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There are well-known procedures for tricking human brains into assigning 3-dimensional perspective to 2D images. Some, like 3D glasses in a theater, work without effort, while others, like the stereogram of a galloping horse below, require some training and many people cannot do it at all:

Galloping Horse Stereogram

We have a good mathematical understanding of 4D, and we routinely use computers to handle calculations in n-dimensional spaces. The problem is that our brain is adapted to create a 3D pseudo-emulation. If you think about it, it's created by neurons spiking: nothing intrinsically 3D about it, rather us Earth-mammals adapted to use 3D because it's useful in a 3D environment.

Now I want to be in a place where a human girl, aged about 12, is able to "see" in 4D. More specifically, I want her to achieve 4D perspective on a 3D object. Assume that a device exists to grant that information to her if she can understand it. It may require effort, like the stereogram, work for a limited 'volume', and she may only be able to sustain it for a brief time (or it may come as natural as breathing).

I know of no human who claims to be able to do this, so this poses a bit of a problem. While it may come down to the wiring of our brain, I'm trying to think of a training program (for the child or even for the infant she once was if we want to take advantage of extra neuroplasticity of newborns) that would engender and boost this ability. I'm thinking computer simulations, minor brain surgery, etc.

How could we train/modify people to see in 4D?

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    $\begingroup$ I suppose we'd need to first understand what 4D is. Imagine you're a 2D creature, living on a sheet of paper. That paper is your entire world, you cannot comprehend anything living outside of it. Now, a baseball pierces through the paper. What do you see? A dot that appears out of nowhere, that then grows larger and larger until it starts shrinking and disappears entirely. Did you understand what 3D is? Can you imagine it? Chances are, you still have no clue what 3D is. And that's how 4D is like for us. $\endgroup$ – Nolonar Sep 19 '15 at 10:29
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    $\begingroup$ Don't we think in 4 dimensions already? Three spacial dimensions plus time. $\endgroup$ – Steve Bird Sep 19 '15 at 10:34
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    $\begingroup$ Aside from what Steve Bird commented above: What is the purpose of four-dimensional rather than three-dimensional thinking? What is the gain for the individual? It's easy to say "it's just one more dimension", but why? Can you give a reasonably concrete example where such an ability would be beneficial? That might make the question easier to answer. $\endgroup$ – a CVn Sep 19 '15 at 13:02
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    $\begingroup$ @SteveBird The time dimension doesn't allow you to walk outside of a locked 3D room. The 4th dimension does. $\endgroup$ – SurpriseDog Nov 7 '19 at 23:23
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    $\begingroup$ The mathematicians answer: first learn to think in N dimensions, then simply let N = 4 $\endgroup$ – jk. Nov 8 '19 at 13:22
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It is fairly easy to depict 4D objects. A variety of possibilities exist, but probably the most intuitive is to use colours. For instance, map the 4th dimensional position of the object to frequencies of light. A purely 3d object is then monochromatic in this view, every element of it occupies exactly one frequency of the spectrum.

A 4-flat object that is shifted along the 4th dimension would be simply a different colour, and so free to coincide with the other object in the same 3d location. Meanwhile an object with depth in the 4th dimension would occupy a range of frequencies and so be a mixture of colours.

What this does not do is make 4D intuitive, which you didn't mention but I suspect is your real objective. Even stereograms do not do this for 3d. All you are doing there is sending visual input that duplicates, to some extent, the visual input the eyes give on a 3d object, fooling the sophisticated built in spatial recognition parts of the brain. 4D spatial recognition parts of the brain simply do not exist. (Try and mentally rotate an arbitrary 3d object and it's easy in arbitrary rotations. Try it for a 4D object and uhhhh) To make them exist, I imagine you probably have to intervene drastically in the early development of the brain, or have some way of doing some serious rewiring....

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    $\begingroup$ Sure, but think about it for a second, not only does a 4d perspective allow you to see all six faces of a cube at the same time, but also every single point within that cube... I don't think color can do that. $\endgroup$ – Serban Tanasa Sep 20 '15 at 0:50
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    $\begingroup$ A stereogram only lets you see one side of a 3d object at a time also, so if you count that as a depiction of a 3d object, then it's little different. $\endgroup$ – Fhnuzoag Sep 20 '15 at 15:06
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If you want to develop a 4D optical sense, you surely need some kind of advanced optical input, that is, an interface to your visual nerve, and a way to train it. Kids usually do that training by cross correlating the optical input with that from the nerves on their hands, I assume. ;)

I guess you are right that a 4D brain is not necessary, although a 3D brain surely will hit its limits with, say, a fifth or sixth spatial dimension.

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