Can you guess a shape from its shadows?

Рет қаралды 463,238

Күн бұрын

This video explores the question of whether you can determine the shape of a 3D object by observing a few of its shadows. The results are surprising and beautifully rich!
I created the slides in PowerPoint, and I wrote the music using a trial version of Ableton Live 11.
For more on this topic, here are some resources from Hideki Tsuiki (‪@hidekitsuiki1551‬) :
- Video showing the 3D prints and their various shadows (Watch This!): • Shadows of Fractal Ima...
- Imaginary cubes webpage: u.kyoto-u.jp/icube
- Imaginary cube sculptures and Latin squares: archive.bridgesmathart.org/20...
- Imaginary cubes and fractals: archive.bridgesmathart.org/20...
- Imaginary cubes and packing puzzles: www.mdpi.com/1999-4893/5/2/273
- Imaginary cubes and hypercubes: link.springer.com/chapter/10....
Special thanks to ‪@feynmanschicken‬ for offering feedback during the creation process.
0:00 - Introduction
2:48 - Imaginary cube patterns
4:32 - Latin squares
5:43 - The replacement trick
6:22 - Fractals
7:44 - Further exploration
#SoME3

Пікірлер: 1 000

@PrimerBlobs 11 ай бұрын

Nice video! Impressive that you also made the music.

@TripPy_Poly 11 ай бұрын

Hi primer, the evolution simulation content creator (idk)

@pirilon78 11 ай бұрын

The man himself

@LadyTink 11 ай бұрын

Imagine having 166 subscribers, and having Primerblobs comment on your video >.< Huge fan btw >.>

@kaininjago6161 11 ай бұрын

it is the man!

@BurningShipFractal 11 ай бұрын

@syriuszb8611 11 ай бұрын

You can make imaginary sphere as simple as just making 3 discs parallel to the shadows they supposed to recreate.

@DisguisedParrot 11 ай бұрын

Indeed, but that really be able to be a stable object? Also love the idea!

@BryanLu0 11 ай бұрын

@@DisguisedParrotYou can intersect the discs and it won't affect the shadows

@tokarak 11 ай бұрын

Notice that you can’t get rid of the three 1-spheres around the centre of the 3-ball, because shifting it will move it outside the 2-ball projection locus in at least one of the other projected dimensions.

@tokarak 11 ай бұрын

Also: the surface area of a unit hollow sphere (a 2-sphere) (SA = 2tau [tau 2pi]); the surface area of 3 disks (3 2-balls) (SA = 3/2tau). In addition, half of the 8 congruent “quadrants” of the 2-spheres can be removed, in a very similar way to the 8 “quadrants” of a cube in the video, getting the SA down to 1 tau.

@tokarak 11 ай бұрын

It’s interesting that we are using surface area, rather volume. But in the video, notice that under infinite recursion of the cube fractal, with the volume halving each time, you also get a shape with 0 volume, but with surface area (which is equal to the area of the projected square). I wonder what this class of fractals is called.

@Dysiode 11 ай бұрын

I gotta say, this felt almost Blue's Clues-y, but as the highest possible praise. The almost uncomfortably long pauses after "can you figure this shape?" and "see if you can make an argument..." got the cogs in my very adult brain churning. Very impressive how you made it more educationally engaging while still keeping the same high level of polish and clear explanation videos from Numberphile and Matt Parker are known for

@woodybob01 11 ай бұрын

I was so f-ing obsessed with blues clues when I was a kid

@vigilantcosmicpenguin8721 10 ай бұрын

This is accurate. Sometimes you need a little Blue's Clues-iness.

@tttITA10 10 күн бұрын

This guy likes when the book goes "we'll leave the demonstration as a puzzle to the reader".

@upsidedown-pug1974 11 ай бұрын

for that circle shadow one, a way you could do it is take any imaginary cube and place sphere inside and remove all parts that extend outside the sphere. This should make imaginary _~circles~_ from the shape and the correct angle

@bengobler 11 ай бұрын

Very clever!

@voidwyrm6149 11 ай бұрын

that should work with all imaginary cubes that are solid and continuous. for "granular" imaginary cubes, you may need to go back and patch up holes in the sillhouettes, and for 2d surfaces, it might not work at all (the intersection of a ball and hollow cube of the same width is just the midpoints of the six faces)

@Luna5829 11 ай бұрын

but it's not technically a circle, because if you zoom in far enough, you can see the zig-zags

@kaininjago6161 11 ай бұрын

@@Luna5829 it would be like taking a slice; there would be no zigzags

@Mikelaxo 11 ай бұрын

That would work with literally any shape in existence

@sokijohnson1309 11 ай бұрын

This video is flawless, not too Mathy, not too long, a clear motivation even for non mathematicians, good interactions, links to go deeper, etc... I hope you get selected, good job! 😊

@0xEARTH 11 ай бұрын

dude the production value here is insane. clean, precise, and simple, while not assuming we don't know anything, with well-made visuals to match; i actually learned something that i might use for something with this! this is absolutely nutty, i'mma stick around to see what else you got!

@hidekitsuiki1551 11 ай бұрын

Thank you for creating an excellent video! Ever since I discovered the H fractal and T fractal, I have been dedicated to raising awareness about them and their significance. I'm thrilled that you've become a part of this endeavor.

@RohitKulan 11 ай бұрын

H and T, your initials!

@wilczus222 11 ай бұрын

Wait, it's the man himself, I just realized!

@vigilantcosmicpenguin8721 10 ай бұрын

'Tis a noble cause.

@weibrot6683 9 ай бұрын

If you want to know what the thumbnail shape would look like, imagine a cylinder with the round parts at the top and bottom, now squish one of the circles together to a line and keep it in the center, that's the shape that would make a circle, triangle and square shadow

@Sir_Gilbert 8 күн бұрын

Imagine like a toothpaste tube on its cap but it's really short

@unknowndeviceconnected 4 күн бұрын

Imagine like the bit of a flathead screwdriver

@user-id2nr1zp1u 11 ай бұрын

I really like the clever trick at 6:00, also the visualization at 6:20 looks like those tiny trinkets or dice you can play around in your hand, they're just so lovely!

@pocarski 11 ай бұрын

I discovered this accidentally when I wanted to see what the bitwise XOR function does to a plane. If we map discrete points in space to (x, y, x XOR y), it forms a Sierpinski pyramid standing on its edge.

@HansLemurson 9 ай бұрын

Wow, that's really cool!

@KarmaKomet 11 ай бұрын

Really wonderful video! Great pacing and kept me interested the whole time.

@scikick 10 ай бұрын

This was captivating. Thank you.

@confusion5025 11 ай бұрын

This was an amazing video, you definitely deserve to win! The question was really interesting, and it was neat to see how the pattern of four cubes could be repeated inside itself to form a tetrahedron. Also, I really liked the animation style, it was aesthetic and also conveyed the mathematical ideas very well.

@j.s.42822 11 ай бұрын

Fantastic content, narration, visuals and music. On top of that, I think that this is one of the few SoME videos that doesn't feel like it sacrificed any rigour in its reasoning. The mathematics is pitched perfectly for what you can communicate in a 10-minute (or so) video, and then executed expertly. Very well done.

@Atomas69 11 ай бұрын

this is really cool! Brings up a cool concept that can be played with in further complexity! I also love how enthusiastic you sound in this, makes anyone watching feel the same sort of wonder, encouraging the discovery of new things this newfound topic

@yassinesafraoui 11 ай бұрын

I love how you make things crazy simple and yet they still feel fascinating, elegant!!

@Llamaful 11 ай бұрын

I started the video thinking it was a very easy and obvious challenge but after the fractal reveal I was hooked. Really good job!

@HazhMcMoor 9 ай бұрын

I didnt watch the video because the thumbnail and title looks lame lmao. When 3b1b said it's about fractals instead, then I watch it.

@mikeebrady 11 ай бұрын

How are you going to use one shape for the thumbnail and then never give the answer to that one in the video? Talk about click bait.

@blobofdespair 11 ай бұрын

This was so well made. The visualizations made it really easy to follow along and the presentation made it so interesting!

@zanop15 11 ай бұрын

I absolutely loved the video! I am, particularly, very enthusiastic whenever the topic is mathematics but I could easily see any of my friends thinking this pretty cool too! The soundtrack, visualisation and explanation is great.

@EleetCanoe 11 ай бұрын

This video is brilliant! I didn’t really get it but the presentation is on another level! If I think hard enough, eventually I think I’ll get the whole shadow thing.

@DisguisedParrot 11 ай бұрын

I love this video soo much!!! You are great at making them

@jonuuh 10 ай бұрын

Really fantastic video, was totally blindsided by the connection to sudoku. Love this style of video and would love to see more!

@hymanimy 11 ай бұрын

This was so awesome. Perfect pauses taken before reveals - kept me as the viewer primed for the next thing.

@Khaktos 11 ай бұрын

I just love how approachable this video is. I could show this to an 8 year old and they would most likely understand everything. But it is not dumbed down, and even an adult can find interesting puzzles and problems behind this idea

@thecalculusofexplanations 11 ай бұрын

This is simply an amazing entry, you deserve a pi creature for this! And a compliment from Primer as well, epic. I will continue to aspire to this level of quality in my own videos. Well done sir.

@Rafael4 11 ай бұрын

Awesome video! Great topic, simply but very well explained, not too long and not too short, very well illustrated, and great sound quality as well (both the music and your voice). Thank you for que high quality content!

@BsktImp 11 ай бұрын

Wow! What a great punchy little video on a fascinating new-to-me topic. Quite aside from the SoME competition, this hits all the right notes on immediacy of engaement, piquing curiosity and leaving me asking more questions. And the revelation that this was done in ye olde PowerPoint _and_ you wrote the music too...! I salute you.

@bengobler 11 ай бұрын

Ha! Old but mighty. The "morph" transition is a workhorse. Without it, I'd have to learn how to animate for real...

@Swagpion 11 ай бұрын

The imaginary sphere is the shape that comes from the intersection of 3 cylinders, each one perpendicular to eachother.

@jules325 11 ай бұрын

I thought at this in first, but it seems to "exceed" if you try to construct it mentally

@amaryllis0 11 ай бұрын

@@jules325 See wikipedia: Steinmetz_solid#Tricylinder

@jetison333 11 ай бұрын

@@jules325 the union of three cylinders is too big, but if you only take the parts that are part of all three cylinders you get a shape that succeeds.

@Swagpion 11 ай бұрын

@@jetison333 yup, I even have a 3D printed version of it. I made it in Tinkercad and got it made real.

@jaki1876 11 ай бұрын

@MemeSwag doesn’t that just make a sphere?

@koendos3 11 ай бұрын

Finally a math video that is refreshingly elegant and suprisingly simple! Love it. Kinda reminds me of those 3d-print, that displays 2 different words, depending on how you look at it.

@Ellen-mw5hn 11 ай бұрын

Great production and so interesting!

@jugglars 11 ай бұрын

So much more than I expected - amazing. Thank you very much

@iau 11 ай бұрын

I think the only thing missing was an animation that showed how the fractal shape actually projects a cube from all directions simultaneously. I understood it _must_ obviously work. But I kinda felt robbed of the satisfaction of seeing it at work, and until I got it I was just _trusting_ your conclusion instead of seeing it with my own eyes.

@davidsorensen2116 10 ай бұрын

Even just the solid imaginary cube shapes before they became fractals would have helped. The demo with the 3D printed model was convincing that at least one possible projection was a square, but didn't show that the projections orthogonal to it were also squares.

@dyld921 11 ай бұрын

The thumbnail is misleading, I think. I was expecting this to be about how to contruct a 3D object from any given set of 3 shadows.

@proxagonal5954 3 ай бұрын

That's not possible unless the object is given to be convex

@williamhoyle6644 3 ай бұрын

The title also plays into that. It was still a very interesting video! But not at all what I was hoping to watch.

@shelly8779 2 ай бұрын

/op womp womp

@FeWi-YT 12 күн бұрын

The thumbnail didn’t even say that and constructing said shape from 3 shadows requires logical thinking

@RocksAreEverythingInTheEnd 7 күн бұрын

I think it's cone with a square cutting it in half.

@tfr 11 ай бұрын

one of the best videos of youtube. keep it up brother. intriguing stuff for sure

@DirectedArt 11 ай бұрын

What an incredible video man Keep it up

@jakobr_ 11 ай бұрын

I thought about this sphere thing before, and surprised myself with the conclusion that the “meet” of three mutually perpendicular cylinders isn’t actually a sphere. You get something resembling a rhombic dodecahedron or cube. I just looked it up, it’s called a “tricylinder”. The sphere is hidden inside this shape, only poking out around three equators. There’s still more to shave off on the corners before it becomes a sphere. I expect this to be the maximal shape that fits in those shadows.

@K-o-R 11 ай бұрын

If you think about the bit outside the cylinder, you see things like two arched ceilings intersecting to form a vaulted ceiling.

@PennyLapin 11 ай бұрын

I really like how excited you sound through the whole video! The best part of these events is that they inspire people to share about the things in mathematics that they love so much!

@SuperLlama88888 11 ай бұрын

So cool, great video!

@skylardeslypere9909 11 ай бұрын

This is a really fun video to watch. It's not difficult to follow, and it explains clearly why the thing that works, actually works. Nice

@BarelyNoticeable 11 ай бұрын

You struck gold with this tone of voice and the animation style and the not dumbed down but simple enough for most to understand scripting in this video, keep doing what you do mate

@latehate1017 11 ай бұрын

Super underrated! I love the 3-d prints you have made for the video too, shows that what we learn in math can be applied in the real world. Continue making more videos!

@bengobler 11 ай бұрын

Thanks! The 3D prints were gifted to me by Hideki Tsuiki (@hidekitsuiki1551). Check out his channel for more videos about these shapes and their other shadows!

@redpepper74 11 ай бұрын

It’s always cool when you can make abstract concepts physical and tactile :)

@UnKnown-xs7jt 11 ай бұрын

Excellent presentation! Thanks for this lesson

@ulrichberger5553 6 ай бұрын

Exceptionally clear!

@bastienpabiot3678 11 ай бұрын

having been lured in by the thumbnail i wonder if you have a solution for the triangle/circle/square shadows?

@sheepy403 11 ай бұрын

Anybody please correct me if I'm wrong, but I don't think there is actually a solution. The defining question I asked myself to reach this conclusion was, "at what elevation can any corner of the square shadow exist?" The triangle shadow tells us that such points can only exist at the lowest height. However, the circular shadow tells us they can only exist at the middle height. Therefore, no point in 3D space could be the corner of that square while also staying within the confines of both the triangle and the circle. Again, if anybody can prove otherwise, I would LOVE to be wrong on this. I hate it when thumbnails present unsolvable puzzles.

@brandonwillnecker8060 11 ай бұрын

@@sheepy403my first guess would be to extend the shapes along their respective axes into a prism shape. Then you could take the triple intersection. I don't know if this would work but it could be tested with something like Blender. The circle and square would produce a cylinder that recreates those two shadows but maybe the triangle messes it up.

@caspianmaclean8122 11 ай бұрын

@@sheepy403I think you're right that it is impossible. I'm curious what is the shape of the largest possible shadow instead of the square, if the triangle and circle shadows are as shown.

@sheepy403 11 ай бұрын

@@caspianmaclean8122 I found the shape, though I don't know if there's a name for it. If we say the square would be 1x1, the shape we get from the triangular prism and cylinder is the following: an ellipse with a major axis of 1 and a minor axis of 0.5, cut in half along the major axis and spaced with a 1x0.5 rectangle. The area of such a shape is 1/2+pi/8

@Sr.Estroncio38 9 ай бұрын

My solution would be a disc as a base then you make a cross section with a thin square and triangle, which should be posible if they have the same hight

@imotvoksim 11 ай бұрын

This was both engaging, well visualized, and well executed from sound and speech perspective! Good job mate 🧠

@brambl3014 11 ай бұрын

Except the horrible click bait thumbnail

@brambl3014 11 ай бұрын

That doesn't even get explained in the video

@rafael_l0321 9 ай бұрын

congrats on a really interesting and entertaining video! I also like your way of presenting

@bowieinc 11 ай бұрын

This was WAY more interesting that I thought it would be:) great job!

@miskolinaccc 11 ай бұрын

Very nice video! The one in the thumbnail is a dizzy to imagine though, I had to sketch it out to see if it was possible and I'm still not sure if it is 😆

@luukthegamer 10 ай бұрын

Its a two faced triangle standing on a disc

@rodomify7681 11 ай бұрын

I remember a while ago finding the shape you get by literally cutting shapes from a cube in all three Directions. The one you get from a circle happens to have the same corner/edge structure as a Rhombic dodecahedron, and if you were to get a shadow from the corner it would end up being a perfect hexagon.

@CookiesRiot 11 ай бұрын

Tricylinder Steinmetz solid. Basically a chonky rhombic dodecahedron.

@santherstat 11 ай бұрын

absolutely fantastic video

@joshl.4358 11 ай бұрын

This was a great video. Well sequenced, well explained, and interesting.

@nubdotdev 11 ай бұрын

This video is awesome! My first exposure to Latin squares was in an abstract algebra class, which got me wondering: What do the fractals of the multiplication tables of all the fundamental groups look like?

@bengobler 11 ай бұрын

That's a great question! I bet there are some nice ones. For starters, the multiplication tables for (Z_2)^n correspond with iterations of the Sierpinski pyramid pattern. The results will depend on how you order the rows and columns (this example uses the natural ordering).

@KevinLubick 11 ай бұрын

Loved the animations and the explanation. A contender for at least an honorable mention in my eyes!

@DroCaMk3 11 ай бұрын

Interesting topic, great visuals, great explanation, catchy background music! Very well done!

@Benny-the-cute-kitty 11 ай бұрын

Amazing video! It was so interesting to watch! 🎉

@ethanwinters1519 11 ай бұрын

Great work on this! What an interesting, yet fun and accessible topic : ) Perfect for SoME3, I'd say. Enjoyed how smooth and charming the visuals were too! Also, in answer to the question you ended on, my first thought was a tricylinder.

@StrawEgg 11 ай бұрын

okay, but... what about the shape from the thumbnail?!

@Luigicat11 11 ай бұрын

Okay so take a cylinder, then angle the circles on the ends so they form the triangle.

@kingsley. 11 ай бұрын

@@Luigicat11how??

@spkrforthedead4844 11 ай бұрын

@@Luigicat11 That doesn't actually work. Try to picture transforming a cylinder into the shape you're suggesting while looking at it from the perspective of the light casting the square shadow. The sides of the square formed by the ends of the cylinder become curves and the other sides get shorter as the ends of the cylinder angle towards each other.

@Luigicat11 11 ай бұрын

@@spkrforthedead4844 You're imagining it wrong then. Or maybe I just worded it poorly. Would it be better to rephrase it as cutting along two lines that intersect on the surface of the cylinder? I could probably draw a picture (a poor-quality one) of the shape I'm thinking of, but I wouldn't be able to show it since you can't attach images in a KZfaq comment.

@Kromiball 11 ай бұрын

Take a cylinder and cut the curved side into a triangle shape

@Xi_Vi 11 ай бұрын

great job explaining! you deserve more recognition!

@Majorfatal1 11 ай бұрын

Nice work!

@blinking_dodo 11 ай бұрын

Okay, but you *still* haven't shown us what shape created the shadow configuration from the thumbnail. Please show me.

@erikhaag4250 11 ай бұрын

in a mathematical "cad", you take the shadows you want to cast, and extrude them into an infinite prism. The set intersection of the 3 prisms will be the biggest set with this property, if the intersection is empty, then there is no set with these shadows. you can also add more shadows and change the angles between the shadows with no lose of generality. there is a video by maker's muse that goes into detail here: kzfaq.info/get/bejne/qJOTgdKX29ycnp8.html

@Tzizenorec 11 ай бұрын

You can't just check if the intersection is empty; you have to check if the intersection of the three sets actually casts those three shadows. In the case of the shadows {square, triangle, circle}, the intersection of those three prisms doesn't actually cast a square shadow (the shadows it casts are triangle, circle, and some sort of lopsided rounded rectangle).

@sirgregsalot 11 ай бұрын

I was really hoping someone would bring up the thumbnail - I was trying to figure how to solve it using CSG intersections. Take the intersection of the square and the triangle to get a triangular prism, and then take the circular intersection orthogonal to that. I can't quite picture the shape, but I can convince myself that it works, and that the order you take the intersections in doesn't matter.

@aspyse 11 ай бұрын

@@sirgregsalot The "base" face of the triangular prism is its only square part. When you intersect this with a cylinder, two of the face's edges are shaved off, making the "lopsided rounded rectangle" shadow they mentioned.

@MilanTheAngel 11 ай бұрын

Seeing this video I assumed you were a big math channel, since the quality is top notch!! You're definitely gonna get there if you make more, its good stuff!

@robharwood3538 11 ай бұрын

Nice one! Good progression from basic concepts to more advanced, in a gradual pacing.

@rubensabatini7265 11 ай бұрын

Wow, the part about latin squares converted into elevation maps is mind blowing, wonderful video!

@JaharNarishma 11 ай бұрын

I recommend you to look into _skyscraper sudoku_ It is a variant of sudoku which uses "elevation" to find the solution.

@Frizou 11 ай бұрын

You have- how much subscribers?? Less than 400?? This is quality well beyond most channels I've seen yet, truly an amazing video and subject! Your editing - both visual and audio - is absolutely amazing, it's captivating but doesn't distract from the subject. I cannot wait to see more

@ZoofyZoof 11 ай бұрын

Well he gained 70 from this video, or more since you said less than 400

@zmaj12321 11 ай бұрын

Wow, great vid!

@fibbooo1123 11 ай бұрын

Awesome video!

@Renteks- 11 ай бұрын

Did I miss the part where you explained what the shape from the thumbnail was? Or was it intentionally omitted?

@tesseract7586 11 ай бұрын

WHOSE THAT POKEMON! its a cube! *sirpinski tetrahedron* AAAAAAAA

@bengobler 11 ай бұрын

Lol!

@tesseract7586 21 күн бұрын

@@bengobler me when i actually get noticed by a youtuber:

@Dan.Incarnate 11 ай бұрын

Amazing video!

@odysseasv7734 11 ай бұрын

This was an amazing video, very consise and motivating to learn more and create some imaginary cubes (or tessaracts ?!). I particularly liked the real life videos of the shapes and the pauses you did to make us think. Good luck in the competition!

@dimethylhexane 11 ай бұрын

this is a great video, but I still really want to know what shapes with shadows like in the thumbnail look like!

@Jellyjam14blas 11 ай бұрын

Awesome video!! Very entertaining and succinct. :D Another neat thing that's related to shapes and shadows is creating complex 3d shapes that spell out different words from different angles! I made one out of Lego with my name and the word hello, and had a lot of trouble fitting an "N" into the shape of an "O" and not having it break apart and fall over haha. I've also played a puzzle board game about Latin squares (though they don't call them that) where you build a tiny city to fit a specific shadow. Sadly I don't remember its name

@mikeflowerdew7877 11 ай бұрын

Very nice! The link to Latin squares was unexpected and very well explained

@jessicawang6558 11 ай бұрын

Well made video! Love the background music : )

@ericpeterson6520 11 ай бұрын

You really buried the lede with the title and thumbnail, I almost didn't click because I assumed it was just going to be that simple guessing game the whole time. Maybe a trio of square shadows and an indication that the mystery shape is NOT a cube would be more eye-catching and illustrate what's interesting about the video better

@ge-orc8286 11 ай бұрын

horrible clickbait. dont get me wrong, it was reasonably interesting, but the case from the thumbnail, with square, circle and triangle was not solved in the video.

@jerecakes1 11 ай бұрын

i genuinely love the catch you introduced at the beginning, i was really flabbergasted and made me watch the entire video which, said video was really concise and full of interesting points and/or realizations what i loved most was the fact that you can interchage the shapes between the cubes, really obvious info once i got it but it sent me how non-square like you can get them to look like LOL

@qy9MC 11 ай бұрын

Impressive! Well explained, interesting and surprising yet simple!? I am truly baffled by quality. Gj. I really liked it.

@JMurph2015 11 ай бұрын

Ok but what about the thumbnail challenge 😅?

@BetaPlayTeam 11 ай бұрын

This video immediatly caught my attention as I had this question of whether you can be sure a object is a sphere by looking at its shadows. Now I need answers!!!

@AndresHughesSalavera 9 ай бұрын

I enjoyed this video immensely

@Uxcis 11 ай бұрын

This has got to be one of the best yourube videos i have watched this year. What a perfect explanation, 10/10

@Yu56-li6dp 11 ай бұрын

One of the best videos i’ve seen in a while, I’m subscribing

@Stroopwafe1 11 ай бұрын

This is incredibly well produced, and the music really adds to the video, impressive! Maybe that's why you were blessed by the YT algorithm

@monsterhunter8595 11 ай бұрын

Very nice, looks like you've got a lot of potential!

@VCLegos 11 ай бұрын

You are a good teacher, this was explained in a very easy to understand way. This is how teaching is supposed to be done, thank you.

@MQWalkman 11 ай бұрын

Everything about this is great! Moments of genuine surprise and excitement.

@MemesNick 9 ай бұрын

Great video! Really enjoyable and the explanation was really clear, well done!!!

@mananbachani5189 10 ай бұрын

This was so good, earned a sub man

@egenriether 11 ай бұрын

Great video and presentation. I look forward to more of your work

@tricanico 11 ай бұрын

This is my favorite so far. Fresh and to the point.

@markkennedy9767 11 ай бұрын

Wow, great video. Perfectly explained

@caspermadlener4191 11 ай бұрын

As an IMO (2022) gold medal winner, I really appreciate elegant arguments like these! At first, I expected a video just explaining why this isn't possible in general, but this is way more elegant! Thinking about latin squares this ways proves that for a latin n×n and m×m square, it is also possible to construct an mn×mn square! Also, notice that it is always possible to "confexify" these fractals, without changing the shadows, if the shadows were confex to begin with. The tetrahedron is the confexification of the Sierpinski fractal, but you can also get the other shape by confexifying the fractal of a more symmetric variant of the minimal 3×3×3 cube than shown in the video.

@bengobler 11 ай бұрын

Excellent! Of course, you can replace the cubes in the Sierpinski pattern with any imaginary cubes of your choice (they don't even need to be all the same). In the video, I say that the Sierpinski pattern pairs well with the tetrahedron, and the reason is precisely that of convexification. Great catch!