triangles) The pentagonal antiprism has all identical vertices but it has two pentagonal faces and 10 triangular, so its dual has all identical faces but the vertices aren't all identical, which is why the faces are kites. Interestingly, the cube can also be viewed as the dual to the triangular antiprism (the octahedron). Notice how the d10 has kind of a zig zag equator. Hold up cube/d6 by antipodal vertices, and you can see the cube as also having a zig zag equator.

[@potatoheadpokemario1931]

there is a hyper Dimond in the third dimension, it's just not a perfect shape, it's called the rhombic dodecahedron

[@alamagordoingordo3047]

I'm so fascinated by geometry in higher dimensions.

[@Dracopol]

The reason for the number-sequence 1, infinity, 5, 6, 3, 3, 3...was already covered in this video. kzbin.info/www/bejne/aKSXhaSMdseZsJY

[@simonwillover4175]

2:35 - I like how we can hear the Neighbor's kids in the background.

[@ComboClass]

I’ve talked to that neighbor, he’s nice. We’ve discussed the noise and are both cool with each other :)

[@Brunoenribeiro]

12:47 "Aaaaaaair..." **squirrel**

[@RandomAmbles]

Quick shout-out to the 720⁰ "angular defect" of all convex polyhedra and the free 3D-model-to-2D-net program Pepakura! Shapes - shapes everywhere!

[@insanitycubed8832]

There are actually a lot more fair dice than that. A fair die is pretty much a polyhedron composed entirely of congruent polygons and have some platonic solid symmetry. That definition may fall short, but for instance catalan solids can make fair dice. Edit: I wonder how many of those are in each dimension. Also forgot my manners. Cool video, and cool squirrel too

[@kikivoorburg]

I vote that we replace “non-convex” with “nonvex”. Now that I think about it I’m not actually sure if non-convex implies they’re concave or if there are shapes that count as neither

[@MrKyltpzyxm]

Salt is a mineral. Salt is a cube. Earth is minerals. Earth is cubes. Vindication for Plato! 😂

[@Gunbudder]

my favorite concept about higher dimension geometry is that the more dimensions you have in a vector, the less meaning its distance from another vector has. or another way of putting it, if you calculate the distance between two n-dimensional points, that distance has less meaning as n grows to infinity. What i mean by "meaning" is that the distance has less information and becomes less useful for analytics. i cam across this when studying machine learning with massive vectors doing something like nearest neighbor. the classic example is a dataset that is a huge list of 22D vectors, each one corresponding to a yes or no. you train your nearest neighbor system on that, then feed it a new 22D vector and see if is closer to the yes or no vectors. Well, my thinking was if 22 dimensions is good (and you literally use an expansion of the distance formula for 2D space) then why not 220 or 22,000,000 dimensions? well i found that the research had already been done, and they found that for sufficiently large dimensions, the accuracy of determining if an unknown vector was "yes" or "no" dropped from 97% to something like 50%, which was worse than the crystal ball method (just guessing). its not too much of an issue though because most practical applications of this method of machine learning use physical parameters of some type. even "big data" has a practical limit on a vector, which is often a person. there are only so many things amazon can measure about a human to determine if they want to buy something or not before they show it to that person. also neural nets just kind of blasted past that method of machine learning at mach 10 in the past few years, so it seems like no one really cares about it anymore anyway lol

[@PhilHibbs]

The ten sided shape is a dodecahedron with two opposing faces extended out to a point.

[@veggiet2009]

More 4D videos!

[@DragonslayerProd]

"Cubes were believed to represent earth" Mojang wants to know your location

[@Skeleton-bs7zy]

Distilled crazy math man

[@OrangeDrink]

I would guess on 2d the shapes that tesselate the plan would only be counted, but regardless watching you smash math to pieces is fun thanks

[@tilnation14]

My problem with calling them hyperdice is that there are a bunch of other shapes, like the catalan solids, that also work. Wolfram has a nice list(I guess this list is exhaustive) of 30 isohedron that would all work as dice (although a few, including the tetrahedron, don't have a 'top' face, making using them as dice more difficult).

[@clown134]

I recently watched a great video dealing with non-con vex polytopes. I forget exactly what it's called but something like " there are 47 shapes "

[@MabInstruments]

A 3D equivalent of the 4D 24 cell is the rhombic dodecahedron, though it isn't regular, of course.

[@asdf30111]

Way I learned to "imagine" 4D for example tesseract, pick the "middle slice" of your shape, in this case we get a cube. Color it purple. Superimpose a red and blue cube on the same exact spot. Then between the three cubes, add more cubes of increasing and lowering amount of red and blue color until you connect the structure together. Though sadly very hard to keep a stable picture or even to rotate it in your mind.

[@christosmani]

Squirrel was a paid actor!😂😂

[@good.citizen]

🎲 thank you good luck.

[@erdmannelchen8829]

12:01 Didn't know that Play Dough is that old.

[@ThisCanBePronounced]

"don't copy any actions from this video" I shouldn't do math? D:

[@ComboClass]

You can copy all the knowledge from the video. Just not the parts with fire and falling clocks and breaking desks :)

[@HomieSeal]

@@ComboClass instructions unclear, calculated how many 3d hyperdice shapes are needed to break my desk and make all of those dice fall over

[@ThisCanBePronounced]

I don't know, desk-breaking looks highly educational. 😜

[@amesstarline5482]

The cube representing earth is fun, considering Minecraft.

[@ezhanyan]

12:48 omg, a squirrel

[@cheeseburgermonkey7104]

There's a great Steam game called 4D Toys that helps you visualize this weird and wacky world with 4 orthogonal dimensions instead of just 3 and shapes in this dimension, it's super cool and if you really wanted to know how to visualize the 4d equivalent of the cube, tetrahedron, octahedron, and other 4d platonic solids, i would definitely recommend it to you

[@kirbymia6209]

You make maths interesting.😆

[@jofx4051]

How to start fun math: *Burning things and messing things*

[@azzteke]

The singular of vertices is vertex, not vertice!

[@malicholson5767]

so care to explain the 48 regular polyhedra described by yan misali i found that video interesting

[@melody3741]

I've heard people say vertexes before but I've never heard people say vertice

[@noahnaugler7611]

That's the thing though, dice only have to be Isohedral, not regular. There are infinitely many Isohedral figures in 3 dimensions, but only 5 regular figures. I'm really curious about the limits of the convex isotopic (cell-transitive*) figures in 4 dimensions. What kinds of fair 4d dice are possible? * I came up with this term myself, based on the etymologies of "isohedral" and "isotoxal", so it may be incorrect, but I couldn't find an alternative in my admittedly brief research

[@kittyn5222]

Rip desk: always dying-

[@dranorter]

What about Catalan dice though??