We can recreate a 3-dimensional picture on a 2-dimensional surface. Why can’t we visualize a 4-dimensional world in a 3-dimensional area?

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We can recreate a 3-dimensional picture on a 2-dimensional surface. Why can’t we visualize a 4-dimensional world in a 3-dimensional area?

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We can. Just how like we can draw a cube (a 3D object) onto a piece of paper (a 2D plane), we can project a 4D object into 3D space. Here’s what a 4D cube (sometimes called a Tesseract) looks like projected in 3D space:

https://en.wikipedia.org/wiki/Tesseract

Just like when we project a cube onto a 2D piece of paper, we lose part of the shape by the projection. We can’t draw a cube that has angles that are all 90 degrees on a 2D piece of paper. Some angles will be skewed. Similarly, the angles between the sides of the tesseract are all actually 90 degrees, but only appear skewed when projected in 3D

The only reason we can properly understand the 2D representation of a 3D object is because we live in a 3D world, so we have a reference point. We’ve never experienced 4D.

That doesn’t stop us from trying though. If you google tesseract you’ll find 3D representations of a 4D object (or, rather 2D representations of the 3D representations, since you’re looking at a 2D screen).

Our entire visual system works by representing a 3D world in 2D, so it’s very intuitive to do this for pictures. As other comments have stated, there are equivalents in higher dimensions, but they’re not nearly as intuitive for us to grasp.

We live in a 3D environment. A 4th dimension, and many more, is something theorized by physicists and mathematicians to make their equations work out, it doesn’t necessarily have a physical meaning.

you can easily recreate a 4dimensional area if you use either time, temperature or colour as your 4th dimension.

only 4 spatial dimensions is a bit iffy but as others have pointed out we can do that as well.

I suspect that we can, but the problem lies in our ability to perceive it, and to subsequently understand it.

We perceive and understand 3D space intuitively, probably because it’s how we process our visual input. It’s more than just the image we see. It’s also how our eye muscles are feeling (and giving feedback), and how our brain is synthesizing it into an understandable model of the world.

We *can* visualise a 4-(spatial-)dimensional world in a 3-dimensional area. There are several games and other media that attempt to do so. The problem is 1) it’s difficult to make sense of what’s happening because our brains have evolved to only understand 3 spatial dimensions, and 2) we haven’t observed a 4th spatial dimension in reality, so all our depictions of it are completely made up.

At best we can look at how a 2D medium can be used to represent a 3D object (eg. a drawing of a cube on paper), the way that 2D projection changes as the object rotates in 3 dimensions etc, and try to extrapolate that behaviour into a 4D object. There can be a little maths and logic involved. But ultimately a 4th spatial dimensions doesn’t appear to exist, we have no idea how one would work if it did exist, so this is just an exercise in creativity.

[This](https://youtu.be/_4ruHJFsb4g) guy does a good job of showing what that would look like

Human imagination is a wonderful thing. Since we perceive the world in 3D, we can look at a flat surface and use spatial cues to infer that depth is depicted. 4D is beyond our perception, hence why we can’t use perception cues gleaned from our everyday life to display it anywhere.

In multidimensional scaling analysis we often visualise higher than 3 dimensional data, but usually through a series of 2 or 3 dimensional graphs (sorry to disappoint). By comparing graphs you can build a mental visualisation of the >3D object/cloud

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We could model it, but we could not “visualize” it, because your retina and therefore your vision is still essentially 2D. But you could use a 3D Printer to print a volume that represents a “flat” projection of a 4-dimensional object. You would not be able to look inside it though.

We’re able to visualise 3D from a 2D picture because that’s literally what we were trained to do from birth: you have one or two 2D pictures from your eyes (depending on whether you closed a eye or not) and your brain is translating that into a 3D object. In particular, our brain uses the shadows and the lighting to help get a better understanding in complex situations.

And in fact, we don’t fully see 3D objects. When you look at a 2D objects, you see the borders and the inside of the objects. When you look at a 3D object, we only see it’s surface and we don’t naturally see inside non-transparent 3D objects.

Both those facts means that representing a 4D object in 3D is very difficult.

(1) We’re not trained to recognize 4D objects. And even less used to see how lighting and shadows work on them.

(2) Since our eyes are still 2D, that means we would be looking at the 2D image of a 3D object representing a 4D hyperobject. Complexity adds up.

(3) Additionally, since we can’t see through 3D objects unless they’re transparent, we would be missing a lot of information that would be hidden inside this 3D representation. You can circumvent this last issue with transparent objects but it still adds to the complexity.