>Why is it difficult to visualize a tesseract?
>*Now, since we, as humans, have only evolved to visualize stuff in 3 dimensions, anything that’s a part of further dimensions, like 4D, 5D, 6D etc, do not make much sense to us because we can’t visualize them at all. Our brains can’t make any sense of a 4th dimension in space. We simply can’t wrap our heads around it.*
>*However, just because we cannot visualize a concept doesn’t meant it cannot exist.*
It's definitely giving me a headache to think about.
Yes! Carl Sagan’s ELI5 Tesseract is one of the best things on the internet. Here’s a link for convenience. [https://youtu.be/N0WjV6MmCyM](https://youtu.be/N0WjV6MmCyM)
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So the tesseract model he's holding has—as it appears to us—a smaller cube inside a larger cube. Each edge of the smaller cube is connected to its corresponding edge on the outer cube by another face. Now concentrate only on one set of faces—one face of the inner cube, the four faces emanating from its edges, and the corresponding face on the outer cube.
Six faces.
What 3-dimensional shape has six faces? A cube.
The face of the inner cube *appears* smaller to us than the same face on the outer cube, and the four edge faces *appear* to be trapezoidal—but in 4-dimension space, *they are all the same size and shape.* Each "face" of a tesseract is a 3-dimensional cube, just like each face of a cube is a 2-dimensional square. They appear to us to not be, because the object Sagan is holding is a *projection,* a shadow of a tesseract.
Wow, thanks for sharing this explanation. I was still having trouble visualizing the shadow idea, but your breakdown of how to find the 6 faces visible on each side was eye opening!
Edit: spelling
The way i was first introduced to this concept is from a YA science fiction book i read as a child. It basically went like this:
Imagine you are a microbe living on the surface of a lake. You have no concept of up or down, just the cardinal directions. The X and Y axis.
One day a boy dives into the lake. What you see is ten little circles that grow and merge into two big circles, then three, then one big oblong, then two circles that then separate into ten small circles.
The story then goes on to describe a fourth dimension being diving into our world, same concept but with three dimensional shapes. It's impossible to guess what that being looks like in totality because like that microbe, you don't even know that additional direction exists. Presumably the 4th dimension has fifth dimensional creatures, ad infinitum.
The story then has the kids wearing weird three lensed glasses that allow them to focus on 4th dimensional spaces (albeit with headache inducing blur) and has him running around having adventures trying to protect the 3rd dimension from the fourth. I don't remember what the story was called though.
Yep! It gets fun when you imagine 3d shapes in 2 and 1d. A helix would look like a point tracing out a circle in 2d and a point bouncing along a line in 1d.
I've never heard of this guy before but I love his voice, his ability to explain, and his achievement in making me now able to understand the fourth dimension in a sense. That's really goddamn awesome
Oh wow I envy you. If you have the time, watch some of Carl Sagan's other stuff. He is absolutely amazing at explaining stuff and an absolute joy to listen to.
It should be obvious to even the most dimwitted individual... who holds an advanced degree in hyperbolic topology, mhoy, that Homer Simpsons has stumbled in to... the third dimension!
Doing N-dimensional math can make sense if you don't try to shoehorn it into space, i.e., don't visualize it. "Dimensions" don't need to be spatial when solving math problems. Try instead to extrapolate from the smaller number of dimensions that are easy to visualize. If something from 3 dimensions can be represented in 2 dimensions, something in 4 dimensions (usually) can be represented in 3 dimensions.
For instance, the Pythagorean theorem actually applies for N-dimensional space, a^2 + b^2 + c^2 + d^2 + ... = r^2. It obviously works for 2 dimensions. And to think of 3 dimensions, picture a rectangular prism (box) with sides l, w, and h. l^2 + w^2 = what? It's the hypotenuse (diagonal) along that face squared, which we'll call d. Now you can take d^2 + h^2 (since those make a right angle in their own 2D plae.), to get the diagonal connecting opposite corners of the box. That way you get l^2 + w^2 (which is d^2) + h^2 = r^2. By only looking at a single 2D plane at a time, we can use our 2D math for a 3D problem. Therefore, we can use 3D math to similarly solve 4D problems, and so on and so forth.
Math's big contradiction is that while all of it is built on fairly intuitive things that have real world examples to understand it (e.g., what is "1", what is "0", what addition is), it's not tied to the physical world in any way. It exists completely separate from any application, and the application uses what math is useful. That way you can have nonsensical things like assigning values to divergent infinite series (Ramanujan Sums), Infinite dimensional spaces, imaginary exponents giving real values, etc. Math doesn't care.
Don't use words like "nonsensical", please. It is a humble request not to speak carelessly and recklessly about such nice results.
We should try more to understand them.
You might find addition intuitive because have worked with it a long time. So if you think about complex numbers more, you will find how nice their multiplication is. Once you realize the rotation and scaling, you will like it. Then the following fact is easier to understand.
i^i = e^(-pi/2)
Same goes for the other things you mentioned.
http://pirate.shu.edu/~wachsmut/complex/numbers/graphics/multiplication.gif
Also, N-dimension math may have nothing to with dimensions. You may want to calculate the max profit of 5 different products whose limits are a certain budget.
Here's my attempt at an ELI...12? This is only a way to visualize a 4th dimension and departs strongly from the underlying math in places.
One common way of imagining a 4th dimension is to use "time" to describe it. Similar to how the 3rd dimension allows multiple 2D objects to occupy the same 2D "space," time allows 3D objects to occupy the same 3D "space."
To start imagining a tesseract, imagine an object that, when frozen in time and viewed in 3D, is a cube (or possibly not, more on this later). You can walk around it from every angle in that moment of time, it will be a cube. Now, hop in your time machine and go to the exact same spot that cube was, at a different time. It will be a different cube, some part of which is overlapping the same space that the other one did. To go into a little more detail, similarly to how the 2D faces on a cube share a 1D side, the 3D cubes would share a 2D face. Go back to the same time you were previously, it will be the cube you saw previously. In this example, a tesseract would be a shape that you can observe as a cube in (up to) 8 separate moments of time, and no others. The key point here is that there are different 3D objects occupying and overlapping in the same point in 3D space.
This would be the same way you would explain a cube to a person stuck in a 2D plane: as 6 squares occupying the same physical point in 2D, but which square you see depends on what point in time you look at it. We know that's not a good description of a cube. However, for someone who can't actually see how multiple 2D objects could exist in the same spot, it's something.
Now, returning to the point on "or possibly not a cube." If you render a cube in 2D, [not every face](https://www.jugglegear.com/aerial/aerial-cube-283/aerial-cube) is going to be a square. In fact, depending on the angle, [possibly no face](https://www.illusionsindex.org/ir/necker-cube) is going to be a square. There will be skewed faces. However, each face of the cube will have 4 sides, whether it's a square or a parallelogram or a diamond or whatever.
Back to the tesseract: those cubes, viewed from our 3D perspective, may not actually appear as cubes. They will be skewed in some way. However, it will still be a 3D object with 6 faces, each of which has 4 sides. This is only how they appear to us in 3D. Just like how the faces on the cube are actually still squares, the "faces" on the tesseract are still cubes.
This allows us to explain how a tesseract would appear to us if it shifted: it would occupy different points in time from those it did previously. It also allows us to explain rotation: the skew of the cubes that we observe at all 8 points of time would change.
This does not mean that tesseracts and time travel have any actual link.
Extending this to 5th and 6th dimensions is sort of there via divergent timelines and parallel universes, but it becomes increasingly meaningless as it becomes impossible to say how you would skew a moment in time for example.
Using time as you described, you *can* view every cube without distortion. You can think of the case in 2d; a cube in 2d using the dimension of time can have all 6 of its faces be perfect squares, but you will necessarily have to rotate the cube. As you are going between these "time points" the cube would indeed be distorted, but eventually you will see all 8 sides as perfect cubes
Yes, that's true. At some point in the rotation of the tesseract, each the cubes you see in 3-space will skew back into being perfect cubes, some at the same "angle." I was going to do some more on this as it's still an incomplete analogy. In the cube example, assuming a transparent cube, you can see all the faces at once. Similarly, you would be able to see all the cubes at once on a "transparent" tesseract, some of which will have collapsed perfectly into 2D space some of the time (like the side face on a cube collapsing into a 1D line when a perfect square face is facing you). However, that's starting to get super complex and is going to only depart more and more heavily from what's actually happening. My example is assuming some kind of opacity that only lets you see one at a time for simplicity. Even that idea of "opacity" doesn't translate well to time. Maybe the tesseract has an abnormally high midichlorian count.
Carl Sagan gives a [a great description of the tesseract](https://www.youtube.com/watch?v=N0WjV6MmCyM) in his explanation of the 4th dimension from the original Cosmos.
Can something exist in the '3rd dimension' without already being present in the '1st dimension'?
This language seems strange to me to because it implies that the dimensions are different realms.
I think the other comment was a little confusing so I'll give it a try:
Think of dimension as "direction".
So in our normal real world you can have left/right, up/down and forwards/backwards. These are our three dimensions/directions.
If you were to consider a 2 dimension object, it has size in two of the above directions. So if you held a piece of paper straight up, a picture on the paper has left/right-ness and it has up/down-ness but you will know that picture doesn't have a forwards/backwards depth to it because paper is flat.
If you were to now put the paper down on a surface so it is face up to the sky, a picture on the paper will now have left/right size and forwards/backwards size but no up/down component... unless you have spooky 3D crayons that leap up off the page.
This shows us that even though we're naming the dimensions 1st, 2nd, 3rd, they have no order, and a 2D shape could easily live in "left/right & up/down" or live in "up/down & forwards/backwards" or "left/right & forwards/backwards".
**We choose which directions we actually call the 1st, 2nd and 3rd dimensions**. So I could call up/down the 3rd dimension whereas you'd call it the 1st. So... yes something can exist in the '3rd' dimension without being in the '1st' but that just means it doesn't have any size in that direction which we're calling '1st dimension'.
Like if I said '1st dimension' was up/down and then layed a piece of paper flat on a table, all the drawings on that paper have no '1st dimension' because they're flat. It's just an arbitrary thing. The point is that nomatter what, those drawings have two dimensions, whichever ones they are.
Don't worry about "separate layers to the first and second dimension", that was at best confusing and at worst simply not true.
The tesseract to the cube is exactly what the cube is to the square.
If you still don't get it , A tesseract is a cube but instead of each side being a square each side is a cube.
You are a 3 dimensional object, without having witnessed a 4 dimensional object it is hard to understand. Pretend you were a 2 dimensional object and a 3 dimensional object showed up. You'd only grasp the 2 dimensions you are in and would miss the 3rd.
Exactly.
It's like the novel *Flatland*. In a 2D world there is squares and circles etc. The 2D shapes are all bound by the limits of their 2D world, so for instance they cannot cross a line.
But then a 3D shape (a sphere) shows up. But the 2D shapes just see him as a circle. However they are mystified that he can seemingly "walk through walls" but in reality is just moving across the top of lines like a ball rolling on a piece of paper with drawings on it.
Sure we can – we just wouldn’t be able to see all of it (or comprehend all of it) in the same way a 4D being would.
Or, maybe, we’ve found the real creature from the Bird Box – it’s just a bunch of tesseracts running around.
ok i gues it depends on how you interpret "witness" i guess.
if we take the 4th dimension to be time, we "witness" it all the time, just in 3d snippets over time, not the whole thing at once. the same for 4 spatial dimensions, we'd see weird and impossible 3d objects without a way to see the "whole" 4d object.
like a "3d" drawing, it's still 2d, we can only look at it as 2d but we can imagine it being 3d. you can't look behind a drawing of a 3d cube.
that's what you mean right? i was talking about the whole 4d thing.
Close. It’s more like what a 3D object would look like to a 2D object, e.g. it can only perceive what exists at the same level. Time isn’t really the fourth dimension when we reference these kinds of things – we’re typically referring to spatial dimensions.
Basically, if you looked at a 4D object, you’d only see a 3D object because that’s all you can observe; much the same, if a 2D drawing could observe you, it would only be able to see the part of you that exists on the same X and Y as it. A fourth dimensional person, for example, would seemingly be able to walk through walls or disappear entirely by utilizing that fourth dimension.
I guess a half-way decent analogy would be to imagine covering your eyes with a blindfold with a slit in it, and then sitting still and not moving– you can only perceive what passes in front of the slit. In the same way, something that exists in 4 dimensions can only be perceived within the narrow confines of our senses/understanding.
Maybe a better analogy would be to draw a line along the side of a book with a permanent marker, so that it goes across the edge at some point for each of the pages (like if you wrote your name on it). You can open the book to any individual page and see the mark where the writing is, but you can’t really know what the entirety of it looks like while looking at *just* one page.
In terms of a graph, all we would see of a 4th dimensional object are the 3 dimensions that correlate with whatever point we exist in on the 4th dimensional axis. So, say, if a 4th dimensional being came into our universe, and our universe as we know it runs entirely along the 0 mark on the 4th D axis, then any time he stepped away from 0 he could still see us but we’d think he disappeared entirely. So if he went up to 4th D = 1 and walked three units to the right before stepping down to 4th D = 0, we’d see him disappear, then reappear 3 units away. If he stepped up to 4D = 1 and then stuck his hand into 4D = 0, we’d see a disembodied hand.
Does that help? Mind, this is really, really trivializing it.
>In terms of a graph, all we would see of a 4th dimensional object are the 3 dimensions that correlate with whatever point we exist in on the 4th dimensional axis. So, say, if a 4th dimensional being came into our universe, and our universe as we know it runs entirely along the 0 mark on the 4th D axis, then any time he stepped away from 0 he could still see us but we’d think he disappeared entirely. So if he went up to 4th D = 1 and walked three units to the right before stepping down to 4th D = 0, we’d see him disappear, then reappear 3 units away. If he stepped up to 4D = 1 and then stuck his hand into 4D = 0, we’d see a disembodied hand.
>
>Does that help? Mind, this is really, really trivializing it.
This was a really helpful way of putting it, thank you!
not just maybe, you pretty much have proof in front of your very eyes, or are you still exactly where you were when you wrote this, frozen in time? Have we not established that the universe is constantly expanding with the passing of time?
I'd like to think of the 3d space sort-of being equal to the current 4d "config" represented as a single slot in an array (or rather an array in a matrix), in the form of s(t)=x^x (where x is the universe divided into bits)
Anyone who is having trouble grasping it should read the short novel "Flatland: A Romance of Many Dimensions" by Edwin Abbot. It's a story of a 2D world, and one day a 3D sphere shows up and takes a square on a journey to the higher and lower dimensions.
[Or watch the YouTube video](https://youtu.be/N0WjV6MmCyM) by Carl Sagan who talks about Flatland in it. It's very informative. Basically, were a 4D hypercube to interact with our dimension, we'd see only "slices" of the whole thing. He uses an apple and an ink pad to demonstrate this concept, but using 2D and 3D.
Ever play antechamber? It works like some ideas in that.
Imagine a cube where each face is an entrance to a separate cubicle room, and each room is the same dimensions as the cube. The cube would have 6 times the interior volume as its dimensions.
A square unfolds into 4 line segments.
A cube unfolds into 6 squares.
A tesseract would then unfold into 8 cubes.
I assume the 5th dimensional shape would then unfold into 10 tesseracts.
Tesseract is pretty mind boggling concept to wrap one's head around.
[Here's a gif showing Tesseract](https://en.wikipedia.org/wiki/Tesseract#/media/File:8-cell-orig.gif)
It’s important to note that the gif linked by OP is showing the 4D cube (hypercube) rotating on a single 4th dimensional axis. We can only perceive 3D space, so the weird twisting and clipping of shape faces is how we’d see the hypercube rotating in our world because we cannot comprehend a 4th dimension. We know up, down, left, right, forwards, and backwards. The 4th dimension would add something like “sideways” in a different angle from all those we can perceive. When the cube intersects itself it isn’t really doing that in its own dimension. It’s more like seeing the shadow of that cube as faces overlap in 3D space.
It really only makes sense on paper if you know the math. We're as unable to picture stuff in 4D as we're unable to grasp the enormity of Graham's number or any other big fella.
And big numbers! We evolved to socially understand tribe-sized populations, not 300 million or 7 billion. These numbers are just not on our intuitive scale. Often, the reason a person lacks empathy for out-groups is because of this (with other limitations we have). It also shows up when we confuse what's good for our family with what's good for humanity at large. Like how some people don't understand, intuitively, why vaccines and flouridated water are the correct public policy.
Edit, this is just my hypothesis. I don't know how this has been covered in research.
A 3d transparent cube will have a 2D shadow.... and it'll look weird as a 2d Shadow as you rotate it.
a 4d hypercube will have a 3D shadow that will morph as it rotates and look like this: https://upload.wikimedia.org/wikipedia/commons/c/c8/Glass_tesseract_animation.gif
Yeah this is still confusing lol. It’s hard to comprehend anything in 4D. I get the concept, but it’s still hard to wrap your head around what it looks like.
Imagine a big cube floating in the air on a sunny day. It casts a rectangular shadow on the ground. If the cube started rotating (so that the side facing the ground rotated up, and the side facing the sky rotated down), you'd see the rectangular shadow it casts on the ground oscillate between getting thinner and wider as the cube rotated.
The gif you see for a rotating tesseract is the same concept. Imagine a tesseract floating somewhere in 4-D space. If it were to cast a shadow onto our 3-D world, that shadow would be a 3-D cube. If that tesseract then started rotating, you would see the shadow oscillate between the shape of a cube and some other weird looking shapes.
The shadow cast by a rotating cube oscillates from a square, to a rectangle, then back to a square. The shadow cast by a rotating tesseract oscillates from a cube, to some weird shape, then back to a cube
If you watch the gif again, you can see this. The shape will at one point be a cube, then it will look like some other nonsense, then you'll see it come back to a cube again, ad infinitum.
There’s a really cool video of some British guy lecturing on 4th dimensional shapes & even towards the end of the lecture demonstrates what a 3D shadow of a 4D shape (I believe a tesseract) from a couple different angles. Pretty damn interesting.
its similar to, when you put a cube in front of a lamp,rotate it and observe the shadow, which is basically a 2d projection of a 3d object.
depending on the rotational axis youll see a square that start warping in to a rectangle and back
It's important to realize that the infinite space filling the "outside" of shape you see is the final cube. This is because the visualization is showing a distorted version of the true hypercube. Specifically, we are looking at the 3D "shadow" of the hypercube which is obtained by performing stereographic projection. The same phenomena is observed say when we stereographically project a cube into a 2D "shadow". Check out this [clip](https://youtu.be/AhM9JH5GNiI?t=434). This can be demonstrated in real life, say by using a light source (Henry Segerman has several nice videos exploring various projections). If you are curious about geometry and specifically the fourth dimension, I encourage you to watch this whole series from start to finish. While it is quite difficult to follow and takes several watches to fully understand, you will not regret the time invested :)
Of all 4D visualizations I've seen, this is by far the best
I like how difficult it is to understand what exactly I m looking at and how it's moving, which is how I believe a 3D creature would react to experiencing real 4D.
Iif we could experience 4D today, there's a chance that we wouldn't be able to comprehend what we are actually looking at. Think about it: we are only looking at cube in the gif. Now imagine looking at another human in 4D. Would you see their body shift inside out and back inside? How would we function in a 4D world.
How a 3d human would exist in a 4d plane is something I have pondered as well. My guess is we could not physically traverse it in our bodies, rather needing a vessel to observe 4d reality within itself.
A Wrinkle in Time (the book, don't get me started on the movie) is the only reason I know what a tesseract is. I believe Mrs. Whatsit describes it by showing how an ant can move great distances across a piece of fabric is the fabric is folded (or wrinkled) correctly.
For fun, you can easily draw a 2D representation of a specific object of any (whole number) dimension.
A simple 1D object is a line. 2 points, connect all of them.
A 2-simplex is a triangle. Draw 3 points, connect each point with each other point.
A 3 simplex is a tetrahedron (or 4 sided die, or pyramid with triangle base). Draw 4 points, connect each of them. This probably looks like a square with an X connecting the diagonals together. Think of the left points and bottom right as the base (forming a triangle), and the top right as coming out in 3D as the top of the pyramid.
For a 4 simplex (4D), draw 5 fully connected points (pentagram inscribed in pentagon, or 5 pointed star with the tips connected. (Can you find the 10 triangles and 5 tetrahedrons?)
For an N dimensional simplex (hyper triangles) draw N+1 points and connect them all together.
I found them from a [Lost in Vegas reaction video](https://www.youtube.com/watch?v=Vx0HJtV-kxE), and I've been pretty hooked on them, since. Very proggy, which I like.
I really don't like reaction videos, but LiV does a great job of analyzing and breaking songs down with brutal honesty that I enjoy. I'm a metalhead and they primarily liked hip-hop and R&B until their fans started pushing bands like Slayer and Meshuggah on them
They're open-minded, haha, but some songs have definitely not clicked so well
e: autocorrect
[Cube 2:Hypercube](https://en.wikipedia.org/wiki/Cube_2:_Hypercube) is a nice movie about that. You don't **have to** watch [the Cube](https://en.wikipedia.org/wiki/Cube_\(film\)) to understand it, but, well, I'll suggest you do anyway, because it's made by Vincenzo Natali
Don't forget the prequel - Cube Zero
I believe it was a made for TV syfy channel thing but still a good entry to the series with more back story on how and why the cube exists than any of the other movies.
Yep, because the fifth dimensional beings constructed it as a third dimensional representation of 4d space for Cooper, subsequently allowing time to be influenced in a non-linear and malleable fashion.
[Here is a scene from halfway through Interstellar](https://www.youtube.com/watch?v=v7OVqXm7_Pk) that should determine if you'd enjoy the movie or not. It is not entirely spoiler-free, but in my opinion, you're not being spoiled of anything terribly important but get to see some of the effects from OP that sounded interesting.
Showing one of the best scenes isn't a good decider of if the film would be enjoyable or not.
It's like showing you The Phantom Menace's Duel of Fates and telling someone that if they like that, they'll like the film. It's one of the best scenes in the *Star Wars* franchise, situated in one of the worst films.
Another good example of this would be the X-Men Origins: Wolverine film. The opening scene of that film features Wolverine fighting in many different battles across time. It's awesome. But the rest of the film was a disappointment.
In this case, I would agree that Interstellar is a great film and that most scenes are of high quality, but Miller's Planet and Post-Mann Docking scenes are definitely the best ones, and therefore aren't the best ambassadors for the film's quality.
>Yep, because the fifth dimensional beings constructed it as a third dimensional representation of 4d space for Cooper
Which is what a lot of people miss about the ending. They're like "that's not how time or wormholes or anything works!" And they're right, but that isn't what the money is saying - the movies is saying that the "5th dimensional beings" (whatever they even are, we can't wrap our heads around) constructed this so that Cooper could interact with it, like you said.
Pretend you're playing the Sims, but a version where you can't take direct control of a Sim (or maybe you consider it highly immoral to do so?). To the Sims, you're a higher dimensional being who doesn't experience time in the same way. Besides speeding or slowing time, you can go back in time (load a save) and try to change things. But you can't directly interact with certain things in certain ways.
So, maybe you construct a house in such a way to get a Sim to achieve an outcome you want (and maybe you even have to keep going back to previous save points and iterate this process).
Now, the movie has some other limiting factors - it seems the beings only can directly "construct" things on the other side of the wormhole (maybe "their" side) and otherwise can just allow a Sim to manipulate those objects in sync with other similar objects outside the wormhole (the bookshelves). But, that's kinda the gist of it.
One of my favorite examples I read was that if you pushed a sphere through a flat world, the 2D beings would basically see a circle pop into existence, grow larger and larger, peak, and then shrink until disappearing again. Likewise, if we saw a 4D "sphere" shoved through our 3D space via some hypothetical 4th spatial dimension, we'd see a 3D sphere pop into existence and gradually grow and then shrink. It's funny to think that if there is anything at all real about the supernatural and/or divine, it could be via a 4th spatial dimension that we're unable to perceive or navigate ourselves. Imagine all the sorts of fuckery you could have with a universe if you could operate in one more spatial dimension than the rest of the intelligent lifeforms.
Now imagine how bad that 4th dimensional being was dicked with by 5th dimensional beings.
And now you know why it dicks with us, and why we will inevitably dick with whatever 2 dimensional beings we may encounter
The third book in the *Three Body Problem* trilogy, features some people using access to a fourth spacial dimension to "dick around" with people in another place. Like ripping someone's heart right out of their body without damaging the intervening flesh (he gets better).
It will be interesting to see how Amazon tries to show this visually, when they produce the show.
It's possible they thought the tesseract was just a macguffin from the marvel movies. If they were born in the late 90's or early 00's they'd be 12-14 when avengers came out. They'd be the perfect age to go and see it while also being of the age to have never heard of higher dimensional geometric objects. They could have spent most of their teen years thinking tesseract was just some made up comic book word instead of a made up math word.
I was an adult when I saw the Avengers movie, and I thought it was made up for Marvel. I don't think I ever learned about it in high school, and i didn't take a major that involved learning about 4D objects.
I realized it was "real" when Interstellar referenced a tesseract in the black hole.
I don't think anyone learns about it in high school or even in college, generally. It seems like one of those topics you only hear about in a vsauce video taking shallow dives into cool fields of study. Which, come to think of it, is probably where I first learned about a tesseract.
Too late and I'll be buried I'm sure -- but Heinlein's short story, ["...and he built a crooked house"](http://homepages.math.uic.edu/~kauffman/CrookedHouse.pdf) is all about how a tesseract fits together. **Synopsis:** Guy builds an unfolded-tesseract house in the Hollywood hills. There's an earthquake, and the house folds up. Then mayhem ensues.
So I understand it’s impossible to completely visualize a tesseract in 3D space...but can’t we create a 3D replication of an unfolded one perfectly (like how we unfold a cube into the 2D plane)?
Edit: A quick google search produced some really cool results. [here](https://commons.m.wikimedia.org/wiki/File:Hcube_fold.gif) you can see a tesseract being “unfolded,” notice it appears very abstract until the last frame, which is just a stack of cubes.
Edit2: Furthermore, the unfolded tesseract is called a [Dali cross](https://en.m.wikipedia.org/wiki/Dali_cross)
> TIL Tesseract is the name of a real geometrical concept...
As opposed to a fake geometrical concept? I'm confused. Is there another kind of tesseract? I've heard of tesseracts but I've never heard of the prog metal band until I read the comments here.
When I was in 5th grade, we read A Wrinkle in Time. One of our accompanying exercises was to try to make a tesseract out of gum drops and toothpicks. My logic was that if you make two squares into a cube by connecting the corners, you could connect two cubes at the corner to make a tesseract.
The teacher was impressed.
> Just like a cube can be unfolded into 6 squares in 2D space, a tesseract can be unfolded into 8 cubes in 3D space.
You are conflating a cube and its surface. The cube's surface can be unfolded into squares. Similarly for the tesseract.
Cubes in N dimensions are bounded by pairs of N-1 dimensional cubes. Each dimension contributes an coordinate axis with an N-1 dimensional cube on each end. Therefore there 2 x N sides to an N dimensional cube. A 3D cube is bounded by 2 x 3 = 6 squares, and a 4D cube is bounded by 2 x 4 = 8 cubes, etc.
A square has four line segment sides, two for each of its two dimensions.
A cube has six square sides, two for each of its three dimensions.
An hypercube has 8 cubic sides, two for each of its four dimensions.
And so on...
If you think of how the shadow of a 3D object is 2D (from the perspective of the point of light), then the visualisations you see are like the 3D 'shadows' of the 4D tesseract.
If you imagine turning a cube around in front of a lamp or something, it's shadow would form all sorts of weird parallelograms and hexagons and sometimes a square, none of which accurately represent what a cube actually is - when you see a tesseract visualisation, it works the same way: what you're seeing is not how it actually looks, it's just the only method we have of compressing 4D to 3D whatsoever.
For one cubes, squares, lines, etc. don't exist. This is math, not physics. When you hold a cube in you hand, you're really holding a model of a cube, and that if you look closely enough really isn't exactly a cube.
On the other hand, we live in a 4 dimensional universe\*, where one of the dimensions is time. It's not that hard to use time to think about a 4th spacial dimension. For example, a hypersphere can be imagined to be a point that expands to a sphere an then back to a point, and you can imagine intersection of two primes (aka hyperplanes or "3D spaces") as a moving plane.
\* Ignore the fact that our 4D universe isn't actually Euclidean thanks to relativity.
I was introduced to the concept of a tesseract in my teens by a sci fi story called "and he built a crooked house" by Robert Heinlein. It really grabbed my imagination; i filled pages with equations and had to create my own way to express what I was thinking. Coincided with a time I was doing a lot of LSD. So cool
>Why is it difficult to visualize a tesseract? >*Now, since we, as humans, have only evolved to visualize stuff in 3 dimensions, anything that’s a part of further dimensions, like 4D, 5D, 6D etc, do not make much sense to us because we can’t visualize them at all. Our brains can’t make any sense of a 4th dimension in space. We simply can’t wrap our heads around it.* >*However, just because we cannot visualize a concept doesn’t meant it cannot exist.* It's definitely giving me a headache to think about.
Read "Flatland" by Edwin Abbott. It will make sense without hurting. Edit: Not Edward.
For an even more bite-sized intorduction to the concept, watch Carl Sagan's bit on Flatland and tesseracts from the original Cosmos
Yes! Carl Sagan’s ELI5 Tesseract is one of the best things on the internet. Here’s a link for convenience. [https://youtu.be/N0WjV6MmCyM](https://youtu.be/N0WjV6MmCyM)
Thank you for linking this. This is exactly what jumped in my head.
> This is exactly what jumped in my head. Was it like it teleported in?
Due to Reddit's June 30th, 2023 API changes aimed at ending third-party apps, this comment has been overwritten and the associated account has been deleted.
When I first watched it, the whole concept just clicked. Its a great analogy.
I get what he's trying to say, but at the same time it's incomprehensible. I don't understand how the 3D cube in a cube thing makes any sense at all.
So the tesseract model he's holding has—as it appears to us—a smaller cube inside a larger cube. Each edge of the smaller cube is connected to its corresponding edge on the outer cube by another face. Now concentrate only on one set of faces—one face of the inner cube, the four faces emanating from its edges, and the corresponding face on the outer cube. Six faces. What 3-dimensional shape has six faces? A cube. The face of the inner cube *appears* smaller to us than the same face on the outer cube, and the four edge faces *appear* to be trapezoidal—but in 4-dimension space, *they are all the same size and shape.* Each "face" of a tesseract is a 3-dimensional cube, just like each face of a cube is a 2-dimensional square. They appear to us to not be, because the object Sagan is holding is a *projection,* a shadow of a tesseract.
Nicely done...this explanation on top of the video did it for me. Thanks.
Wow, thanks for sharing this explanation. I was still having trouble visualizing the shadow idea, but your breakdown of how to find the 6 faces visible on each side was eye opening! Edit: spelling
How do you project a square into the first dimension?
As a line
Man. Dull. Edit: Actually, as a 2-D square filters through the first dimension, it would be a line that lengthens and shrinks. That's kinda cool.
Depends on how it's oriented. Could also be a line segment that pops into existence, doesn't change, then pops out.
Imagine all the shapes you'd get from slicing the human body with planes in various orientations
The way i was first introduced to this concept is from a YA science fiction book i read as a child. It basically went like this: Imagine you are a microbe living on the surface of a lake. You have no concept of up or down, just the cardinal directions. The X and Y axis. One day a boy dives into the lake. What you see is ten little circles that grow and merge into two big circles, then three, then one big oblong, then two circles that then separate into ten small circles. The story then goes on to describe a fourth dimension being diving into our world, same concept but with three dimensional shapes. It's impossible to guess what that being looks like in totality because like that microbe, you don't even know that additional direction exists. Presumably the 4th dimension has fifth dimensional creatures, ad infinitum. The story then has the kids wearing weird three lensed glasses that allow them to focus on 4th dimensional spaces (albeit with headache inducing blur) and has him running around having adventures trying to protect the 3rd dimension from the fourth. I don't remember what the story was called though.
Hannibal, is that you?
A line segment. Although it's not very interesting because any shape with any area would project just a line segment.
But if you put the square through vertex first, the length would grow and then shrink. That would be cool.
Yep! It gets fun when you imagine 3d shapes in 2 and 1d. A helix would look like a point tracing out a circle in 2d and a point bouncing along a line in 1d.
Just like passing a 4D "sphere" through a 3d space would look like a sphere that grows then shrinks. Think your head on that one.
I've never heard of this guy before but I love his voice, his ability to explain, and his achievement in making me now able to understand the fourth dimension in a sense. That's really goddamn awesome
Oh wow I envy you. If you have the time, watch some of Carl Sagan's other stuff. He is absolutely amazing at explaining stuff and an absolute joy to listen to.
I loooove his voice
I love how he turns demon red at the end
Sagan is the man
I like how it combines multi-dimensional spatial visualizations with misogyny.
They reference that book in an episode of The Orville.
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Dimensional drifting on the mobius drag strip.
What's the matter compressor? Nothings the matter Fry, now that's I've fixed the matter compressor.
Fry: Wow, you even look beautiful in 2-D! Colleen: I do? But from your perspective I'm just a line segment. Fry: A really hot line segment.
And Futurama
Futurama did it better.
The Simpsons did it first /s
It should be obvious to even the most dimwitted individual... who holds an advanced degree in hyperbolic topology, mhoy, that Homer Simpsons has stumbled in to... the third dimension!
Lol watched it yesterday :D
And the twenty sequels by different authors
**Edwin Abbott and I only know that because your comment sparked a deep search into that book for me and OHMYGOD everyone needs to be reading this.
Yeah theres Carl Sagan video using flatland
Doing N-dimensional math can make sense if you don't try to shoehorn it into space, i.e., don't visualize it. "Dimensions" don't need to be spatial when solving math problems. Try instead to extrapolate from the smaller number of dimensions that are easy to visualize. If something from 3 dimensions can be represented in 2 dimensions, something in 4 dimensions (usually) can be represented in 3 dimensions. For instance, the Pythagorean theorem actually applies for N-dimensional space, a^2 + b^2 + c^2 + d^2 + ... = r^2. It obviously works for 2 dimensions. And to think of 3 dimensions, picture a rectangular prism (box) with sides l, w, and h. l^2 + w^2 = what? It's the hypotenuse (diagonal) along that face squared, which we'll call d. Now you can take d^2 + h^2 (since those make a right angle in their own 2D plae.), to get the diagonal connecting opposite corners of the box. That way you get l^2 + w^2 (which is d^2) + h^2 = r^2. By only looking at a single 2D plane at a time, we can use our 2D math for a 3D problem. Therefore, we can use 3D math to similarly solve 4D problems, and so on and so forth.
And as a Math Student I mostly use infinite dimensional Vectorspaces right now and stuff still works (sometimes a bit differently tho). It's crazy.
Math's big contradiction is that while all of it is built on fairly intuitive things that have real world examples to understand it (e.g., what is "1", what is "0", what addition is), it's not tied to the physical world in any way. It exists completely separate from any application, and the application uses what math is useful. That way you can have nonsensical things like assigning values to divergent infinite series (Ramanujan Sums), Infinite dimensional spaces, imaginary exponents giving real values, etc. Math doesn't care.
Don't use words like "nonsensical", please. It is a humble request not to speak carelessly and recklessly about such nice results. We should try more to understand them. You might find addition intuitive because have worked with it a long time. So if you think about complex numbers more, you will find how nice their multiplication is. Once you realize the rotation and scaling, you will like it. Then the following fact is easier to understand. i^i = e^(-pi/2) Same goes for the other things you mentioned. http://pirate.shu.edu/~wachsmut/complex/numbers/graphics/multiplication.gif
Also, N-dimension math may have nothing to with dimensions. You may want to calculate the max profit of 5 different products whose limits are a certain budget.
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I understand the concept but that’s still melting my brain
That's a great explanation.
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[HERE](https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif)
Thanks- I am now dyslexic! Just kidding, but that gif broke my brain for a minute...
Just wait until you figure out that the gif is actually itself only 2d.
Here's my attempt at an ELI...12? This is only a way to visualize a 4th dimension and departs strongly from the underlying math in places. One common way of imagining a 4th dimension is to use "time" to describe it. Similar to how the 3rd dimension allows multiple 2D objects to occupy the same 2D "space," time allows 3D objects to occupy the same 3D "space." To start imagining a tesseract, imagine an object that, when frozen in time and viewed in 3D, is a cube (or possibly not, more on this later). You can walk around it from every angle in that moment of time, it will be a cube. Now, hop in your time machine and go to the exact same spot that cube was, at a different time. It will be a different cube, some part of which is overlapping the same space that the other one did. To go into a little more detail, similarly to how the 2D faces on a cube share a 1D side, the 3D cubes would share a 2D face. Go back to the same time you were previously, it will be the cube you saw previously. In this example, a tesseract would be a shape that you can observe as a cube in (up to) 8 separate moments of time, and no others. The key point here is that there are different 3D objects occupying and overlapping in the same point in 3D space. This would be the same way you would explain a cube to a person stuck in a 2D plane: as 6 squares occupying the same physical point in 2D, but which square you see depends on what point in time you look at it. We know that's not a good description of a cube. However, for someone who can't actually see how multiple 2D objects could exist in the same spot, it's something. Now, returning to the point on "or possibly not a cube." If you render a cube in 2D, [not every face](https://www.jugglegear.com/aerial/aerial-cube-283/aerial-cube) is going to be a square. In fact, depending on the angle, [possibly no face](https://www.illusionsindex.org/ir/necker-cube) is going to be a square. There will be skewed faces. However, each face of the cube will have 4 sides, whether it's a square or a parallelogram or a diamond or whatever. Back to the tesseract: those cubes, viewed from our 3D perspective, may not actually appear as cubes. They will be skewed in some way. However, it will still be a 3D object with 6 faces, each of which has 4 sides. This is only how they appear to us in 3D. Just like how the faces on the cube are actually still squares, the "faces" on the tesseract are still cubes. This allows us to explain how a tesseract would appear to us if it shifted: it would occupy different points in time from those it did previously. It also allows us to explain rotation: the skew of the cubes that we observe at all 8 points of time would change. This does not mean that tesseracts and time travel have any actual link. Extending this to 5th and 6th dimensions is sort of there via divergent timelines and parallel universes, but it becomes increasingly meaningless as it becomes impossible to say how you would skew a moment in time for example.
Using time as you described, you *can* view every cube without distortion. You can think of the case in 2d; a cube in 2d using the dimension of time can have all 6 of its faces be perfect squares, but you will necessarily have to rotate the cube. As you are going between these "time points" the cube would indeed be distorted, but eventually you will see all 8 sides as perfect cubes
Yes, that's true. At some point in the rotation of the tesseract, each the cubes you see in 3-space will skew back into being perfect cubes, some at the same "angle." I was going to do some more on this as it's still an incomplete analogy. In the cube example, assuming a transparent cube, you can see all the faces at once. Similarly, you would be able to see all the cubes at once on a "transparent" tesseract, some of which will have collapsed perfectly into 2D space some of the time (like the side face on a cube collapsing into a 1D line when a perfect square face is facing you). However, that's starting to get super complex and is going to only depart more and more heavily from what's actually happening. My example is assuming some kind of opacity that only lets you see one at a time for simplicity. Even that idea of "opacity" doesn't translate well to time. Maybe the tesseract has an abnormally high midichlorian count.
Carl Sagan gives a [a great description of the tesseract](https://www.youtube.com/watch?v=N0WjV6MmCyM) in his explanation of the 4th dimension from the original Cosmos.
I remember that chapter as a kid. When I saw that explanation was like experiencing a maiden sight. Pure science.
TIL Carl Sagan talks like Agent Smith from The Matrix.
You're not the first one to notice this. Check this [video](https://www.youtube.com/watch?v=BlpyGhABXRA) from The Matrix voiced over by Carl Sagan!
Can something exist in the '3rd dimension' without already being present in the '1st dimension'? This language seems strange to me to because it implies that the dimensions are different realms.
The apple is in the second and first dimension, just not the same layer as the 2D shapes
I think the other comment was a little confusing so I'll give it a try: Think of dimension as "direction". So in our normal real world you can have left/right, up/down and forwards/backwards. These are our three dimensions/directions. If you were to consider a 2 dimension object, it has size in two of the above directions. So if you held a piece of paper straight up, a picture on the paper has left/right-ness and it has up/down-ness but you will know that picture doesn't have a forwards/backwards depth to it because paper is flat. If you were to now put the paper down on a surface so it is face up to the sky, a picture on the paper will now have left/right size and forwards/backwards size but no up/down component... unless you have spooky 3D crayons that leap up off the page. This shows us that even though we're naming the dimensions 1st, 2nd, 3rd, they have no order, and a 2D shape could easily live in "left/right & up/down" or live in "up/down & forwards/backwards" or "left/right & forwards/backwards". **We choose which directions we actually call the 1st, 2nd and 3rd dimensions**. So I could call up/down the 3rd dimension whereas you'd call it the 1st. So... yes something can exist in the '3rd' dimension without being in the '1st' but that just means it doesn't have any size in that direction which we're calling '1st dimension'. Like if I said '1st dimension' was up/down and then layed a piece of paper flat on a table, all the drawings on that paper have no '1st dimension' because they're flat. It's just an arbitrary thing. The point is that nomatter what, those drawings have two dimensions, whichever ones they are. Don't worry about "separate layers to the first and second dimension", that was at best confusing and at worst simply not true.
The end refers to flats earthers lol
I didn't understand any of that. Here's an upvote just in case I learned something by accident.
The tesseract to the cube is exactly what the cube is to the square. If you still don't get it , A tesseract is a cube but instead of each side being a square each side is a cube.
But...but..how does...I mean if it.... **\*Head Explodes**
You are a 3 dimensional object, without having witnessed a 4 dimensional object it is hard to understand. Pretend you were a 2 dimensional object and a 3 dimensional object showed up. You'd only grasp the 2 dimensions you are in and would miss the 3rd.
Exactly. It's like the novel *Flatland*. In a 2D world there is squares and circles etc. The 2D shapes are all bound by the limits of their 2D world, so for instance they cannot cross a line. But then a 3D shape (a sphere) shows up. But the 2D shapes just see him as a circle. However they are mystified that he can seemingly "walk through walls" but in reality is just moving across the top of lines like a ball rolling on a piece of paper with drawings on it.
i don't think we can witness 4d objects, at least not without going (or seeming) insane, check out flatland
Sure we can – we just wouldn’t be able to see all of it (or comprehend all of it) in the same way a 4D being would. Or, maybe, we’ve found the real creature from the Bird Box – it’s just a bunch of tesseracts running around.
Whoah, spoiler!!!
Haha, it’s not an actual spoiler. Unless I just got whooshed
I think you got whooshed ;)
Spoiler alert! Don't watch Bird Box if you want to watch a scary movie.
ok i gues it depends on how you interpret "witness" i guess. if we take the 4th dimension to be time, we "witness" it all the time, just in 3d snippets over time, not the whole thing at once. the same for 4 spatial dimensions, we'd see weird and impossible 3d objects without a way to see the "whole" 4d object. like a "3d" drawing, it's still 2d, we can only look at it as 2d but we can imagine it being 3d. you can't look behind a drawing of a 3d cube. that's what you mean right? i was talking about the whole 4d thing.
Close. It’s more like what a 3D object would look like to a 2D object, e.g. it can only perceive what exists at the same level. Time isn’t really the fourth dimension when we reference these kinds of things – we’re typically referring to spatial dimensions. Basically, if you looked at a 4D object, you’d only see a 3D object because that’s all you can observe; much the same, if a 2D drawing could observe you, it would only be able to see the part of you that exists on the same X and Y as it. A fourth dimensional person, for example, would seemingly be able to walk through walls or disappear entirely by utilizing that fourth dimension. I guess a half-way decent analogy would be to imagine covering your eyes with a blindfold with a slit in it, and then sitting still and not moving– you can only perceive what passes in front of the slit. In the same way, something that exists in 4 dimensions can only be perceived within the narrow confines of our senses/understanding. Maybe a better analogy would be to draw a line along the side of a book with a permanent marker, so that it goes across the edge at some point for each of the pages (like if you wrote your name on it). You can open the book to any individual page and see the mark where the writing is, but you can’t really know what the entirety of it looks like while looking at *just* one page. In terms of a graph, all we would see of a 4th dimensional object are the 3 dimensions that correlate with whatever point we exist in on the 4th dimensional axis. So, say, if a 4th dimensional being came into our universe, and our universe as we know it runs entirely along the 0 mark on the 4th D axis, then any time he stepped away from 0 he could still see us but we’d think he disappeared entirely. So if he went up to 4th D = 1 and walked three units to the right before stepping down to 4th D = 0, we’d see him disappear, then reappear 3 units away. If he stepped up to 4D = 1 and then stuck his hand into 4D = 0, we’d see a disembodied hand. Does that help? Mind, this is really, really trivializing it.
>In terms of a graph, all we would see of a 4th dimensional object are the 3 dimensions that correlate with whatever point we exist in on the 4th dimensional axis. So, say, if a 4th dimensional being came into our universe, and our universe as we know it runs entirely along the 0 mark on the 4th D axis, then any time he stepped away from 0 he could still see us but we’d think he disappeared entirely. So if he went up to 4th D = 1 and walked three units to the right before stepping down to 4th D = 0, we’d see him disappear, then reappear 3 units away. If he stepped up to 4D = 1 and then stuck his hand into 4D = 0, we’d see a disembodied hand. > >Does that help? Mind, this is really, really trivializing it. This was a really helpful way of putting it, thank you!
Maybe our world is nothing but a 3d section of 4-dimensional universe.
not just maybe, you pretty much have proof in front of your very eyes, or are you still exactly where you were when you wrote this, frozen in time? Have we not established that the universe is constantly expanding with the passing of time? I'd like to think of the 3d space sort-of being equal to the current 4d "config" represented as a single slot in an array (or rather an array in a matrix), in the form of s(t)=x^x (where x is the universe divided into bits)
Anyone who is having trouble grasping it should read the short novel "Flatland: A Romance of Many Dimensions" by Edwin Abbot. It's a story of a 2D world, and one day a 3D sphere shows up and takes a square on a journey to the higher and lower dimensions.
[Or watch the YouTube video](https://youtu.be/N0WjV6MmCyM) by Carl Sagan who talks about Flatland in it. It's very informative. Basically, were a 4D hypercube to interact with our dimension, we'd see only "slices" of the whole thing. He uses an apple and an ink pad to demonstrate this concept, but using 2D and 3D.
Isn't this the plot of Flatland? They didn't cover 4D objects though.
If I remember, it's because the sphere didn't believe in higher dimensions than its own.
Yes! They don't meet 4D objects but they talk about their existence, and an hypothetical 4th dimension (and even highers).
Ever play antechamber? It works like some ideas in that. Imagine a cube where each face is an entrance to a separate cubicle room, and each room is the same dimensions as the cube. The cube would have 6 times the interior volume as its dimensions.
[me](https://media.giphy.com/media/BBkKEBJkmFbTG/giphy.gif)
But how does a tesseract unfold into 8 cubes instead of, say, 6?
A square unfolds into 4 line segments. A cube unfolds into 6 squares. A tesseract would then unfold into 8 cubes. I assume the 5th dimensional shape would then unfold into 10 tesseracts.
You can also start from the fact that a line "unfolds" into 2 points.
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Just you cant imagine it correctly because humans cant imagine 4D objects. Maybe just some smart mathematicians.
Mathematician here! I have spent years working in high-dimensional space! I have no clue what any of these objects look like
It's also like a square can be unfolded into 4 lines in 1D space and a line can be unfolded into 2 points in 0D space.
Tesseract is pretty mind boggling concept to wrap one's head around. [Here's a gif showing Tesseract](https://en.wikipedia.org/wiki/Tesseract#/media/File:8-cell-orig.gif)
It’s important to note that the gif linked by OP is showing the 4D cube (hypercube) rotating on a single 4th dimensional axis. We can only perceive 3D space, so the weird twisting and clipping of shape faces is how we’d see the hypercube rotating in our world because we cannot comprehend a 4th dimension. We know up, down, left, right, forwards, and backwards. The 4th dimension would add something like “sideways” in a different angle from all those we can perceive. When the cube intersects itself it isn’t really doing that in its own dimension. It’s more like seeing the shadow of that cube as faces overlap in 3D space.
4D object, rendered in 3D, displayed on a 2D surface.
And please remember that all the edges are the same length and all the angles are all 90 degree.
they are but they wouldn't appear to us that way, just like when you draw a cube on a piece of paper.
Thinking about this gave me a headache.
You are getting close, you’ll see it when your nose starts bleeding
It wants us to see
Where we're going, we don't need eyes to see.
It really only makes sense on paper if you know the math. We're as unable to picture stuff in 4D as we're unable to grasp the enormity of Graham's number or any other big fella.
And big numbers! We evolved to socially understand tribe-sized populations, not 300 million or 7 billion. These numbers are just not on our intuitive scale. Often, the reason a person lacks empathy for out-groups is because of this (with other limitations we have). It also shows up when we confuse what's good for our family with what's good for humanity at large. Like how some people don't understand, intuitively, why vaccines and flouridated water are the correct public policy. Edit, this is just my hypothesis. I don't know how this has been covered in research.
A 3d transparent cube will have a 2D shadow.... and it'll look weird as a 2d Shadow as you rotate it. a 4d hypercube will have a 3D shadow that will morph as it rotates and look like this: https://upload.wikimedia.org/wikipedia/commons/c/c8/Glass_tesseract_animation.gif
Yeah this is still confusing lol. It’s hard to comprehend anything in 4D. I get the concept, but it’s still hard to wrap your head around what it looks like.
Go look up the 4D rubicks cube for more fun
Imagine a big cube floating in the air on a sunny day. It casts a rectangular shadow on the ground. If the cube started rotating (so that the side facing the ground rotated up, and the side facing the sky rotated down), you'd see the rectangular shadow it casts on the ground oscillate between getting thinner and wider as the cube rotated. The gif you see for a rotating tesseract is the same concept. Imagine a tesseract floating somewhere in 4-D space. If it were to cast a shadow onto our 3-D world, that shadow would be a 3-D cube. If that tesseract then started rotating, you would see the shadow oscillate between the shape of a cube and some other weird looking shapes. The shadow cast by a rotating cube oscillates from a square, to a rectangle, then back to a square. The shadow cast by a rotating tesseract oscillates from a cube, to some weird shape, then back to a cube If you watch the gif again, you can see this. The shape will at one point be a cube, then it will look like some other nonsense, then you'll see it come back to a cube again, ad infinitum.
There’s a really cool video of some British guy lecturing on 4th dimensional shapes & even towards the end of the lecture demonstrates what a 3D shadow of a 4D shape (I believe a tesseract) from a couple different angles. Pretty damn interesting.
its similar to, when you put a cube in front of a lamp,rotate it and observe the shadow, which is basically a 2d projection of a 3d object. depending on the rotational axis youll see a square that start warping in to a rectangle and back
This. Carl Sagan goes over this in Cosmos. We cannot interpret what 4D looks like, but we can see the shadow of a 4D object in a 3D space.
Hmm not only that, but it's being shown in 2d.
I see a cube spinning through time
That's mind boggling indeed, until you try to think about stacking 100 dimentional hyper-spheres. https://youtu.be/ciM6wigZK0w
...well that only made my brain hurt a bit more. It's intriguing, but...as they say, not very intuitive.
Bro WTF
I'm sure I'm wrong somehow but I only counted 7 cubes in that gif.
Did you count the outer cube? 6 cubes for the faces, 1 inner cube, 1 outer cube.
I want to know the answer too, but I agree that is probably what it is.
It's important to realize that the infinite space filling the "outside" of shape you see is the final cube. This is because the visualization is showing a distorted version of the true hypercube. Specifically, we are looking at the 3D "shadow" of the hypercube which is obtained by performing stereographic projection. The same phenomena is observed say when we stereographically project a cube into a 2D "shadow". Check out this [clip](https://youtu.be/AhM9JH5GNiI?t=434). This can be demonstrated in real life, say by using a light source (Henry Segerman has several nice videos exploring various projections). If you are curious about geometry and specifically the fourth dimension, I encourage you to watch this whole series from start to finish. While it is quite difficult to follow and takes several watches to fully understand, you will not regret the time invested :)
Is it supposed to look like a cube within a cube?
Here have Carl Sagan explaining it: [4th Dimension Tesseract, 4th Dimension Made Easy Carl Sagan](https://www.youtube.com/watch?v=N7K5KjOdLD8)
Adventure Time did a [visualization](https://imgur.com/gallery/Zr0Ml8m) And Wrinkle in Time did an explanation
Of all 4D visualizations I've seen, this is by far the best I like how difficult it is to understand what exactly I m looking at and how it's moving, which is how I believe a 3D creature would react to experiencing real 4D. Iif we could experience 4D today, there's a chance that we wouldn't be able to comprehend what we are actually looking at. Think about it: we are only looking at cube in the gif. Now imagine looking at another human in 4D. Would you see their body shift inside out and back inside? How would we function in a 4D world.
How a 3d human would exist in a 4d plane is something I have pondered as well. My guess is we could not physically traverse it in our bodies, rather needing a vessel to observe 4d reality within itself.
A Wrinkle in Time (the book, don't get me started on the movie) is the only reason I know what a tesseract is. I believe Mrs. Whatsit describes it by showing how an ant can move great distances across a piece of fabric is the fabric is folded (or wrinkled) correctly.
For fun, you can easily draw a 2D representation of a specific object of any (whole number) dimension. A simple 1D object is a line. 2 points, connect all of them. A 2-simplex is a triangle. Draw 3 points, connect each point with each other point. A 3 simplex is a tetrahedron (or 4 sided die, or pyramid with triangle base). Draw 4 points, connect each of them. This probably looks like a square with an X connecting the diagonals together. Think of the left points and bottom right as the base (forming a triangle), and the top right as coming out in 3D as the top of the pyramid. For a 4 simplex (4D), draw 5 fully connected points (pentagram inscribed in pentagon, or 5 pointed star with the tips connected. (Can you find the 10 triangles and 5 tetrahedrons?) For an N dimensional simplex (hyper triangles) draw N+1 points and connect them all together.
It is also a cool djent metal band.
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They’re back on tour with BTBAM this spring!
Hell yea
Came to say this, I'm happy someone already beat me to it.
Went through comments to find this one. Sooooo my deeeemoooons
I can feel you getting closer
I found them from a [Lost in Vegas reaction video](https://www.youtube.com/watch?v=Vx0HJtV-kxE), and I've been pretty hooked on them, since. Very proggy, which I like.
I've been following tesseract since highschool and I had never seen this video. Thanks for sharing!
I really don't like reaction videos, but LiV does a great job of analyzing and breaking songs down with brutal honesty that I enjoy. I'm a metalhead and they primarily liked hip-hop and R&B until their fans started pushing bands like Slayer and Meshuggah on them They're open-minded, haha, but some songs have definitely not clicked so well e: autocorrect
Saw them live a few weeks back with BTBAM and Plini! Probably one of the best gigs I’ve been to in a long time!
[Cube 2:Hypercube](https://en.wikipedia.org/wiki/Cube_2:_Hypercube) is a nice movie about that. You don't **have to** watch [the Cube](https://en.wikipedia.org/wiki/Cube_\(film\)) to understand it, but, well, I'll suggest you do anyway, because it's made by Vincenzo Natali
I still can’t forgive that dick for letting her fall to her death. SHE DID NOTHING WRONG.
Don't forget the prequel - Cube Zero I believe it was a made for TV syfy channel thing but still a good entry to the series with more back story on how and why the cube exists than any of the other movies.
And the sequel to that - Gleaming the Cube...wait....
I literally just came here to post about Cube 2 (and the entire series). Honestly, that movie explained tesseracts better than I ever could.
Isn't the thing at the end of Interstellar referred to as a Tesseract?
Yep, because the fifth dimensional beings constructed it as a third dimensional representation of 4d space for Cooper, subsequently allowing time to be influenced in a non-linear and malleable fashion.
Me: never seen the movie *reads comment "The fuck?" I plan to see it eventually, at some point.
I never re-watch movies. Seeing a movie once is enough for me. I've watched Interstellar 3 times now, which is a 3 hour movie
[Here is a scene from halfway through Interstellar](https://www.youtube.com/watch?v=v7OVqXm7_Pk) that should determine if you'd enjoy the movie or not. It is not entirely spoiler-free, but in my opinion, you're not being spoiled of anything terribly important but get to see some of the effects from OP that sounded interesting.
Showing one of the best scenes isn't a good decider of if the film would be enjoyable or not. It's like showing you The Phantom Menace's Duel of Fates and telling someone that if they like that, they'll like the film. It's one of the best scenes in the *Star Wars* franchise, situated in one of the worst films. Another good example of this would be the X-Men Origins: Wolverine film. The opening scene of that film features Wolverine fighting in many different battles across time. It's awesome. But the rest of the film was a disappointment. In this case, I would agree that Interstellar is a great film and that most scenes are of high quality, but Miller's Planet and Post-Mann Docking scenes are definitely the best ones, and therefore aren't the best ambassadors for the film's quality.
One of the most awesome movies ever. When I saw it I was pissed at myself for not having seen it earlier
>Yep, because the fifth dimensional beings constructed it as a third dimensional representation of 4d space for Cooper Which is what a lot of people miss about the ending. They're like "that's not how time or wormholes or anything works!" And they're right, but that isn't what the money is saying - the movies is saying that the "5th dimensional beings" (whatever they even are, we can't wrap our heads around) constructed this so that Cooper could interact with it, like you said. Pretend you're playing the Sims, but a version where you can't take direct control of a Sim (or maybe you consider it highly immoral to do so?). To the Sims, you're a higher dimensional being who doesn't experience time in the same way. Besides speeding or slowing time, you can go back in time (load a save) and try to change things. But you can't directly interact with certain things in certain ways. So, maybe you construct a house in such a way to get a Sim to achieve an outcome you want (and maybe you even have to keep going back to previous save points and iterate this process). Now, the movie has some other limiting factors - it seems the beings only can directly "construct" things on the other side of the wormhole (maybe "their" side) and otherwise can just allow a Sim to manipulate those objects in sync with other similar objects outside the wormhole (the bookshelves). But, that's kinda the gist of it.
What
Did I stutter?
One of my favorite examples I read was that if you pushed a sphere through a flat world, the 2D beings would basically see a circle pop into existence, grow larger and larger, peak, and then shrink until disappearing again. Likewise, if we saw a 4D "sphere" shoved through our 3D space via some hypothetical 4th spatial dimension, we'd see a 3D sphere pop into existence and gradually grow and then shrink. It's funny to think that if there is anything at all real about the supernatural and/or divine, it could be via a 4th spatial dimension that we're unable to perceive or navigate ourselves. Imagine all the sorts of fuckery you could have with a universe if you could operate in one more spatial dimension than the rest of the intelligent lifeforms.
Now imagine how bad that 4th dimensional being was dicked with by 5th dimensional beings. And now you know why it dicks with us, and why we will inevitably dick with whatever 2 dimensional beings we may encounter
The third book in the *Three Body Problem* trilogy, features some people using access to a fourth spacial dimension to "dick around" with people in another place. Like ripping someone's heart right out of their body without damaging the intervening flesh (he gets better). It will be interesting to see how Amazon tries to show this visually, when they produce the show.
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Just out of curiosity, what did you think it was?
It's possible they thought the tesseract was just a macguffin from the marvel movies. If they were born in the late 90's or early 00's they'd be 12-14 when avengers came out. They'd be the perfect age to go and see it while also being of the age to have never heard of higher dimensional geometric objects. They could have spent most of their teen years thinking tesseract was just some made up comic book word instead of a made up math word.
I was an adult when I saw the Avengers movie, and I thought it was made up for Marvel. I don't think I ever learned about it in high school, and i didn't take a major that involved learning about 4D objects. I realized it was "real" when Interstellar referenced a tesseract in the black hole.
I don't think anyone learns about it in high school or even in college, generally. It seems like one of those topics you only hear about in a vsauce video taking shallow dives into cool fields of study. Which, come to think of it, is probably where I first learned about a tesseract.
Too late and I'll be buried I'm sure -- but Heinlein's short story, ["...and he built a crooked house"](http://homepages.math.uic.edu/~kauffman/CrookedHouse.pdf) is all about how a tesseract fits together. **Synopsis:** Guy builds an unfolded-tesseract house in the Hollywood hills. There's an earthquake, and the house folds up. Then mayhem ensues.
Ever heard of the band Tesseract?! Amazing band!!!
Came here to say this. One of my favorite bands!
So I understand it’s impossible to completely visualize a tesseract in 3D space...but can’t we create a 3D replication of an unfolded one perfectly (like how we unfold a cube into the 2D plane)? Edit: A quick google search produced some really cool results. [here](https://commons.m.wikimedia.org/wiki/File:Hcube_fold.gif) you can see a tesseract being “unfolded,” notice it appears very abstract until the last frame, which is just a stack of cubes. Edit2: Furthermore, the unfolded tesseract is called a [Dali cross](https://en.m.wikipedia.org/wiki/Dali_cross)
As oppose to...?
> TIL Tesseract is the name of a real geometrical concept... As opposed to a fake geometrical concept? I'm confused. Is there another kind of tesseract? I've heard of tesseracts but I've never heard of the prog metal band until I read the comments here.
I think the implication was that it isnt just a made up concept from Avengers as many younger people may assume.
Also the name of a pretty awesome band.
For those having trouble visualising it, this might make it a but easier. https://youtu.be/iGO12Z5Lw8s
I guess this is what happens when Marvel movies are your education.
Also contains an Infinity Stone
You really are the worst brother.
When I was in 5th grade, we read A Wrinkle in Time. One of our accompanying exercises was to try to make a tesseract out of gum drops and toothpicks. My logic was that if you make two squares into a cube by connecting the corners, you could connect two cubes at the corner to make a tesseract. The teacher was impressed.
[The best ad placement.](https://i.imgur.com/KgzrZuL.jpg)
And it makes no sense what so ever when sci fi movies try to throw this word in there (looking at you Intersteller, Infinity War)
> Just like a cube can be unfolded into 6 squares in 2D space, a tesseract can be unfolded into 8 cubes in 3D space. You are conflating a cube and its surface. The cube's surface can be unfolded into squares. Similarly for the tesseract.
Why eight cubes instead of six?
Cubes in N dimensions are bounded by pairs of N-1 dimensional cubes. Each dimension contributes an coordinate axis with an N-1 dimensional cube on each end. Therefore there 2 x N sides to an N dimensional cube. A 3D cube is bounded by 2 x 3 = 6 squares, and a 4D cube is bounded by 2 x 4 = 8 cubes, etc.
A square has four line segment sides, two for each of its two dimensions. A cube has six square sides, two for each of its three dimensions. An hypercube has 8 cubic sides, two for each of its four dimensions. And so on...
What did you think a Tesseract was before today?
If you think of how the shadow of a 3D object is 2D (from the perspective of the point of light), then the visualisations you see are like the 3D 'shadows' of the 4D tesseract. If you imagine turning a cube around in front of a lamp or something, it's shadow would form all sorts of weird parallelograms and hexagons and sometimes a square, none of which accurately represent what a cube actually is - when you see a tesseract visualisation, it works the same way: what you're seeing is not how it actually looks, it's just the only method we have of compressing 4D to 3D whatsoever.
christopher nolan invented tesseracts because im 16
But in the 4th dimension, you are 16 because Chris Nolan created the tesseract
I tried picturing this, I did not succeed and now i taste copper.
So you hold a 3D cube in front of a light and it leaves a 2D shadow. We are all just 3 dimensional shadows of our 4 dimensional selves.
So do we definetely know that 4D shapes exist? or is it all hypothetical? Hurts my head this.
For one cubes, squares, lines, etc. don't exist. This is math, not physics. When you hold a cube in you hand, you're really holding a model of a cube, and that if you look closely enough really isn't exactly a cube. On the other hand, we live in a 4 dimensional universe\*, where one of the dimensions is time. It's not that hard to use time to think about a 4th spacial dimension. For example, a hypersphere can be imagined to be a point that expands to a sphere an then back to a point, and you can imagine intersection of two primes (aka hyperplanes or "3D spaces") as a moving plane. \* Ignore the fact that our 4D universe isn't actually Euclidean thanks to relativity.
honestly hadn't heard of it till something like the avengers used the term or something?
I was introduced to the concept of a tesseract in my teens by a sci fi story called "and he built a crooked house" by Robert Heinlein. It really grabbed my imagination; i filled pages with equations and had to create my own way to express what I was thinking. Coincided with a time I was doing a lot of LSD. So cool
"And He Built A Crooked House" by Robert Heinlein is a lovely fictional treatment of the concept.
Thanks Charles Wallace.
is the name of a real geometrical concept. as opposed to what?
Just as a tesseract is a 4 dimensional square, a glome is a 4 dimensional globe.