Can you guess a shape from its shadows?

[@PrimerBlobs]

Nice video! Impressive that you also made the music.

[@TripPy_Poly]

Hi primer, the evolution simulation content creator (idk)

[@pirilon78]

The man himself

[@LadyTink]

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

[@kaininjago6161]

it is the man!

[@BurningShipFractal]

Hi

[@syriuszb8611]

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

[@DisguisedParrot]

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

[@BryanLu0]

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

[@tokarak]

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]

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]

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]

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]

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

[@vigilantcosmicpenguin8721]

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

[@tttITA10]

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

[@pineapplequeen13]

I was thinking something similar! I thought it felt a bit like a children's program asking the audience for help, but in a very endearing way. It makes the video engaging and accessible for all ages

[@sokijohnson1309]

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! 😊

[@weibrot6683]

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]

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

[@unknowndeviceconnected]

Imagine like the bit of a flathead screwdriver

[@CuantumQ]

I would describe it as a triangular prism, but cut using a round cookie cutter

[@nekomimicatears]

Kind of like a cone but with a line instead of a point

[@iiiiii-w8h]

It's literally called a conoid. It has a wikipedia article. It's volume is 1/2 πr²h

[@iau]

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]

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.

[@Random_Nobody_Official]

I still don't get how a tetrahedron (or either of the other 2 shapes) can make a square shadow

[@mrosskne]

​@@Random_Nobody_Official0:56.

[@upsidedown-pug1974]

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]

Very clever!

[@voidwyrm6149]

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]

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

[@kaininjago6161]

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

[@Mikelaxo]

That would work with literally any shape in existence

[@hidekitsuiki1551]

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]

H and T, your initials!

[@wilczus222]

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

[@vigilantcosmicpenguin8721]

'Tis a noble cause.

[@Khaktos]

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

[@FernTheRobot]

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!

[@j.s.42822]

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.

[@demitrischoenwald1436]

This was so amazingly comprehensive that it made complete sense. And I didn't get bored listening to a bunch of nonsense, because it used a simple mathematical concept to explain how it could work alternatively. Very cleanly done.

[@0xEARTH]

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!

[@blacklistnr1]

@5:50 The reason I care is that I got tricked by your shenanigans in the beginning, now I can rewatch the video and go "That is clearly an iteration of a fractal with minimal cubes!"

[@Llamaful]

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]

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.

[@pocarski]

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]

Wow, that's really cool!

[@KarmaKomet]

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

[@mastershooter64]

the premise is kinda like "Can you hear the shape of a drum" which gave rise to spectral geometry

[@bengobler]

That's a great connection!

[@EleetCanoe]

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.

[@caspermadlener4191]

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]

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!

[@thecalculusofexplanations]

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.

[@iniaimanharith2344]

5:40 Maybe the latin squares work because since each row and column have all the numbers, which means there isn't a gap on the shadows.

[@jakobr_]

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]

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

[@jonuuh]

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

[@koendos3]

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.

[@tfr]

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

[@latehate1017]

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]

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]

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

[@aepokkvulpex]

That's right, this shape fits in the square shadow!

[@DisguisedParrot]

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

[@kitty_foreststreamvods4116]

Ngl this minimal cube idea makes sudoku easier

[@dyld921]

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]

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

[@williamhoyle6644]

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

[@shelly8779]

/op womp womp

[@FeWi-YT]

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

[@Not_an_expert7]

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

[@AZALI00013]

amazing video !!!!!

[@confusion5025]

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.

[@BsktImp]

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]

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

[@Swagpion]

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

[@jules325]

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

[@amaryllis0]

@@jules325 See wikipedia: Steinmetz_solid#Tricylinder

[@jetison333]

@@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]

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

[@jaki1876]

@MemeSwag doesn’t that just make a sphere?

[@TheScienceNerd100]

Saw the thumbnail and instantly went into Inventor Pro to CAD up what it would look like. Might 3D print it for fun if I get my printer working again.

[@miranhaespada1381]

7:51 if o use 3 circles (each perpendicularly to a light) with the same radius as the sphere it would work, since we have in the sphere (kinda) infinite layers perpendicular to each light, but just the layer with the bigger radius is needed for that specific shadow Join the 3 circles and I have the 3 shadows the same way

[@flippedmasterguardian]

This video has instantly become one of my favorite videos on the website. It's been a long time since I've ever been so baffled by something so simply complex. Love it, hope there's more in the future!

[@bastienpabiot3678]

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

[@sheepy403]

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]

​@@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]

​@@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]

@@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]

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

[@loucrawford5690]

I saw the thumbnail and in my head I heard: "iiiiiits a Pikachu!"

[@BarelyNoticeable]

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

[@zanop15]

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.

[@imotvoksim]

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

[@brambl3014]

Except the horrible click bait thumbnail

[@brambl3014]

That doesn't even get explained in the video

[@Stroopwafe1]

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

[@ConceptsIlluminated]

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

[@random832]

the easiest example of "three circular shadows, but not a sphere" is the Steinmetz solid - the intersection of three cylinders.

[@rodomify7681]

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]

Tricylinder Steinmetz solid. Basically a chonky rhombic dodecahedron.

[@BetaPlayTeam]

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!!!

[@hey_hey4394]

I spent hours trying to solve the puzzle in the thumbnail just for you not to mention it at all in the video. Is it even possible?

[@VCLegos]

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.

[@rubensabatini7265]

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

[@JaharNarishma]

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

[@MemeAnt]

This is very interesting. It’s rare I find a video like this. Fractals are cool, and I love geometry. Cudos.

[@blinking_dodo]

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

[@mrosskne]

A triangular plane, circular plane, and square plane intersecting at right angles.

[@jessicawang6558]

Well made video! Love the background music : )

[@Renteks-]

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

[@Autoskip]

I saw part of the trick from the start - all of my answers were along the lines of "a cube would work" - but I didn't realise that a serpenski pyramid could do it. Once I saw it though, it made sense, and I managed to figure out that it was minimal shortly before you spelled it out. As for the 3 circles option, it's a bit unusual in that the most common expectation (the sphere) isn't the option with the most volume - that title belongs to the overlap of three perpendicular cylinders, which is a shape similar to a rounded rhombic dodecahedron, and has 8 sharp corners (and six points where four faces meet collinearly, which could be called corners, but are completely flat at the exact point that they meet). Also, I clicked hoping for an exploration as to why the circle/square/triangle shape from the thumbnail was impossible, but this was pretty interesting too.

[@PennyLapin]

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!

[@mikeflowerdew7877]

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

[@StrawEgg]

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

[@Luigicat11]

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

[@kingsley.]

@@Luigicat11how??

[@spkrforthedead4844]

@@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]

@@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 KZbin comment.

[@Kromiball]

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

[@ebenenspinne4713]

One possible shape would be a disc, which is at an angle (maybe 45°, but I'm not sure). The highest point of the disc could be at the bottom left side of a cube imagined in the place of the sphere and the highest point would be on the top right side.

[@ethanwinters1519]

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.

[@manowartank8784]

Clicked this video only to say the thumbnail is impossible. You clickbaited us! The video was nice thought...

[@miskolinaccc]

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]

Its a two faced triangle standing on a disc

[@blacklight683]

5:30its full and there is 5of each number which is just what you need in a 5×5cube

[@erikhaag4250]

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: kzbin.info/www/bejne/qF7Gf6Kcq9trpM0

[@Tzizenorec]

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]

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]

@@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.

[@user-yn1oi6iv1f]

7:56 It's 3 flat (2d) circles combined, center of every circle is center of the whole shape.

[@tesseract7586]

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

[@bengobler]

Lol!

[@tesseract7586]

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

[@nino-ciampa]

Great content! I’m so ready for the sequel 😤😤😤 liked and subscribed!

[@nubdotdev]

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]

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).

[@deanthe3684]

In middle school, I once cadded a shape which was essentially a cube cut by a cylinder on each axis to have 3 circular shadows. It also had the property of rolling on a grid. It looked like a hackey sack sewn of 12 rhombi. Pretty cool shape.

[@Frizou]

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]

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

[@RecursiveTriforce]

Thank you for making a video with my 3-dimensional cousin and playing with him! He loved the experience, even if he thought the lights were a bit blinding.

[@dimethylhexane]

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

[@Alt-Key_Here]

Underrated Channel

[@ge-orc8286]

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.

[@pawedziedzic3250]

Im disappointed the impossible shape from the thumbnail didnt make an appearance

[@RyanEglitis]

Clickbait - never shows the actual answer from the thumbnail

[@lucaslevelups9592]

5:35 Say we only look at one row of a Latin square and try make that into a cube. While the process still works it leaves all these holes. But if we then pick one column and layer it on top of our row, because there is 1 number per column, and the amount of said numbers matches the dimensions of the grid, the remaining column will fill in the gaps, repeat that for every spot on our row and it will cast a full square shadow. This reasoning could also be applied to every direction, therefore casting a square shadow on all 3 walls.

[@Jellyjam14blas]

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

[@MilanMilan0000]

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!

[@JMurph2015]

Ok but what about the thumbnail challenge 😅?

[@mange_1_kiwim1k96]

This reminds me of tomography (reconstructing an object from its projections). The main difference is that in tomography you typically know the "width" of the shadow, i.e. the object is semi-translucent and so at each point, the shadow can be of any color between white and black.

[@mikeebrady]

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.

[@S8EdgyVA]

Don’t worry, that shape is totally real, there was an entire sketch about it in an old geometry teaching series in my country

[@S8EdgyVA]

Hello, is this the Recovery Home for Crushed Childhoods?

[@hymanimy]

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

[@ericpeterson6520]

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

[@skylardeslypere9909]

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

[@Monkeymario.]

0:23 pyramid

[@Hi-Im-Noob-uwu]

Priminx (that's my first thought....)

[@IDI0T203]

Dude

[@tejuv6725]

it tingles my brain by how wonderful this field of mathematics is and how it's all connected . Great job , really appreciate you man.

[@insu_na]

Cool video, but I'm slightly disappointed that you didn't show an object that produces the shadows from the thumbnail

[@bowieinc]

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

[@JohnnyboiOrlin]

Did you intentional chose an impossible shadow for your thumbnail? Awesome video by the way! 😊

[@jerecakes1]

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

[@Rafael4]

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!

[@ryan-tabar]

Great work! I love the fact you used PowerPoint for this. A lot of people just do not know the full power of that software.

[@Atomas69]

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

[@robharwood3538]

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

[@holzmann8443]

I just wanted to play the game from the thumbnail. A thumbnail that turns out to be a lie. Ben, why are you like this?

[@tntobi80]

You only having 111 subs despite the quality of your videos is unbelievable I think your channel will start growing rapidly soon, at least i hope it will

[@salxrn]

when you were talking about how there was one cube in each row, column and stack I was thinking about sudoku and then you immediately say exactly what I'm thinking about

[@hamzamotara4304]

Such high quality and understandable content.