We humans live in a three-dimensional world. As a three-dimensional creature, there is nothing more terrifying than a four-dimensional creature. To us, they are like gods, and if they are a little evil, they can destroy and destroy us at will.
In the face of four-dimensional creatures, humans do not have any physical equipment and psychological equipment, so any higher dimensional creatures have an innate tactical advantage for us.
The comic book collection Tales of the Uncanny (published 1993) is relatively accurate in describing cross-dimensional warfare. In the story "It Comes From... High dimensional space! 》（It Came from... Higher Space!), a creature from four-dimensional space attacks a victim in three-dimensional space.
The monster in the comics is made up of many suspended, separated limbs. It flies through the air and makes all sorts of deformations.
Whenever this monster appeared, a wisp of smoke first appeared in mid-air, and then the smoke expanded like a balloon into a three-dimensional body. The author portrays it as a donut made of meat, which is really disgusting.
Imagine a three-dimensional creature attacking a two-dimensional creature, and we can explain why the body of a four-dimensional monster can be separated.
Being in three-dimensional space means that we can move in 3 different directions: left and right, front and back, up and down, while two-dimensional creatures can only move in two directions: they are confined to a plane.
Let's imagine a completely flat, two-dimensional alien creature, let's name it hypoflatical, and assume that the universe in which they live is an extremely thin plane. To us, their universe is like a piece of paper. We can quietly approach their universes from above or below.
Since they have no concept of the third dimension, they don't know where we are.
Three-dimensional space provides us with the perfect camouflage. It's time to launch our terrorist attack on two-dimensional creatures, and all we need to do is enter their two-dimensional universe from the third direction.
Now switch to the perspective of the Flat Star People. As our fingers penetrate their two-dimensional world, they see several suspended disks gradually getting larger and closer.
Then, when the palm enters the two-dimensional world, the discs merge. The Flat Star people can only see the two-dimensional section of our hands, which is like a meatloaf flying in the air.
(The hand next to the two-dimensional plane will not be found.) Only when the hand gradually passes through the two-dimensional world, the part where it intersects the two-dimensional world will be seen by the flat star people)
It's no use hiding: as three-dimensional creatures, we can see the entire two-dimensional world like a floor plan—and that's how the four-dimensional creatures in Alan Moore's story see our three-dimensional world.
Therefore, the Flat Star People could not hide behind something, nor could they lock themselves in a safe corner. For us, getting in and out of a closed two-dimensional world is as simple as going in and out of a two-dimensional square.
We can also see clearly the interior of two-dimensional beings: all the organs in their bodies are displayed in front of us, at our mercy.
This is exactly the threat that four-dimensional beings pose to three-dimensional creatures: they lurk next to the three-dimensional universe, quietly observing our every move and all the parts of our body; they can easily enter our bodies and kill us from within.
These monsters are our nightmare, but fortunately, there is no evidence that four-dimensional creatures exist, but it doesn't hurt to know more about four-dimensional space, just in case.
Suppose we are benevolent three-dimensional beings, with the goodwill of cross-dimensional communication, we want to show our three-dimensional world to the flat star people—to explain to them what the world outside of them is like, for example, to show them the three-dimensional cube.
If we move cubes into the two-dimensional world of flat stars, they will only see a variety of different cross-sections. The most boring way is to push into one side of the cube, so that they can only see a square suddenly appear and then suddenly disappear.
(What the Flat Star People see when a three-dimensional cube starts from one side and crosses a two-dimensional plane)
A slightly more interesting way is to first push in a prism so that they will see a rectangle grow from scratch, slowly getting larger, and then slowly shrinking to nothing.
But the most interesting way is to push in a vertex first, and the flat star will see a triangle grow from scratch, slowly getting larger, and then the triangle will change in shape and eventually shrink to nothing. This is nothing new for flat people who are confined to the plane.
(What the flat star people see when a three-dimensional cube starts from a vertex and crosses a two-dimensional plane)
And we'll also find that this approach is related to a mathematical problem we mentioned:
If we could keep track of all the areas covered by the record change section, it would be a perfect regular hexagon; not only that, but it would be exactly the cubic section map we encountered when solving prince Rupert's cube through the puzzle.
Now, let's think carefully about what happens when a four-dimensional hypercube travels through our three-dimensional world.
Suppose a friendly four-dimensional creature also shows us a four-dimensional hypercube with the goodwill of cross-dimensional communication. As the hypercube travels through our three-dimensional world, we see a series of three-dimensional cross-sections of it.
Observe the four-dimensional pattern of mood gradually irritable
The most boring way is still to start crossing from the surface, we can only see an ordinary three-dimensional cube suddenly appear, and then suddenly disappear.
If the hypercube starts with the edges traversing, things become much more interesting. We will see a triangular prism grow from scratch, slowly become larger, then deform into a hexaprism, then become a triangular prism in the opposite direction of the previous one, and finally slowly disappear.
The most interesting thing is still that the hypercube travels through our three-dimensional world from a vertex.
At this point, a tetrahedron appears out of thin air and grows evenly, then into a strange figure of hexagons and triangles, for a moment it becomes a regular octahedron, and then it changes backwards back to the previous shapes (but in the opposite direction) until it shrinks and disappears.
(How the three-dimensional section changes as a four-dimensional cube begins to pass through our world from the edges.) All the spaces it covers will form a hexagonal prism)
I think that's what makes it a tall image.
Let's explore again the various shapes that arise during the crossing. If you divide a four-dimensional cube into exactly the same two halves and look for the largest three-dimensional section, you get a regular octahedral section, which is the dual shape of the three-dimensional cube.
In the process of traveling through three-dimensional space, all the space covered by the four-dimensional cube forms a rhombic dodecahedron .
However, none of this tells us the full picture of the four-dimensional cube. It's not helpful to say more, let's make models of these shapes together.
(How the three-dimensional section changes as a four-dimensional cube begins to travel through our world from a vertex.) Interestingly: during the crossing, all the space covered by the four-dimensional cube forms a diamond-shaped dodecahedron)
Now we can build a four-dimensional cube with a straw. As with the Spatial Fill Vier-Phelan model, we still use colored straws, but here, different colors represent different directions.
Let's start with low dimensions and then keep increasing the number of dimensions. Making a one-dimensional shape is very easy: a straw is a one-dimensional shape. We use red for the only direction.
(one-dimensional line)
Making two-dimensional graphics also doesn't add much difficulty. As before, we use a curved straw brush to connect the ends of the red straw (for horizontal orientation) and the blue straw (for vertical orientation).
At this point we can move freely in both directions. In fact, for the flat star people, this is also a breeze.
We can understand two-dimensional squares this way: copy the edges of one dimension into two, and then connect them with two sides in the other direction.
(two-dimensional square)
Similarly, a three-dimensional cube can be understood as copying a two-dimensional square once and then connecting the vertices of the two squares through a new edge in the third direction.
Therefore, you only need to make another square according to the color scheme just now, and then use another color eyedropper (such as green) to connect the vertices of the two squares, and you can get a three-dimensional cube model.
Out of habit, you might make the model upright. However, in order to show the cube to the flat star, we can squash the cube on a flat surface.
(3D cube)
To make the cube four-dimensional, we copy a cube, and then connect the vertices of the two cubes with another color eyedropper (such as yellow), which represents the fourth direction. This model is similar to the cube we showed to the flat star people.
When we show the three-dimensional cube, the second square should have been lifted away from the two-dimensional plane to become three-dimensional, but we pressed it next to the first square.
Here, the second cube is supposed to be lifted away from our three-dimensional surface into the fourth dimension, but we let it land next to the first cube.
This model is a complete four-dimensional cube, only flattened to fit our poor three-dimensional world.
(4D Hypercube)
The flat version of the 3D cube has the same shape as the projection of the cube. If we use light to project a three-dimensional cube onto a plane, we can see its two-dimensional projection, and this is exactly the three-dimensional world that flat stars see.
A benevolent four-dimensional creature can project a four-dimensional figure in the same way, showing us its three-dimensional projection: the four-dimensional straw cube model you make is a three-dimensional projection of the four-dimensional cube.
But there's another way we can get projections: by introducing perspective. If you shrink the second square of a three-dimensional cube a little so that it hovers inside the first square, you get a flat projection of the cube, with no intersecting sides to each other.
Similarly, we can also make the second three-dimensional cube of the four-dimensional cube smaller, so that it is suspended inside the first cube, so as to obtain a projection of the four-dimensional cube in three-dimensional space, and the faces will not intersect with each other.
Does it feel like déjà vu? You've really seen a similar four-dimensional cube before this — a cube-like soap bubble. At that time, you had inadvertently made a three-dimensional projection of a four-dimensional cube with a soap film.
In addition to studying high-dimensional graphs through perspective, we can also study the projection of a rotating cube.
To obtain a two-dimensional projection of a rotating 3D cube, dyeing one of the opposing faces with color and leaving the other 4 adjacent faces blank can help us track the rotation of the cube.
(Add a little perspective to avoid self-intersection of the graph)
We can also use this rotating projection to show the three-dimensional cube to the flat star people. Although the cube rotates in three dimensions next to the two-dimensional universe of the flat star, the flat man is able to make judgments through the projection of the square.
Shadows are getting bigger and bigger, indicating that the cubes are moving closer to them; conversely, the projections are getting smaller and smaller, indicating that the cubes are moving away from them.
Unfortunately, in their view, the two squares are constantly passing through each other. None of the explanations allow the flat star to see that the two squares do not intersect in higher dimensions, but are in different positions before and after.
(Rotate the 2D projection of the 3D cube and rotate the 3D projection of the 4D cube, and the upper and lower two pictures rotate half a circle at the same time)
Similarly, we can project rotating four-dimensional cubes into a three-dimensional world. We still stain one of the opposite cubes, and the other cubes that act as a link are transparent.
As this four-dimensional cube rotates outside of our three-dimensional universe, each cube becomes larger as it gets closer to us and smaller as it gets farther away from us.
Unfortunately, like the Peace Star, in our view, the two cubes are constantly intersecting, but in fact, these two cubes belong to the relationship between the front and back in the four-dimensional space, but we can't see this.
When a two-dimensional face of a three-dimensional cube is just perpendicular to the two-dimensional plane on which the shadow is located, its projection disappears briefly.
(A four-dimensional Rubik's Cube without restoration)
Similarly, when a three-dimensional cube of a four-dimensional cube happens to be perpendicular to our three-dimensional universe, it also briefly disappears from our line of sight.
Incredible Fourth Dimension!
If you want to try to turn a four-dimensional cube yourself, I suggest you try to study rubik's cube.
A 3D Cube requires moving a two-dimensional patch of the same color to the same two-dimensional face of a three-dimensional cube, but the four-dimensional cube is a little different, and you need to move a three-dimensional patch of the same color to the same three-dimensional face of a four-dimensional cube.
You can't transform a four-dimensional cube directly, but you can do it indirectly by dragging and dropping the three-dimensional projection of a four-dimensional cube, which can be found online.
Your mind will be particularly dizzy when playing, because the 3D projection can move in the three directions we are familiar with, but the 4D cube can also rotate in the direction of the fourth orthogonal (that is, the intersection angle is at a right angle) (on the website by holding down the shift key).
It's a little bit worse, though, because you can only see the three-dimensional projection of the four-dimensional cube on the computer screen, so you're actually dealing with the two-dimensional projection of the three-dimensional projection of the four-dimensional cube, and I can only wish you good luck.
<h1 class="pgc-h-arrow-right" > recommended reading:
Mathematics has long been seen as synonymous with education, and it should have been synonymous with fun and exploration. As we solve the puzzles one by one and put the math games into practice, we can quickly appreciate the beauty of mathematics that most people have never seen before. </h1>
We will be able to practice things that are beyond the intuition of ordinary people. In this game, mathematics shows us graphics and objects beyond infinity that exist only in 196,883 dimensional space. From the fourth dimension to the transcendental number, we will gradually understand.
This article is excerpted from the Houlang book "What Can We Do in Four-Dimensional Space", and [Meet Mathematics] has been authorized.