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The issue is with the "sheet of paper" concept. Spoilers ahead.

In the story, a two-dimensional plane is shot into the solar system and as it interacts with three-dimensional matter, it converts the three-dimensional matter into two dimensions. The author describes that the shapes of the three dimensional objects are "unfolded" onto the two-dimensional plane - that no surface is hidden. The author also indicates that the shape of objects, such as people remain recognizable.

Three sections from the book:

It was possible to distinguish every individual on Revelation: They were laid out side by side, holding hands, every single cell in their body exposed to space in two dimensions.

Both gas giants had been two-dimensionalized. Uranus’s orbit was outside Saturn’s, but since Uranus was currently on the other side of the Sun, Saturn had fallen into the two-dimensional plane first. The giant planets ought to look like circles after collapsing, but due to the angle of view from Pluto, they appeared as ovals.

The precision of the drawing was at the level of the individual atom. Every atom in the original three-dimensional space was projected onto its corresponding place in two-dimensional space according to ironclad laws. The basic principles governing this drawing were that there could be no overlap and no hidden parts, and every single detail had to be laid out on the plane.

The problem is that if the unfolding occurs down to the atomic scale and the plane is truly two dimensions (which it is described as having zero/non distinguishable height), then the individual atoms have to unfold. An atom unfolding its internal shapes would be massive. The author has already established the behavior in the first book, The Three Body Problem, when the Trisolarans unfolded a proton which filled their sky:

The second try unfolds the proton into three dimensions, and large, reflective geometric solids fill the sky, “as though a giant child had emptied a box of building blocks in the firmament,”

Due to this, I believe if an object were to unfold at the atomic level - first it would not be recognizable on the two-dimensional plane and secondly would take up a massive portion on the plane. I don't have the math, but my guess is that an object such as a spaceship would occupy a two-dimensional plane much larger than the solar system.

We can do a simple thought experiment:

Let's say we have a three-dimensional cube that is made up of two-dimensional sides (the sides have no thickness, only length and width). Unfolding the cube to two dimensions, we see six two-dimensional sides. But, we also have to account for the other sides of the two-dimensional sides, meaning there are now twelve "sides" on our two-dimensional plane. For complex objects, you can see how this would get significantly more complicated and the shapes' areas on the place would get exponentially larger. Getting down to the atomic level (because atoms are three dimensional) and the scale would be unfathomable.

And I believe the same would happen in reverse. If a 3D object was to go into 4D space, I believe the scale effect would occur in the opposite direction. An object would become infinitely smaller - but when the spaceship (I forgot the name) entered 4D space, for all intents and purposes, it remained the same/unchanged.

All that said, am I correct in this hypothesis? And at the very least, is there an issue with Cixin Liu's world building between the size of the Sophons and the effect on converting three-dimensional elements to two dimensions?

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    This sounds like a question about real-world science, which would be off-topic. Even if it were not, it would have a disappointing answer: no true two-dimensional objects are known to exist in the real world, and as such no method of turning higher-dimensional objects into two-dimensional objects is known (nor is likely to exist). An author can come up with whatever ideas they like, without one being more or less incorrect than another.
    – Adamant
    Sep 22, 2021 at 2:45
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    Do also note that mathematically, there is no unique way of cutting and deforming a closed two-dimensional manifold embedded in 3D space into a plane shape, nor is there any unique projection of most 3D objects into a plane
    – Adamant
    Sep 22, 2021 at 2:57
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    @Adamant It's not real-world science, but it is mathematics. However, Jason is primarily asking if the story is internally consistent, not whether the author's explanations clash with real-world maths (or physics).
    – PM 2Ring
    Sep 22, 2021 at 5:37
  • Let's take a simple case. Is it possible to map every point on a sphere onto a circle and retain all of the relationships of its points to one another? Sep 22, 2021 at 15:56

2 Answers 2

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Generally the perception of being able to see inside of a 3D object is how things work from a 4th dimensional perspective. In a sentence every point in a volume is a surface to a 4D light-ray.

To render a 3D object in two dimensions would be to slice it and tile the slices, which is inconsistent with the notion of "The author also indicates that the shape of objects, such as people remain recognizable." Like this anatomy video doesn't seem recognizable to me:

But the tiling effect would definitely occupy as much area as wanted. And an atomic scale slice would be a map roughly the size of the universe. Which is consistent with "when the Trisolarans unfolded a proton which filled their sky:"

Lastly with your cube analogy I suppose if unfolding just created a surface with a view of each of the 6 space perspectives that would produce a multiangle view in 2D - but not infinite or internal like a 4D perspective would be.

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Yes. It is not possible to map a three-dimensional object onto two dimensions without distortion.

To take a test case, what happens when a sphere is mapped onto a plane? It is possible to do this in such a way that every point of the sphere corresponds to one on the plane. However, it is not possible to do so without distortion. For instance, you can't take the peel of an orange and flatten it into a circle. To "unfold" the surface of the sphere requires tearing it, and the problem only intensifies when you consider the sphere's interior. In Liu Cixin's chilling scenario, the universe is flattened in such a way as to leave objects recognizable (which leaves the interesting possibility of reversing the process), but in reality the distortion would be so great as to damage the two-dimensionalized objects severely.

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    It's made clear that the process is not reversible. The visually flattened object is merely an after-image of something that has been fundamentally downshifted a dimension
    – Valorum
    Sep 22, 2021 at 16:30
  • @Valorum that's an interesting point. That the observers are viewing two dimensional space from three dimensional space, and that any "image" may not be representational of what's occuring in two dimensional space
    – Jason
    Sep 22, 2021 at 21:53

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