The Mysterious Hyperdice Sequence

[@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

[@MeshremMath]

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

[@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.

[@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.

[@stickfiftyfive]

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

[@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

[@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"?

[@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!

[@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

[@darkstar342]

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

[@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!

[@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.

[@thatssokwekwe]

This channel should be required viewing for any mathlete

[@Metaller1982]

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

[@DecayedPony]

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

[@evancherpeski1876]

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

[@tjeepert9782]

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

[@sirludicrous7823]

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

[@chinmayjoharle9065]

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

[@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.

[@qmoyer]

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

[@jesnoggle13]

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

[@vanevarnel7131]

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

[@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!

[@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.

[@azimuth4850]

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

[@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

[@ПлатонАнтонов-о9я]

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

[@rhodesmusicofficial]

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

[@publiconions6313]

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

[@frimi8593]

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

[@natalie5947]

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

[@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.

[@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.

[@jmm1233]

Nice cameo for the squirrel

[@james10492]

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

[@Dracopol]

12:47 A squirrel scurried by!

[@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

[@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

[@wesleydeng71]

12:47 A squirrel visits your backyard!

[@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

[@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 😊

[@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

[@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!

[@Kreypossukr]

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

[@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

[@thejaywalker8922]

I love your videos man!

[@iranzijlion7674]

Thanks man, your videos are helpful. Keep delivering 👏

[@sillypilly1234]

Great video! lots of interesting stuff and entertaining

[@MabInstruments]

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

[@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,

[@PhilHibbs]

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

[@marveloussoftware1417]

Very cool!!

[@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.

[@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 😎

[@BlueMayze]

combo class ost mixtape when domotro?

[@ComboClass]

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

[@BlueMayze]

@@ComboClass sweet

[@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)

[@potatoheadpokemario1931]

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

[@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

[@alamagordoingordo3047]

I'm so fascinated by geometry in higher dimensions.

[@steelegagnon5273]

can’t wait for 4d 😝😝

[@FirstLast-oe2jm]

incredibly fun video as always

[@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

[@OrangeDrink]

First hyperdice. Imagine the shape of infinite -1 size, a circle difference from a triangle would a walk across verticis even be visible/measurable. My assumption is that all variables include if your whole perception was greater than one unit the answer is no, but if your perception was an infitesimal or slightly greater then it would be not perceived

[@erdmannelchen8829]

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

[@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).

[@brandonstokes5927]

great video as usual

[@katiebarber407]

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 "

[@sthubbar]

Thank you.

[@epicdaniel508]

This is amazing

[@cec4]

this is top quality stuff

[@november666]

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

[@MrKyltpzyxm]

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

[@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

[@jesnoggle13]

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

[@joaomrtins]

12:47 esquilo!!!

[@malicholson5767]

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

[@veggiet2009]

More 4D videos!

[@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 :)

[@DragonslayerProd]

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

[@Brunoenribeiro]

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

[@denischen8196]

Is it proven that there are only 3 regular convex polytopes in higher dimensions? Is there any hidden 5D or 6D polytope that has yet to be discovered?

[@boghag]

It's proven. They don't fit around the vertices anymore, but you can always make a Simplex (Tetrahedron equivalent), a Hypercube and the Dual of a Hypercube

[@dranorter]

What about Catalan dice though??

[@DIPHENHYDRAMINEMAN]

I love this guy :)

[@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

[@amesstarline5482]

The cube representing earth is fun, considering Minecraft.

[@christosmani]

Squirrel was a paid actor!😂😂

[@Skeleton-bs7zy]

Distilled crazy math man

[@jofx4051]

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

[@ezhanyan]

12:48 omg, a squirrel

[@Jacob-yg7lz]

I just noticed that the 5 pointed star is the dual shape to the pentagon

[@GorduneDelaine]

having dreams where I'm a clock and you shatter and burn me and I only have one mark that says 0 and while the hand moves between 0 and 0 I am too scared to wonder how many subdivisions I have so I close my eyes and let the fire take me

[@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.

[@matthewhafner962]

That's a lot of 3's. Did Data subliminally screw something up?

[@melody3741]

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

[@good.citizen]

🎲 thank you good luck.

[@maht0x]

even god plays dice, Plato said