Hypercubist
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HyperCubist Origin Story
🕑 Added 2025-04-19 05:02:04 +0000 UTCHi all! Ciretose2 asked a question I thought would make a great introductory post, so here goes:
"I'd love to know more about what led you to your conceptualization of 4D. It's on another level, no pun intended. The deck of cards visualization and that matryoshka projection are inspired."
I thought I'd tell you a bit about me and where these ideas came from. I came from a Mechanical Engineering background, and have been tutoring math and physics for close to three decades. My first exposure to 4D as a kid was the cube-in-a-cube tesseract model (I think I saw the clip from Carl Sagan's Cosmos, or someone explained it to me), which, like a lot of us, never made any sense to me.
Years later I started thinking about time as the 4th dimension, and imagined a cube existing over a certain time duration (like I show in pt 1), and realized that the tesseract model could represent the cube at the two endpoints of the time interval. That was good enough and I moved on.
But one day while playing with a problem from basic calculus, I realized you could solve it with a very concrete, geometric approach by taking it from a 2D area problem to a 3D volume problem, and that you could extend this to approach to 4D or beyond (this is the video I made about this problem: https://www.youtube.com/watch?v=DRzmT-E0ncQ). But to do so you have to think of 4D spatially, not as time. So I became hyper focused on trying to visualize 4D spatially.
I watched lots of explainers on YouTube, mostly saying the same things - "it's impossible for us to truly grasp", "we can only see 3D shadows", etc. I started thinking in terms of the "slice" or "cross-section" method, and came across the app 4D Toys which uses the same approach (https://www.youtube.com/watch?v=0t4aKJuKP0Q), which gave me a ton of insight (I highly recommend playing with it). Then I found this page (https://www.qfbox.info/4d/vis/vis), a virtual encyclopedia of 4D shapes, and learned a ton. I still haven't gone through it all.
Around then I started experimenting with my own 4D models using the educational math software that some of my students used, GeoGebra https://www.geogebra.org. I tried to re-create models and animations of 4D shapes I'd seen online, and got pretty good at it, though it took me a while to figure out 4D perspective. Basically I was modeling 4D by manipulating a 3D graphing program (I may do a how-to video on this to show how you can make your own basic 4D models). I spent a lot of time on this during the Covid lockdown, in addition to dabbling in gameDev with Unity (a game engine) as well.
One YouTube series that was particularly inspirational was The Lazy Engineer. In part 2 of his 4D series (https://www.youtube.com/watch?v=4URVJ3D8e8k) he flattens out the Earth to make room for an extra dimension - a trick I've seen elsewhere as well, especially in the context of General Relativity. He uses this plane to show the cross-section method (what 3D cross sections of 4D shapes look like when intersected with our universe). But that trick got me thinking, and it inspired the Deck-Of-Cards model - an entire continuum of squashed 3D planes printed or displayed on cards, each representing a full 3D space. Now I had a way to visualize a hypersphere, which lined up with another influential 4D video from Cool Ideas: https://www.youtube.com/watch?v=RFK2_dAZvLo.
After experimenting with my GeoGebra models, I realized the Deck-of-Cards was just a special case of a more general framework, with the axes lined up in a particular way. But I was still thinking in terms of 3D 'projections' of 4D shapes, because that's the conventional view, and was literally what I was doing in GeoGebra (these projections are well understood mathematically btw, not my own invention). But while playing with 4D rotation, something clicked in my brain - you don't have to interpret these as 3D projections - I was automatically visualizing two depth dimensions, and could rotate with either one independently. Philosophically, it didn't make sense to say we could see 3D but only projections of 4D, as we only see 2D projections of either. I spent a lot of time trying to understand this as a coherent 4D visual framework - interpreted as a 2D view of 4D space, as opposed to our usual 2D view of 3D space. That's where the plane-of-sight idea came from. And like the deck-of-cards, I've never seen that interpretation/model presented anywhere else (though I could very well be wrong).
After getting into MANY debates in various YouTube comments sections, I realized that 1) most people have no clue about what geometric 4D really is (other than time), and 2) even people well-versed in 4D are convinced it's impossible to to truly visualize. I started thinking about how I would present my take on all this, as I thought I had some unique conceptual ideas and interpretations that by then had been percolating for a couple years.
I had never made a video before, but 3Blue1Brown (simply the best math channel on YouTube https://www.3blue1brown.com) had started doing his Summer of Math Expo, encouraging educators and math fans to make their own content. I started with some of my 'alternative' takes on calculus problems (like the one above), which had some initial success with a math-focused audience. Then I laid low for a while, after only making 2-3 videos.
Finally I started writing and assembling my first 4D videos. The first one blew up much faster than I could have imagined, and I started revamping the (mostly done) 2nd. I learned how to do the same things I was doing in GeoGebra with Unity, but with more complex models (like the Russian doll, an idea that came while writing part 3). After the 2nd video came out, I realized had to address some issues I was saving for later (like 4D perspective and how Freddy sees), and started a new video from scratch. That, the holidays, and the LA Fires (not to mention the unstable political situation) all contributed to taking extremely long to get part 3 finished, and I'm grateful for everyone's patience.
And that brings us up to now! Part 3 seems to be well received, I've finally started the Patreon, and I have lots planned for future videos. In addition to the ongoing 4D material, I'll continue to make math (and physics) content, especially as it relates to 4D. As I get better with 3D (er, 4D) modeling, I'll release simple interactive demos here on the Patreon to workshop them. If you have any ideas for things you'd like to see in future videos, please feel free to share! I'm figuring this all out as I go along, but I'm thrilled you all are here taking part. Thank you all for your support!!
-HC
Comments
Gooi Chyi
The fact you mentioned - that we only truly see the world in 2D, or a projection of 3D onto 2D. In this sense, "flattening" a 4D object into 3D space makes sense. And then you can further flatten it into a 2D projection, which we can all work with on our screens. Even though it will be much easier to visualize a 4D object using a 3D or "volumetric" projection, these holographic displays are still early in stages of development and not quite ready for commercial production, so access to one of those will be fairly difficult. But one day, I'd like to see one and use it to visualize 4D objects better. Using 2D screens is the best tool we have today - we could even use a split eye 2D technique to visualize 3D depth, making 4D objects more intuitive to understand using our existing 3D framework in the real world. Weeks ago, I was pondering whether flattening a 3D object into 2D would have actually helped us to understand how 3D works, because in effect, both of our eyes only see a slightly different, "flattened" 3D world projected into 2D. Then our brains can take these two images and compute the "depth" aspect, which is inherently absent from 2D projections, but nonetheless we can feel it very much with our senses. What if, we take this analogy and apply it to Earth itself? We live in a 3D world, but it doesn't mean we get to see everything all at once, but when we zoom out and see Earth as a sphere, and yes we are still in a 3D world, but in fact, we are "flattening" our day-to-day lives living in cities, countries into the 2D surface on a 3D sphere. But we are still 3D creatures living on the seemingly flattened crust of the Earth, so what does this mean? Does it mean that seeing Earth from space afar gives us an "extra" spatial dimension - that now we can see "everything" on Earth? Even if it is just seeing one side of the Earth, we can already see much more than we can see if we were to live in it. So wouldn't Earth itself be a "3D" projection of a 4D space? I'm not quite sure this holds up, until we can "stack" multiple crusts of Earth together, much like putting Earths within Earths - forming a "hypersphere". If so, it will also be easier to understand because we already have an intuitive understanding of how our view is "flattened" when seen from a car, then plane, then from a space sattelite, and so going into different "cross sections" of the Earth would just mean descending into the mantel, or going above the clouds.
Ted
Hi! I've definitely seen videos that acknowledge flatlanders have a 1D view, but it's usually meant to show or imply that they couldn't possibly visualize a cube. Of course, they can't see cubes the WAY we do, just like squares. Hence the maze example, which shows a 1D view carries much more info for them. People have a hard time accepting that we can't see 3D, as it's hard to imagine what TRUE 3D vision is. They get hung up on the idea that depth perception = 3D vision. "And so we already have a framework for visualizing a hidden dimension." Great way to put it! Haven't read 3BP yet, but it's on my list, mostly for the 4D stuff. Now that I think on it, 3B1B's quaternion video / demo really stirred me into 4D vis again. It was the first time I really GOT quaternions, but I don't think the whole stereographic projection thing is really necessary once you've learned to interpret 4D projections. The fact that they're so practical (and confusing) for gameDev makes them a ripe topic to explore further. As far as hypersphere goes, it's probably my favorite application of the deck-of-cards. Just think of it like this - every slice is a normal sphere, just like every slice of a sphere is a circle. They grow from a point, to a sphere, to the 'equatorial sphere', and back down to a point again. In the deck, each card shows one spherical slice. The 'width' of the deck is the same as the diameter of the hypersphere. The REAL fun is trying to imagine what it's like to live on the 3D surface of the hypersphere (which is possibly the shape of our universe) - we'll have lot of fun with that idea in later videos. I honestly think that 4D is having a bit of a renaissance right now - whether from the recent games (4D golf, 4D Miner, Miegakure), 3BP, quaternions, more pop-sci awareness of General Relativity (like in Interstellar), etc. I'm really hoping to add to the conversation and give people a one-stop comprehensive deep-dive. Glad you're here - thanks for the support! (What kind of gameDev are you into btw? So far I'm just using it for animating my demos).
Naajaw
Love this lore! I've also recently jumped into Unity, and got caught up on quaternions—I'm very excited to see your take on it. 1blue3brown's interactive website has only gotten me so far... and of course you don't need to conceptualize hyperspheres to use quaternions, but ever since I first encountered them, it's been an intellectual challenge of mine to really understand these things. It's difficult for me to extrapolate your ideas to spheres because they don't have the rigid edges like polyhedrons. Maybe it'll click after my brain has time to chew on it. I can empathize with your experience to you up until you had that calculus insight (or, rather, calculus-avoidant). I'm no math tutor, though; I went down the path of software engineering, but game dev has given me the opportunity to dive back into my true love for math. Point being, I've likewise spent a fair amount of time watching dimension-related stuff as the internet was growing up with me. As far as I can comprehend what you've relayed so far, the most profound—and as far as I've seen, unique—thing about your approach is ironically something so obvious that I'm amazed that no one else thought to point it out (including myself!) I jokingly alluded in my comment on youtube: you're literally the only one I've seen to recognize that a flatlander's perspective is 1D, not 2D. That has always bugged me in other videos. But I never thought to extrapolate that idea by asserting that *we* are the ones that see in 2D. We've been working with a hidden dimension all along! And so we already have a framework for visualizing a hidden dimension. Brilliant stuff. Another thing that just clicked for me was a part of the Three Body Problem books—I think in the 3rd book, there's a subplot where humans end up in a 4D "puddle", and he describes how the characters get the hang of not accidentally reaching into each other's innards by accidentally moving in a direction that didn't exist in normal 3D space. The people who go there are driven crazy with claustrophobia when they return. Not sure if you've read them, but he has some very interesting ideas on the subject of spatial dimensions. The story is a real bummer, though. Pardon the wall of text. Just very excited by your content, and delighted to join your patreon to help support it! Keep it up!
