# Why does 4D navigation work like this?

At one point in Cixin Liu's novel "Death's End", humans enter a bubble of four-dimensional space.

The author explains that, as their equipment is only designed for three dimensions, they are always one coordinate short and therefore cannot accurately measure any distances. Later on, they carefully approach a four-dimensional object, and it is said to first appear as a small dot for a long time, just to suddenly pop up right in front of the human expedition.

Why is that? If I imagine things one dimension below, a two-dimensional being might not see a three-dimensional object at all, but if it is visible (because it intersects with the 2D being's plane of sight), it certainly will not suddenly be much closer to the viewer than an instant before.

Does this make sense within the universe of that book, either there or later in the novel?

EDIT: To be clear, once they have entered four-dimensional space, the protagonists do not see just a three-dimensional slice, they appear to perceive all four dimensions for some reason, as is described, for example, in this passage:

One could see the interior of solids, such as the bulkheads or a piece of metal or a rock - one could see all the cross sections at once!

• I don't know the book, but you may get a similar effect in nautical (mostly 2D) navigation as well: What looks like a small object on your 2D plane of sight (the ocean surface) can actually be much bigger in 3D and may "suddenly pop up" if either you submerge below surface (intentionally or not) or the object emerges (on its own or because of the tides). Could that be a reasonable explanation for your question? – Annatar Feb 12 at 12:45
• There's an actual video example of this. If you seach online, you can find videos of icebergs suddenly rolling over, and what looks like a berg a fair distance away suddenly appears much closer as submerged sections rotate up into the field of view closer to the viewer. – Keith Morrison Feb 12 at 17:31
• @Annatar: I don't think that's what's happening, as the preceding pages in the book make it abundantly clear the humans' vision somehow happens in four dimensions - for any given three-dimensional object, they can see the entirety of the surface and all of its contents, outside and inside. – O. R. Mapper Feb 12 at 17:38
• I don't think human beings could even survive in 4-dimensional space. Things like the heat equation and the wave equation, not to mention the quantum mechanical equations they're based in, would have entirely different solutions, and so biological systems that evolved to work in 3D would experience serious problems as soon as their components weren't all in a single 3D slice of the 4D space. Or possibly sooner. – Adamant Feb 14 at 4:53
• @Adamant: I agree, and I also found it curious while reading that the author makes a big deal about how the 3D spaceship does not appear as a sealed body in 4D any more and they directly see space withuot having to wear a spacesuit plus that the protagonists can evidently access the interior of objects that are sealed in 3D, by moving toward a direction that does not exist in 3D. Thus, evidently, they leave their 3D slice of the 4D space, but the atmosphere in the ship does not come with them. – O. R. Mapper Feb 14 at 11:34

## 3 Answers

It does make sense, if one takes the analogy you started with, and run a little further with it.

Imagine a circular being living on an infinite two dimensional plane, lets call him "CircleMan". From your "3D-perspective", you can look down on CircleMan, and see both his surroundings, and his internal organs, where as, to any other Circle-beings, he looks perfectly normal, sense organs on the outside, everything else inside his circle of skin.

You decide to push a part of yourself into his dimension, and slowly push your fingers through his plane of existence. From CircleMan's perspective, he sees four tiny circles appear in front of him, out of no-where, which rapidly expand into four large circles, in front, to his left, and right, all separate, but somehow, moving and acting together as one.

As you continue to push your fingers, then hand and then finally wrist through his plane, CircleMan will see the four circles joined by a fifth circle, briefly (your thumb), and then suddenly all the circles will suddenly merge into a single giant oval (palm / wrist), massively increasing in size without warning.

In the same way, 3D-beings, in a 4-dimensional space will only be able to perceive a "slice" of reality. That "slice" can and will change drastically, as the actual 4D object moves and rotates in 4D space, and the 3D beings move in their planes of existence, drastically changing its 3D cross-sectional appearance, as perceived by the 3D beings.

The humans of your story are only seeing 3D slices of the true 4D object, from a certain point of view, surfaces and interior all mish-mashed together and rather incomprehensible. Its rather like looking at an x-ray or MRI scan of a human being (a 2D slice of human, if you will), and trying to figure out their hair and skin color. It just can't be done, as too much (3D) information has been lost if you can only see a slice of the original thing.

As they approach the object, and as the object itself may be moving or rotating, and indeed, the entire 3D-space of the humans may be moving or rotating within the bubbles 4D-space, their apparent view of the object can and will change drastically, moment by moment, in ways that don't make much sense from a 3D-perspective, but would make perfect sense if they could see "outside" the limits of their 3D-senses.

• Good point about seeing interiors still being a 3D view of a 4D object. – DavidW Feb 12 at 21:03
• I'm not convinced they are seeing just a 3D slice (even though that is what I would actually expect to happen). Like I quoted from the novel, "one could see all the cross sections at once" seems to imply precisely that they do not see just a slice, but the full 4D view. – O. R. Mapper Feb 12 at 21:23
• Cross-section: noun 1. a surface or shape exposed by making a straight cut through something. It literally means they are seeing slices of the object, 3D views of the 4D object. Many of them, sure, but still only slices of the true 4D object. – Tristan Chappell Feb 12 at 21:27
• how seeing the inside of things works. Take a cube. Every single point in that cube has a coordinate between 0,0,0 to let's say 1,1,1. To access a point in the center at 0.5, 0.5, 0.5 in 3D we would need to pass through some other point of the cube to get to it - thus we do not see the inside of solid things just the outside. however to an observer at a 4D vantage point - their coordinate at 0,0,0,1 does not have to pass through any point of the cube to access the interior points. A sight line from 0,0,0,1 to 0.5, 0.5, 0.5, 0 will only intersect at exactly the center point – lucasbachmann Feb 13 at 2:56
• @TristanChappell: "Many of them, sure, but still only slices of the true 4D object." - no, all of them, at once. At least in my understanding, that precisely means they are seeing the 4D object as a whole, not just individual slices, one or a few at a time. – O. R. Mapper Feb 13 at 8:04

As you clarified in a comment the seeing the insides is a 4D perspective on a 3D object. "One could see the interior of solids, such as the bulkheads or a piece of metal or a rock - one could see all the cross sections at once!"

The only way a 4D object can suddenly expand is if is moving and a slice of it in 3D is differently shaped than a nearby slice. I'm not familiar with this book just geometry - but it seems odd that if they can see in a 4D sense as the quote indicates - that they cannot perceive 4D objects properly.

Additionally if the navigation is cropping a dimension in the coordinates it could make the object seem artificially closer or farther. Distance being square root((x1-x2)^2+(y1-y2)^2+.....). For example draw a line. and then a point on the line and a second point not on the line (like a right angle triangle with sides 1:2). If your distance is just based on the distance of the x or y coordinates those will be 1 or 2 units. But if suddenly the true distance was perceived I suppose something you thought was 2 units away could suddenly be 1.4 (sqrt2) units away. However I strongly suspect the book would be describing a iceberg effect mentioned by others - that a 4D shape sliding into 3D will wildly look different from moment to moment. see

• Did you mean to edit your other answer or actually post this as new answer? – TheLethalCarrot Feb 13 at 17:15
• I apologize if there is posting etiquette I've crossed. The content strikes me as a "new" answer as it is focused on distances not slices. – lucasbachmann Feb 13 at 17:42
• It’s really up to you, if you feel it’s different enough for it’s own answer it’s fine. Just wanted to make sure you didn’t mean to edit the other one. – TheLethalCarrot Feb 13 at 18:04