4D World (IV): Vision of 2D Beings

Many literary and cinematic works have mentioned fictional two-dimensional flat worlds and their inhabitants, because people are very interested in various effects brought by dimensional changes, such as physical laws, biological and mechanical structural design, etc. Today we’ll look at something people often overlook: the vision of two-dimensional beings. We won’t stop at the descriptive level, but will truly experience the first-person perspective of flat beings with the help of computers. Of course, we care about the experience of 2D people to do a pre-excercise for understanding the vision of four-dimensional beings.

Eyes of 2D Beings

First, let’s assume a two-dimensional world (a.k.a Planiverse). This is a circular planet inhabited by 2D people. Their eyes, like ours, see objects by sensing light. We assume that the 2D world also has electromagnetic waves (this can exist! Only the electric field is a vector, while the magnetic field is a scalar), and we’ll only consider geometric optics. Where does light come from? Of course, it’s emitted by the star - the sun in this world and other artificial light sources. Note that all light rays can only propagate within the flat world. So our most common smiley face emoji is not a 2D being—because the smiley face’s eyes are inside its brain and can’t receive any light! Its mouth is also inside its head and can’t eat anything! So the eyes of 2D beings should be on the surface of their heads. Their eyes’ photosensitive layer (a.k.a retina) is a one-dimensional curved segment, so what they see is one-dimensional.

Dimensional Reduction Challenge

For example, above is a simple scene: a square placed on the ground. In the eyes of a 2D person, it looks like the left image below:

This view feels like looking at the entire extruded scene from the side. Why does it feel this way? Because 2D people living in the $xy$ plane have absolutely no sense of the z-axis direction. Their field of view is an absolute one-dimensional straight line. Here I must clarify one important point: two-dimensional beings have no perception of dimensions on the z-axis, they don’t feel their one-dimensional vision is “narrow”, and they never care about or can never imagine existence outside their one-dimensional view. Although we think they’re pitiful living on a plane without thickness, they’re like children in poor mountainous areas without electricity or internet—they can still live happily without knowing the outside world (but their worldview completely changes after seeing the outside world, refer to a Chinese TV show “X-Change”). So the different widths of the two color bands on the left side of the image above make no difference. I drew them with width just for convenience, because if I really drew them as infinitely thin ideal lines, they would be so thin that we (as 3D beings) couldn’t see them. You can imagine that if 4D people tried to imagine what we see, they would be amazed that our field of view is an infinitely thin ideal plane that can’t even be seen, yet we actually feel fine!

Now let’s try to challenge some slightly more complex 2d scenes.

Did you figure out the answer? I believe this shouldn’t be too difficult.

Three

Two

One

Here’s the answer: Click here

Stereo Vision

Why do humans have two eyes? Because the two eyes are in slightly different positions, the visual disparity can be calculated by the brain to determine how far objects are from us, creating a sense of depth. Actually, two-dimensional beings also have two eyes, and similar to our stereo vision, they can also perceive the distance of objects. Let’s try looking at their left and right eye images: if you can cross your eyes, you should be able to easily overlap the left and right images below. If you can’t, no worries. Put your finger in the center of the screen, stare at your fingertip, then gradually move your hand toward your eyes. During this process, make sure to keep your left and right lines of sight horizontal. Using your peripheral vision, you’ll notice the content on the screen behind starts to shift and becomes a bit blurry. When the two images on the screen just overlap, you’ll be amazed to find that this overlapped image floats up! This floating position is roughly the same as your current fingertip position. Then you can see the stereoscopic image.

Do Three-Dimensional People Have the Ability to See Through Interiors?

But now I’m going to tell you that the scene you just saw might actually be like this:

Or like this:

Both are possible! Because 2D beings can’t see inside, who knows what’s hidden inside!
Actually, these images all have some issues: light can only illuminate outter surfaces, so the inside of objects should always be black, but the materials inside are definite. If you cut open the object, light can get in, so you can think of our coloring method as coloring by material, not what can actually be seen in the 2D world. Note that air in the 2D world is transparent, but we colored air black according to “material.” The black that 2D beings actually see is the black background at infinity, not the color of air. Actually, we in the 3D world can’t see the 2D world at all, because light in that world cannot reach the third dimension to be observed by us. So the 2D worlds we draw are just schematic diagrams. Since they’re schematic diagrams, the drawing methods can vary greatly. For example, the scene just now could also be drawn like this: (The checkerboard pattern represents the transparent color of air)

Dimensional Elevation Attack?

Now we tell the two-dimensional beings that there’s actually a 3D world. But note that all electromagnetic waves in the two-dimensional world can only propagate on that plane, which includes all intermolecular forces, so we assume that worlds of different dimensions cannot physically interact directly, but can send information to each other through certain means. How can we let them see our three-dimensional world? The simplest method is to send photos to the 2D people. Since 2D people’s vision is one-dimensional, their displays are also made one-dimensional. Even if 2D engineers develop two-dimensional display screens, or directly print photos with a 2D printer, they can’t see the “interior” of the photos.

The 2D person walks around the photo and finds it’s just a square with many colored patterns on each side: as for the graphics in the photo, they have no chance to see them.

What can we do about this? Only being able to see the edges of photos is too frustrating. 2D people can tear the photos apart bit by bit to gradually see inside, like how we peel onions, but this method is too troublesome. 2D engineers are clever—if they make the photo material semi-transparent, won’t they be able to see inside?

Transparent Photo Method

Let’s pretend to be 2D people and try to look at semi-transparent 2D photos from their perspective. Let’s start with a simple geometric photo: a blue diamond drawn on a white canvas.

Let’s try to look at this transparent photo from the side:

I’ve drawn the left and right eye images, please view them using the method taught earlier. 2D people also have two eyes to determine the distance of objects, which is calculated by the brain through the visual disparity of the two eyes, just like us. Using stereo vision makes it slightly better. But actually, 2D beings can only see a line like this:

Even with stereo vision, they can’t see clearly, can they? Now the interior of this diamond is filled with uniform pure blue, with no light variation, and overlaid with the surrounding white background, it appears even more blurry. 2D beings have difficulty knowing its shape, but they at least know there’s a blue area in the center of the photo, surrounded by white.

And our Lena photo becomes like this:

Cutting line Scanning Method

Although this method is theoretically feasible, actual photos contain too much information, and 2D beings cannot distinguish these stacked pixels well. There’s another method: since compressing 2D directly to 1D results in too much information causing unclear vision, we might as well just look at images on some cutting lines, indirectly reading all the content on the screen by scanning the entire image with cross-sections, a bit like the CT scanning process. Although this method can see each pixel clearly, the positional relationships of these pixels are far less direct and clear than the first method. Is there a method that combines the advantages of both? I’m afraid it’s hard to find.

Now let’s try to operate on 2D people’s computers from their perspective, viewing some 3D scenes in first person. Let’s first return to a normal 2D screen to look at this 3D scene, then look at it again on 2D people’s 1D screen. Best viewed in Chrome. If you’re using a phone, skip ahead and look at the images below…
Key Instructions:

• 3D Control: Keyboard WASD for character movement up/down/left/right, move mouse left/right or press keyboard JL to look at different horizontal directions; scroll mouse wheel or press keyboard IK to look at different vertical directions (look up/down). Not using mouse movement in all directions to control orientation is also to get closer to the 2D person’s experience: they can only move their mouse on the one-dimensional ground. (Using the scroll wheel is also to adapt to our future operations in 4D space)
• View Control: Left/Right arrow keys / Keyboard ZX to rotate 2D view

We make two cuts horizontally and vertically through the center of our field of view, and draw these intercepted pixels separately for 2D people to see. Note that these cross-section color bands are also made in pairs for left and right eyes. Our ordinary photos can only show up/down/left/right, with front/back directional information already lost. But to better help 2D beings understand the 3D world, we use the kind of dual cameras with slightly different positions used for shooting 3D movies, selecting corresponding cross-sections on the two images respectively, thus allowing 2D people to have front/back stereo vision.
I’m sure we’re very familiar with 3D control operations, but note that real 2D people can only see the images in the top three rows of color bands. Only by constantly rotating the 2D image can they roughly know the content in the 2D view… I estimate 2D beings would mainly rely on the first two rows of cross-sections.

What Kind of Intuition Do 2D People Have When Seeing These Color Bands?

We can directly see 2D and have intuition for imagining 3D space; 2D beings can directly see 1D and have intuition for imagining 2D space. Since they can sense one more dimension through left/right eye stereo vision, it’s necessary to mention which dimensions each part of the current image has. First, 2D people only know up/down/front/back, with no concept of left/right (so strictly speaking, their two eyes should be called upper eye and lower eye). I’ve marked the directions they perceive in the image below:

Actually, this image is cut along the bright purple vertical line in the photo below. Through stereo vision, 2D people can know the sky and green ground, but what puzzles them is a gray circle floating in mid-air. Below, besides the shadow of the gray circle, there’s also a shadow of an object they can’t find in their field of view. This is impossible in the 2D world, but of course we know it’s just a cross-section of the 3D world, so it’s not strange: the circle can float because it’s part of a gray arch, supported from another dimension that 2D people can’t see; the extra shadow is from a cube that doesn’t intersect with the vertical cross-section at all, but its shadow is cast on the cross-section.

2D beings who have seen the cross-section color band of that horizontal black line will feel like they’re floating in “space” because there’s no vertical direction on this color band. As for why I say it’s like “space,” you’ll also be able to experience it when you go to 4D.

Finally, let’s annotate the directions on the semi-transparent photo: note that the directions here are not absolute, they can rotate the photo to interchange the positions of up/down/left/right.

Although the semi-transparent photo observation method is the most direct, natural, simple and crude, it’s too blurry. I haven’t found any intuition for it.
There are some interesting phenomena worth noting:

• Although we show 2D beings cross-sections of photos (projecting 3D scenes to 2D), 2D people feel it’s more like looking at 1D photos of 2D cross-sections of 3D scenes (2D cross-section scenes projected to 1D);
• When looking around left and right, 2D people feel like they’re looking around the 2D cross-section of the 3D scene shown in the first row of color bands (horizontal cross-section), while the 2D cross-section shown in the second row of color bands (vertical cross-section) constantly changes;
• When looking up and down, 2D people feel like they’re looking up and down at the 2D cross-section of the 3D scene shown in the second row of color bands (vertical cross-section), while the 2D cross-section shown in the first row of color bands (horizontal cross-section) constantly changes;

Will We Also Be confused by Dimensional Elevation?

Well, the dimensional reduction from 3D to 2D experience tour comes to an end for now. But suddenly one day, a being claiming to be a four-dimensional person wants to show humanity three-dimensional photos transmitted from the four-dimensional world! Our experience will be extremely similar to that of two-dimensional beings! (To be continued… The fish that dries up the sea is not here)