Anon does the impossible

But if one wants to represent a tesseract- a “4D cube” on paper- one simply has to draw a cube with a second cube within it, and the two connected. There. You have a 4D graphic representation.

Paper, computer screen, tv, whatever. Pretty much every form of graphic representation we have (asides from sculpting and such) is a 2D projection. A picture you take of a cute dog is 2 dimensional. The data is generally stored in a “flat” format. When you view it- you will most likely view it on a “flat format” like a screen or paper. But you can tell your dog is 3D. This is because your mind uses cues- the same way we create “photorealistic” art or “3D” on paper. Using what is essentially optical illusion- cues are included like lighting, shading, forced perspective and such- that your brain interprets as “3D.” In fact- most optical studies and current knowledge hold the human eye captures images in 2 dimensions and the brain translates them to 3 dimensions.

This is part of why people with loss of vision in one eye have little or no depth perception. If you focus on an object and close one eye, then quickly alternate and open the closed eye, and close the formerly open eye- and repeat- you will see that the image shifts. How much it shifts on which eye depends a lot on which if either of your eyes is “dominant.” That demonstrable shift is a sort of easy example of how each of your eyes sees a different picture- largely because they are physically distanced from each other on opposite halves of you head- but in some part for some people due to having differences in visual acuity between eyes.

So your brain takes these two pictures and puts them together, and using the difference between them it helps your brain calculate the 3rd dimension- depth. This mechanism is one of the major reasons that a 2D picture can be seen as 3D- your brain doesn’t care or really know of a thing is or isn’t 3D- it merely applies the “rules” or “expectations” it has to whatever you see and processes it by the number of dimensions it appears to have. Or only is this how optical illusions mainly work- it’s also a major factor in why CGI often looks fake or “wrong.”

There are many cues your brain is looking for- things it has learned to expect from the world. So relatively small differences in shading or the behavior of light, a surface that refracts or reflects or absorbs light a way the brain catches as odd, small movements or after effects of movements that are missing or wrong... subconsciously you pick up on these.

So back to the subject though- a “3D” computer image does in fact have 3 virtual dimensions. That becomes a complex subject- but let’s just say that it’s largely irrelevant since you will likely be viewing on a 2D screen, and your eyes will only actually capture 2 dimensions and the brain will create the 3rd if the artist/math wizard did their job right. So showing a 4D object on paper isn’t hard, either way you do it- but most 4D representations are more realistically captured through moving picture so that one an get a better view of the extra dimension to better comprehend what they are looking at isn’t 2D or even 3D. Although you’re still perceiving the 4D object as though it were in a 3D space via a 2D space.

Virtual experimentation has shown that humans do appear, or at least some humans do appear, to be able to acclimate to a 4D space in a way that at least allows them navigation- an ability to move through the 4D space in a 3D manner.

To our “man on a piece of paper” example- to the man on the paper- you can appear and disappear at will. You can seem to move an object from one place to another through teleportation. If you were to place an apple in a box and place that into his 2D space- he would see a line, a wall. If he moved around it it would be a square. He could not get the Apple without destroying the box- even though the box is wide open. You could reach in to the top of the box and pull the Apple right out. To him- that would be baffling, amazing.

You could walk around his entire world, put a finger down one place, and then on the opposite side of his universe, within moments. You could crumple his world into a ball- it would all the there but he could not see it. If you punch a hole in the paper- to him it would be a either a blank spot the same as the edges of the paper, or a line, a point.

So a 4D being and 4D objects very likely could be said to behave the same way, or analogous. This is concept often called “dimensional analogy” and our man living on paper is an oft used and well known, if slightly imperfect, example.

The the question itself is doomed from the start. If I put a dot on a paper that could be called a representation of 4D space. If I wrote and equation that most people wouldn’t understand or at least be able to usefully visualize, that is a representation of 4D space- and with any “representation” there is a measure of skill. The 4 year old that draws grandma looking like a squid crossed with a stop sign- that’s a representation, a likeness of grandma- it just isn’t a very skilled one, or to be kind is a more abstract one.

But ultimately, the question of perception comes in to play and what it is that you’re really asking for- how you want a 4D space represented and to what perception from what perspective is rather important. In the end regardless of what you put on the paper it will end up being a somewhat poor representation- we are limited in that regard.