Four-dimensional aliens.
I had a very interesting exchange of ideas with some UFO enthusiasts in the Fediverse, which I would like to represent here because it was so beautiful that it seems absurd to me to lose it when I delete the old posts. The question was: but if aliens were beings with 4 spatial dimensions, what would happen?
Be careful, I didn't say "four dimensions plus time", as the facetious old man who invented relativity said. I mean four spatial dimensions. In mathematics there is no problem in extending the number of spatial dimensions. What we call “Euclidean Geometry” as we know it is a 3-dimensional Hilbert Space.
And Hilbert spaces, in gometry, which for mathematicians is algebra, can be extended to any number of dimensions. Obviously, then you have to resign yourself to the rules of the place, in the sense that if we take the dimensions to infinity, very funny things happen, but even in four-dimensional spaces things aren't easy right away.
The question was, therefore, whether we could see (regardless of the neurological question), in the most perceptive sense of the term, a 4-dimensional alien. (In MIB there was one at 5, but I assure you that four are more than enough to give unpleasant surprises, as we will see).
So, now let's try to solve it in a simple way, first we define what we need to see in a geometric sense, then we train in the spaces we know (up to three dimensions), and when we have warmed up our brain and understood how to proceed, we go to four dimensions.
My personal challenge is that it shouldn't take more than high school to understand.
What do we need to “see” someone using sight? Let's say that light proceeds in a straight line, and therefore we need a straight line that goes from every point of the "thing" (or at least of its surface, but we will see the difference later, and it is not trivial), but let's say that we need a straight line that intersects both a point of the eye and the geometric object.
For convenience, let's say that the eye is a "focal point", so we reduce it to a single point, and that light is always a straight line without curves, deviations, refractions, reflections, etc.
It is a beautiful sunny day and something appears in the air in front of us. Do we see it?
“Seeing” requires that this line intersects both the focal point and the object, in one or more points.
So it's a problem of intersections.
What does geometry (i.e. algebra) say about intersections and dimensions? Is there something that connects them? Of course there is. It seems complicated, but don't run away, we'll make it simple.
https://it.wikipedia.org/wiki/Teorema_delle_intersezioni_dimensionali
What does he say? It says that if we are in an n-dimensional world, two objects have dimensions p and q, their intersection I has dimension I = p+qn.
As it is, you will be tempted to escape, so now we warm up a little, train our brains and when we have managed to understand the thing, we move on to 4 dimensions.
Let's start training.
We take it for granted that we know that:
- zero dimensions are a point
- one dimension means a straight line (a straight line)
- two dimensional means a plane (a non-curved surface)
- three dimensions means a volume (a cube)
These are convenient assumptions, but they are enough for us to understand.
Let's start with a simple example that we all understand.
Dimension 2
Two lines on the plane always meet at one point.
Then we have:
- n = 2 (the space we are in has dimension 2)
- p = 1 (the first line has dimension 1)
- q = 1 (the second line has dimension 1)
I = p + q -n = 0
Therefore, the two lines meet in a space that has zero dimension. A point. But we already know that. Ultimately, if we are on a plane and change angle we can always see all the points of a straight line. Well.
The first line, p, is the ray of light we use to "see".
Let's move on to the examples in dimension 3.
Dimension 3
Can we see a straight line in a 3D space?
- n = 3 (we are in a 3d space)
- p = 1 (our light ray)
- q = 1 (the line we want to see)
I = p +q – n, which is 1.
Not only can we see it, but it also appears that the two lines intersect in a straight line. What does it mean? Geometrically this "excess" means that we can define a single plane using two intersecting lines, let's take this "surplus" of dimensions like this. At the moment.
Can we see a surface in 3D space?
- n = 3 (we are in a 3d space)
- p = 1 (always a ray of light going in a straight line)
- q = 2 (our plan)
I = p + q – n = 0
So yes, we can see every point of our plan.
Can we see a volume in 3D space?
Here, pay attention to how it ends, it's crucial.
- n = 3 (we are in a 3d space)
- p = 1 (always a ray of light going in a straight line)
- q = 3 (our cube)
So let's apply the theorem:
I = p + q -n = -1
Dirty slut. We can't see the cube. And you will say "what the fuck are you talking about, I can see cubes very well, and so can cubists!". Real. But it's not true. That “-1” means that you have to remove a dimension from n if you really want to SEE the cube, or, said differently, it means that you only see n -1 dimensions of the cube, that is, a surface.
Simply put, you don't see the cube: you see its surface facing you. Which has two dimensions. And lo and behold, n + I = 2.
When this calculation gives us a negative number, then we have to remove a dimension to get the intersection. Our light ray does not hit the cube, but its surface. For this reason, we CANNOT see all the faces of the cube at the same time, much less its volume.
We can imagine it, think it, but not see it.
In fact, if you could "see" the whole cube, with I having zero dimension, you would see the cube like this:
The cubist would be so bizarre that I won't tell you. But this is also fantastic, because it means that a “3:2 map” is possible, that is, it is possible to represent three-dimensional things on a plane in two dimensions. Painters, photographers, etc. do it. what they do is called “projective space”.
https://it.wikipedia.org/wiki/Spazio_proiettivo
We are ready to move to four dimensions.
Dimension 4
Imagine waking up tomorrow in a four-dimensional world. Open your eyes, e?
Can we see a straight line, in a 4D world?
- n = 4 (we woke up in a bad place)
- p = 1 (our eyes always use light rays)
- q = 1 (there is a line in front of us)
I = p + q – n = – 2
Minus two is a mess, because it means that we can only see the line if it is on the same two-dimensional plane as our eye. Imagine having a sheet of paper in our 3D world, and saying that from a point on the sheet of paper you can reach the straight line only if your eye is a point on the same sheet. From the outside, i.e. looking at the sheet, the line does not appear.
Here's what would happen in a 4D world if you tried to see a straight line that isn't on the same uncurved surface as your eye. Strange things start to happen.
Or if you prefer, in a 4D world two lines do not always meet. They meet in very particular conditions.
Can we see a plane, in a 4d world?
- n = 4 (we woke up in a bad place)
- p = 1 (our eyes always use light rays)
- q = 2 (there is a flat surface in front of us)
I = p + q – n = – 1
We can't always see it. We can only see it if we are inside a three-dimensional cube together with the plane. Just as in the 3D world we can exist but we are not always on the same plane as the rest (height also exists), in the 4D world we can only see something if we are in the same volume.
That is: not always. And just as in 3D we only see the surface of the cube, in 4D we would only see one side of the plane. We would see the plan as a line. Always at the side, in short. But this line would not be what we encounter in 3D. We see it as a line, but it is a surface, that is, a line with an area, and not just a length.
More and more bizarre.
Can we see a cube, in a 4d world?
- n = 4 (we woke up in a bad place)
- p = 1 (our eyes always use light rays)
- q = 3 (there is a cube in front of us)
I = p + q – n = 0
Oh, great. Apparently we can see the cube. WITHOUT decreasing the size. That is, we see all the faces at the same time. As in the image above. We don't have to downsize anything. The intersection with our ray of light is a point, anywhere on the surface.
Everything that has three dimensions appears “flayed” and open to us.
Can we see a hyper-cube, in a 4d world?
Can we see the alien, if we are also in his 4D world , using our usual ocular apparatus?
- n = 4 (we woke up in a bad place)
- p = 1 (our eyes always use light rays)
- q = 4 (there is a hyper-cube in front of us)
I = p + q – n = -1
No we can not. As in the case of the 3D cube when we are in 3D, we can only see the surface facing us. In the case of 3D, we can only see the volumes facing us, which intersect the straight line we use to see.
Our brain cannot imagine 4D worlds, so let's not go further. But this tells us one thing: a 4-dimensional sculptor could create, in theory, a 3-dimensional sculpture that represents a 4D object, just as a human painter represents a three-dimensional object on a canvas. There is therefore, in the 4d world, a "4:3 map", which does not exist here.
But let's be clear: only in HIS 4D world. Not from us.
Having clarified how we think, let's get to the real problem . We are in our 3D space, and suddenly someone comes to visit us. As if we were going to visit a flat universe, as if on a sheet of paper. Okay.
So we would be in our 3d space, the object would have 4 spatial dimensions, and we look at it. What happen?
- n = 3 (we woke up in our usual world)
- p = 1 (our eyes always use light rays)
- q = 4 (there is a hyper-cube in front of us)
I = p + q – n = -2
So in order to see it we would have to be on the same straight line. I'm not saying it's two little drawings on a sheet of paper. I'm talking about being two points on a line. That is, we would only see the lines of that object that are not parallel to the ray of light.
That is, our ray of light ONLY intercepts the edges of the hypercube, the lines where the surfaces meet.
Just as we only see the surfaces of a cube facing us, in this case we would only see the edges of the hypercube facing us. The edge in 3D geometry is the line where two faces of the cube meet).
But we're doing it wrong. We are taking it for granted that if two sides of the cube intersect in a straight line, the same happens in the 4d world. How are the edges of a hypercube made? How do two planes intersect in the 4d world?
- n = 4 (we are talking about the 4d world)
- p = 2 (one face/surface of the hypercube)
- q = 2 (one face/surface of the hypercube)
I = p + q – n = 0
In that world, two surfaces intersect at one point.
So, when the 4D alien comes to us, we only see dots, which are the edges of the hypercube facing us. They are the only things that intersect our light rays. For them these "points" have a length as a property, almost as if they were straight lines, but for us they are dots.
What does it mean to see many dots? We do this when we observe an almost transparent vapor or a rarefied but colored gas. Since we are in a 3D world, if the light maintains its properties, such as scattering and more, we could see a diaphanous image in the air. But maybe.
Everything OK? A diaphanous cloud?
No. This object would have strange capabilities.
First of all, it could rotate around an axis that does not exist for us. In this case we would see it disappear into thin air. But it would have another incredible ability.
If you put yourself on a line you are a point. You may “see” another dot, as long as there are no dots in between. If there is a point between the two, the light from the first point stops at the middle point. Therefore we cannot see the third point on the line.
If we now paint this identical situation on a plane, and look from a point outside the line, the topological limit no longer exists: a point outside the line can see all the points on the line.
Now let's take a plan. We draw a square, and inside the square we put a point. If you are a point outside the square, but still on that plane, you cannot look inside the closed square. The light will stop on the perimeter of the square.
But if we are three-dimensional beings, we can see everything on a 2d plane. Just don't be on that floor.
Now let's imagine being locked inside a room with no exits. No other 3D being can see us. The light stops on the walls of the room.
But from a point in 4d, as long as you are not inside the room , the alien does not have this limit. He sees everything. For him, the closed 3D room, which holds us prisoners, is open, like we look at a closed square on a sheet of paper.
Therefore, we would see a diaphanous object, a sort of transparent but visible cloud of gas (like a cloud or cigarette smoke), capable of entering and exiting closed volumes, and capable of suddenly disappearing.
If one day aliens living in a 4D geometry decide to visit us, those who meet them better not believe in ghosts.
LOL.