Demystifying the 4D Rubik's Cube

One of the things that makes it difficult to even try the Rubik's Cube 40 is that it is so confusing and daunting. I mean, how can you even begin to understand something like this, much less solve it well? This video will do just that. I'm not. I'm not going to teach you how to solve it in this video, but I hope to give you a better intuition about how this puzzle works so that it's easy when you solve it first of all. What do I mean by four dimensions? What is a dimension? The dimension of a mathematical space or object is informally defined as the minimum number of coordinates needed to specify any point within it which probably doesn't make much sense, but imagine you have a universe that is basically online, all objects and creatures, all matter must exist within it. line Let's say you want to use numbers to identify any location on this world line, how many numbers would you need to name any point on this line?

You only need one, obviously there's more information you need to set this coordinate as an origin point where you define a zero and ones, how much of this line you need to go before counting by one, but once you've calculated them you only need one number and you can identify any location on this line, that's why we call it one-dimensional universe, so we only describe the place that is one-dimensional, but dimensions can be used to describe both the space itself and the objects that occupy the space. things that can exist in a line the universe would be an infinite line or a line segment basically something that only has length but no width or height, but now let's say that the world is more than just a line, let's say that it is a simple universe, now a single number would not be enough to identify any location, I mean how would you differentiate between this point and At this point, with our previous method, you would have to call both locations as one, so we introduced the second number which solves our problem because we need two numbers to name any location on this plane.

We call this space two-dimensional. The things that exist here would not be If they are lines, there will be 2D shapes that have both length and width or length and height whatever you want to call it, such as squares and circles, among other more complex geometries, if both the line and the flat universe exist simultaneously, what would you see if you were? trapped in 1d universe and saw a 2d object pass through 1d space. My best guess is that you would see a line appear and disappear from existence because a line is all you can see in a 1d universe, let's add a third dimension to our hypothetical notice of how every time we add another dimension it is like adding another direction in the one we can move that is perpendicular to the previous direction, well two directions to be exact, but we'll call opposite directions just one for the sake of this video, we can represent this. direction using a line making three lines that are all perpendicular to each other, in fact, the maximum number of perpendicular lines that can pass through a single point is the number of spatial dimensions there are in a three-dimensional world, you cannot have more than three lines perpendiculars passing through a single point now four dimensions what happens then well using what we have been doing we would have to imagine a point that is completely outside of a three dimensional world we would also have to be able to understand four lines passing through a single point that are all perpendicular to each other and that is incredibly difficult, if not impossible, because we live in a three-dimensional world and we are restricted by this, if the existence of an object exceeds the three dimensions that we have, we will not get a complete view. from him or any of you, what if there is a way to flatten the geometry so that it takes on a lower dimensional shape?

Let's call this process projection. A projection is a mapping of a set onto a subset, which probably sounds meaningless if you're not familiar with the mathematical terms, and to be clear, I don't fully understand all the mathematical concepts related to projection. What I present here is just an explanation that helped me understand the 4d

cube

better, so yes, anyway this is a better way to look at it, it's like casting a shadow when you shine a light on an object and it has a shadow. Basically, you've created a 2D projection of a 3D object. We can take a 40 Rubik's

cube

created using mathematics and then have our computers. project it in 3D so we can have a visual view of it, but first we'll try to project a 3D Rubik's Cube into 2D to understand what we see in 4D, so we have a 3D hollow roux here.

We have a light source right in front of us. on the green side is our projector uh, we also have a wall to project on. By the way, I got this idea from another 4d video that I'll put in the description. I will also link other 4d videos that I found while researching for In this video, we turn on the light and it reveals a shadow on the wall. Usually the shadows are gray or dark, but I made it so that the shadows show the color depending on which side it is from, to better differentiate between each side.

Notice how the green side is. The closest to the light source is superimposed on the other faces. If the green side were opaque, it would actually obstruct the view of the other faces, so we hide that face and now we are left with 5 of the 6 faces we will draw. the lines that represent the two dimensions of these dimensions we can see in the projected image, let's call them x and y axes, the third line we will call z axis, we can't actually see, uh, not from the perspective of this. 2D image, no matter how much we move or rotate in 2D, we will never be able to see this third dimension, even if we made it to appear on the wall, it would not be the real third dimension.

One way to tell is that it is not perpendicular. to the other two dimensions of our projection we can see that the closed oven of the Rubik's cube of the light source becomes scale, it is represented as an interior exterior relationship, blue is in the center and is the smallest because it is the far side. Of the light we show in the cube, the green side we just hidden is the closest to the light, so it is the largest. If we take points and move them across the z axis, we could see the outline of the cube. axis and see what our projection does to represent this third dimension.

We can visualize the outline of the cube through a wireframe cube. These edges that appear diagonal are actually perpendicular to the x and y axes. You may think I'm pointing out the obvious here. but these things will help us understand Rubix cube 4d later. What happens if we try to rotate this Rubik's cube again? The side closest to the light source obstructing the view is always hidden. Okay, pretty normal, let's move the faces away from each other and then. Watch it again, trust me. This will make sense later. Watch how the faces push against each other.

The center face goes to one side and then disappears as you turn further at the other end. Another face comes into view and then takes center stage. In fact, let's go to the 4D Rubik's Cube. The 4D cube we are seeing here is the result of a projection made by the computer. Think of it as a shadow of a real 4D Rubik's Cube. It looks like a group of three by three by three small ones. cubes floating in space and that matches what our projected 3D Rubik's cube looked like, a group of three by three squares floating in space, also remember the big face we had to hide, the same thing happened here, one face had have to hide because otherwise you obstruct the view of the rest of the puzzle there are actually eight of these so-called faces one of them is always hidden this is what it would look like if you could see the eighth side now let's rotate the puzzle of course you can you can just do click, drag and rotate this way, but that doesn't show the full scope of the types of rotations possible.

What you can do is Ctrl-click any of these outer groups of cubes. What you see may look like what we saw previously with Rubik's 3D. projected cube in 2d putting a wireframe to outline the outline of this 4d cube and looking at these diagonal edges these are the lines that represent the fourth dimension or the fourth axis after having seen the 3d cube projected in 2d this should be a little more easy to understand and yes, these edges are technically perpendicular to the rest of these edges, but they won't look like that because of the projection.

I feel like every time someone talks about a 40 to 3D projection, they have to mention that you're technically looking at 4D. object projected in 3d and then projected in 2d so that it can be displayed on a computer screen first from these 3 by 3x3 cubes. I've been calling them faces or sides, but they're actually called cells. Each of these little cubes are 4d stickers. The Rubik's cube has 3D stickers and then we have the pieces like in the 3D cube. The pieces have stickers on them, but here the piece itself is invisible, so we only see the stickers so they appear to float in space.

There are four types of pieces of an additional type, then in the 3D Rubik's cube we have the center pieces of one color or a seed piece and they are located in the center of each cell. We have the two color or two cps face pieces, their placement resembles the center pieces in the 3D Cube, but make no mistake, these are all pieces that contain two stickers. I'll connect the stickers that belong to the same face piece and then we have the three-color border pieces into a 3D cube. A three-color piece would be a corner piece, but here.

It is a border piece, these stickers are placed similarly to the border pieces in the 3D cube, hence the name. This is how each of the border stickers come together to form border pieces and the new piece that you get in 4d, uh, the four color corner piece of these stickers. Occupy the location similar to a corner piece in the 3D cube and here are all the stickers connected once again for a twist in this puzzle, just click on any sticker other than the sticker of a center piece, a left click It's a counterclockwise turn and a right click.

Spin is a clockwise turn on the 3D Rubik's Cube. Each side rotates clockwise or counterclockwise. In this puzzle, each cell can rotate in many more ways because they are cubes. To better understand these turns, just think of a line passing through the label you just clicked on. and across the center sticker, that line is what the cube rotates about, it does the same thing, but when you do a spin on the cell, it doesn't just rotate the cell, it brings the entire layer with it because we're moving pieces, not just stickers. just look at a few moves of a move to deduce what move was just made, just look at where the stickers are and where they belong, you can see these white stickers and these screen stickers are swapped, these cyan stickers and the purple ones are swapped and the yellows and blues are swapped, well we have a move that does that, if you look at the rotation of an edge, that's exactly what it does.

This is the sticker we need to click on, this is where we want the surrounding stickers to rotate and because these stickers are swapped, it doesn't matter which direction we go, whether it's clockwise or counterclockwise. clockwise, let's look at another one here, we see that these green, cyan and white stickers are in the three cycles, the same as the blue purple and the yellow, and we have a move to fix that one corner rotation of a cube makes exactly that, so the thicker we click or the thicker the opposite side is in this puzzle, there are always two opposite stickers that make the same turn, this time the direction matters the white The stickers have to turn in the opposite direction to clockwise to return to the original location, so you need to rotate this sticker counterclockwise.

Well, the last one, you can probably guess what kind of twist is being seen, since we only have one that we haven't seen. and you can probably quickly know what move to make in this turn by clicking on this face sticker. In general, spins involving face pieces are the easiest to understand because they are like the typical rotations of a cube that you would see in a 3D Rubik's Cube. With so many possible terms we can create, we definitely need an updated system for writing these moves here. We need a name for each sticker that we can click to name each sticker.

First we will name each cell. I'll put them on the screen. of each sticker will be just the combination of these cell names, for example this sticker is in cell r, so its first letter will be r. It is also part of a two color piece where its other sticker is in cell u, so its name will be If the other sticker of the same piece will be called, when it reaches three and four color pieces, you will face the problem How to sort the letters, just know that the first letter has to be the cell that your specific sticker is in. in the other letters you can arrange them in any order you want, forFor example, this label f ​​is part of a three-color part, its other labels are in cells t and r, so you can call it ftr or frt if you just mean whole part in general um and not any particular decal, then you can Just put them, put the letters in the order you want.

Please note that all of these decals have connected decals that are part of the hidden cell, so whenever you do, whenever you check them out. make sure there is a k somewhere in its name so this sticker is rk and this sticker you can call it r u f k or um rkuf or something like that even as we move on to 4d and higher dimension puzzles yes there are 5d60 and 70 yes you are ready. The challenge is that we don't have to waste our 3D outputs. There are a few ways we can apply our 3D algs in this puzzle.

First of all, we can convert each move into k stickers um by k stickers, I mean rk lk fk uk dk stickers. and bk any sticker with a k as the second letter, there are six as I just mentioned and we can assign each of the six faces of the Rubik's cube to one of these stickers, so this conversion is like this when we make an r on the cube Rubik. It's like making a rk on the 40

rubik

's cube, let's try to do it soon using this conversion, it basically takes the three layers that go from inside to outside or outside to inside and treat the corresponding pieces as a unit, if that doesn't make sense, just You will take a look if we ignore the purple cell in the center and take out a 3D Rubik's cube.

You can see that it matches perfectly for this comparison. I deliberately placed the six standard Rubik's Cube colors in the outer cells and we can actually derive other ways of applying three outputs to For example, if we do a rotation and bring cell u to the location of cell t, notice what happens to those k stickers that we were clicking on, they become u stickers, which means that rk becomes r u fk becomes fu and uk becomes tu, so now if we do the same soon the algorithm uses again the Using stickers, the result is that now the three horizontal layers are treated as one unit like a 3D Rubik's Cube instead of the three outer to inner layers if we ignore the und cells and only look at the cells in the middle.

I can see that the results completely match a 3D Rubik's cube. Note that I have my yellow cell at cell location T, which makes the comparison possible, so we have the label k and use sticker conversions before continuing. I will show how these movements correspond. For spins performed on a 3D cube, another type of conversion I'll show you is called rkt. It's quite powerful and you wouldn't use it much as a beginner, but it would definitely come into play if you try something like cfop for

rubik

's cube this technique treats an entire cell like a 3d rubik's cube and is based on the constraint of rotating only one side of the rubik's cube plus the use of cube rotations, let's go to a 3d rubik's cube, probably everyone recognizes this case of oil, in fact, if you do the six moves outwards, the cube will skip the yellow color, but remember one restriction : we can only rotate one side and do only q rotations to access the other sides so our first move will be f we do a rotation of the main cube and then we do a rum the next move is an r so we rotate back and we do an r, then we have a u, so we do a z rotation and then another r, and so on.

This might seem useless in a 3D Rubik's cube, but in 4D it has huge benefits if we just have to do our spins we can dramatically reduce the unintended consequences of our algorithms, so let's go back to the 4d cube and for the sake of simplicity , we will choose cell t as our rubik's cube. Please note that while a cell normally refers to the 3x3 by 3 set of stickers in this specific case when I say sell I also mean the surrounding layer of stickers for our turn we can use rt or rk but we will use rk because it is easier click and that's where the name rkd comes from, the t refers to cell t so this time let's make a permanent t so an r is just an rk to make a u you just make a tf for the rotation and then another rk and then you rotate back and then r prime u prime r prime f r2 u prime r prime u prime r u r prime f prime when you master it you can eliminate unnecessary rotations and go directly to the face you want to rotate, but now I suggest that every time you do a rotation you undo that rotation before doing a different rotation that way you are less likely to miss now if we look at cell t you will see that it is exactly a t-perm pattern, using rkt you can reduce the damage of its soft parts only to the r layer, whether or not the r layer will align at the end of an algorithm depends on whether there are an equal number of clockwise and counterclockwise movements. clock and there is a way to predict it if I did an r and then u prime then the layer will align because it's one clockwise move and one counterclockwise move but if I add another r prime now you will misalign the layer again if I add an f, the layer aligns again, um, a double flip like r2 or f2. complicates things a little, you can consider two turns clockwise or two turns counterclockwise.

I will say that in general if there are more clockwise turns then consider the double counterclockwise turn if there are more counterclockwise turns and write it down as in clockwise if clockwise and counterclockwise turn out to be the same, then it doesn't matter which side you count it on, let's try an algorithm, for example, a permanent j , you have five clockwise turns and eight counterclockwise turns, there is a double turn that will be added to the clockwise turns. um, that makes seven clockwise turns versus eight counterclockwise turns, so we know that the r layer will be misaligned one turn counterclockwise.

A disadvantage of using rkt is that you really have to know the algorithm consciously if you normally rely only on muscle memory to run it. You may have to relearn those algorithms. You can also perform rkt from a different angle if we push the cell u towards the center. Look what happens to the old cell t and the label rk becomes a cell d in the label ru. do the same rkt as rud when using cell d as a 3d cube the orientation doesn't change this is still the top face and this is still the r face but the projection makes it hard to see how the ru label controls the r face let's try r u r prime u prime using this as this is something I really use in resolutions so to do an r now you do a r u to rotate the face u to the right you would do a df and then r prime u prime yes We move cell d back to the location of cell t.

You can see we just did rkt from a different angle for this last bit. I'll go over some features of the program on that 4d magic cube um. I'll go over how to set it up. custom colors in the puzzle and how to allow the computer to automate algorithms using macros to change the colors, it's pretty easy, but you'll have to do a little research to find the hex or rgb values ​​for the colors you want. I recommend a Wikipedia page called web colors. Go there. and find the 8 colors you want, then write down their hexadecimal values ​​or their RGB values.

Interestingly you can even get a mix of hex and rgb and this will still work in the same folder where your magic q4d is and create a text file called face colors. put your hex values ​​or your rgb values ​​in there if it's a hex value put a pound sign before it and have the eight numbers one after another if it's an rgb value have a comma between the r g and b values ​​and then uh and I have a space between the different rgb values ​​to get colors for certain cells, you will have to experiment.

What I do know is that opposite positions correspond to opposite cells, so the first color and color a will be opposite to the second and the second will be opposite and so on, if you want to use only the colors that I use, I will put the mine in the description. Okay, this is the last thing we're almost done to record a macro. All you have to do is click Start and Stop Macro. definition or control m, then choose three reference stickers. These stickers you click will help the program determine at what angle to apply the algorithm.

I recommend using the same reference stickers for all your macros. Once you choose your reference stickers, then do your algorithm along this. You can cancel the process at any time by pressing the Escape key and then when you are done with your sequence, simply click Start and Stop Macro Definition again and then you can name your macro to use these macros. All you have to do is click on the macro you want. Click on the reference stickers and then the program will perform the algorithm for you if you discover some interesting algorithms that you don't want to miss.

You can save these macros to a file if you want to apply a setting. move but I don't want to have to undo them you can also have the program do it for you just click start macro setup moves um and then you do setup moves and then you apply the macros then the program will undo your settings for you you choose whether you want to use macros or not, most of the time the algs I use are pretty manageable but if you have a really long switch then you might consider using this, hopefully this will give you a bit more information to understand how it works A Rubik's cube can be scaled to higher dimensions.

If you want an actual guide to solving this puzzle, I'll put some links in the description. I might also post a tutorial to solve it with this one later. Well, see you soon.