Is it possible for us to really understand tetraspace?

[Jay]

When Fred sees a drawing of a square, he only sees at most 2 sides (lines) at reconstructs the remaining two in his mind to form the square. When we see a drawing of a cube (the non hollow one), we see at most 3 sides (squares) and reconstruct the other 3 to form the cube. I assume when Emily sees a drawing of a tesseract, she sees at most 4 sides (cubes) and reconstructs the other 4 in her mind to make the tesseract.

If Fred saw the drawing of the cube, and see saw two lines, we would probably infer a square, or rectangle, or whatever. Even if he oriented the drawing to see 3 sides, he would infer a pentagon, or some irregular 2d figure. (I'm basing this on "Flatland"). Even if you told him if was a drawing of a cube, he would gain no understanding from looking at it. So when we try to represent tesseracts on 2d pieces of paper or computer monitors, they really give no insight to the true object.

And think about it. Do you think you could understand a drawing of a cube if you couldn't see inside the squares? So how could you understand a drawing of a tesseract if you can't see inside cubes? And even though you know what a cube looks like, if I showed turned it so that you could only see it from a linear perspective, you wouldn't be able to infer the cube from the lines you saw.

Inferring a tesseract from planes =
Inferring a cube from lines =
Inferring a square from a point

[RQ]

But if Fred can understand a square as what it is then, he should be able to understand what a hollow cube drawn in his bionian universe would relatively look like to us.

[alkaline]

why should he be able to do that? Fred wouldn't be able to understand the way things look to us.

[Geosphere]

Fred would never be able to understand it.

If you see a shadow of a circle, can you tell me if the casting object is a circle, a sphere or a hollow sphere, or an oblate held at an angle, or a cylinder perpendicular to the light source?

[Jay]

If you see a shadow of a circle, can you tell me if the casting object is a circle, a sphere or a hollow sphere, or an oblate held at an angle, or a cylinder perpendicular to the light source?

Exactly, the fact that the number of shapes increases with the dimension means that different higher dimensional objects may be conveyed the same way in lower dimensions. So its even harder to infer a 4d structure from a 2d drawing, since the drawing could represent a 2d slice of many varying 4d objects.

I'm not trying to be pessimistic. I wish tesseracts could be drawn on 2d paper. I just don't think it's possible or practical for trying to understand the concept.

[Polyhedron Dude]

One neat thing about the tesseract, its 32 edges would feel pointy, like the cube's corners do. Also it's 24 square faces would feel like the cube's edges.

Polyhedron Dude

[Jay]

One neat thing about the tesseract, its 32 edges would feel pointy, like the cube's corners do. Also it's 24 square faces would feel like the cube's edges.

That depends on who's touching it. To Emily, the point edges, linear edges and planar will all feel sharp because they are cutting into her realmic skin.

The planar edges of a tesseract wouldn't feel sharp to us because our skin is planar. Just like the linear edges wouldn't feel sharp to Fred because his skin linear.

[RQ]

the shadow of a circle is the same as that to a cylinder only in the third dimension

[RQ]

Well, actually I agree on the fact that even if Fred could understand the possible shape of a hollow cube drawn in his universe, he wouldn't understand it. Perhaps if we could draw in a hologram, we might be able to achieve something.

[PWrong]

I know this topic is finished, but I have an idea

You don't need a hologram at all. I made a model of a hypercube with toothpicks. It's just like drawing one. You build two cubes and connect them at each point (with Blutack). It doesn't look much different from the drawing, but you can look at it from different angles. I guess it's the same as what a tesseract drawn on a swock would look like to a tetronian. If Emily's looking at me right now, she probably recognises my model.

Anyway, I think I figured out why Fred can't draw a cube. I can see the inside of my model because it doesn't have a surface. If I put paper over each of the 24 faces, it would just look like a wierd shape.

For Fred, a line is a surface. So Fred can't draw a cube because all the lines are hidden. Even if he constructed a model, he wouldn't be able to see the whole thing because the lines on the outside would be in the way. But he could draw a cube on his paper with points. Or he could use see-through lines to make a model. That's like us using glass or cellophane on each face instead of paper.

I think some of our expectations are too high. I don't think an empty tesseract would look any more spectacular to a tetronian than a model or a drawing of it looks to us.

The only problem is you can't put anything inside the hypercube, or imagine something being inside. It's hard to imagine that the 8 cubes we can see on the hypercube are actually surrounding the space inside.

[RQ]

It's not really the fact that Fred can't see the hidden lines, but he just can't see it the way we do even if he could recognize the possible shape of the hollow cube. If you notice, the hollow cube has two places where lines meet that the don't join (hollow cube, drawn on a 2D paper). So fred would only assume they were corners, and even if he didn't, it still wouldn't give him the perspective he needs.

[PWrong]

I see what you mean. But there's still no point in waiting for holograms to be invented. Although maybe a virtual reality simulation would work.

[RQ]

Yes, well, maybe we can just look at a cube in 3D with its edges made out of straw.

[PWrong]

What I meant was, we don't need a hologram of a hypercube because we can just build a model like the one I made out of toothpicks. A hologram wouldn't be any better because we still see it in 3D. But if we could somehow put 4D images into the brain from a computer, bypassing the eyes completely, we would eventually adapt to seeing a simulated 4D universe. Babies see upside down for the first few weeks until the brain adapts and switches it the right way up.

[arsenic]

It is possible to understand 4d geometry

[whiteonriceboy]

a 2-dimensional being can perhaps make a diagram of a 3D object, but there is no way that he can view the diagram from his 2D plane. in the same way, how can we hope to draw an accurate representation of a 4D object on a 2D plane and not even in the 3rd dimension and expect it to be accurate?

[RQ]

Although Fred wouldn't be able to visualize or understand the direction of the 3rd dimension, he should be able to understand how to make a square that would be on top of another one, that would look like a hollow cube to us, but just like two squares in each other to Fred. We can use the same technique, although we would not be able to understand (definition of 3rd dimension) we would know that it is only 3D in realmspace.

[whiteonriceboy]

I know i can't figure out even what the hypercube is supposed to look like or mean, but i can't understand how anyone could make a picure of a 4D box-thing on a flat plane. do you know how?

[BClaw]

I believe that it's impossible for us to *truly* understand the 4th dimension. We can *infer* a lot, and get a feel for it, but the math, or visualizations of 4D objects is not the same as *seeing* one. A blind person can understand the concept of color vision, can tell you what color red and yellow make together, but will still never truly understand what color is, or how you can tell what color something is at a distance without 'hearing' it. I think we are similary handicapped in out understanding of the fourth dimension, though not nearly as much so in our 3D world as it would be in a 2D world. We square the amount of information we can display just by being in a higher dimension than poor Fred, and Emily's ability to ascertain visual complexity must be truly amazing, as hers is the square of ours! Let's not forget some of the other ascpects of visualization that can help us out though, like color, or movement, or opacity. People who have no depth perception are often able to simulate this by moving their heads to see how far things 'jump' in their view. If they move a lot, the objects must be close. Fred could get a lot of information out of a display that rotates the cube, has different colors assigned to the endpoints, and is transparent enough so that all the point and lines contribute a bit to the overall image he's seeing. It's still not *really* a cube, but he still gain a lot of insights into how a cube might work. Same with us, only more so, since the plane we see contains so much more information than the lines Fred sees. Sure we see depth, but so does Fred, and that's not truly 3D in the sense that we can't see what's behind an opaque object, like Emily can with her superior view on our world. We can see Fred's world better than he himself can, so I submit he sees things as all linear, while we see things that are planar.

[PWrong]

Actually, if you think about it, we can't actually understand living in 2D space either. I can imagine seeing a coloured line, but not the same way Fred does. Consider this question, is the coloured line vertical or horizontal? To Fred, it should be neither. In theory, the line should be infinitely thin, but I can't imagine an infinitely thin line. Finally, when I imagine seeing only a line, I see blackness on either side of it. I can't imagine not having that black space, or another colour.

[pat]

I can imagine seeing a coloured line, but not the same way Fred does. Consider this question, is the coloured line vertical or horizontal? To Fred, it should be neither.

When you see a line painted on the ground in a parking lot, is it horizontal or vertical? The answer is clearly 'horizontal'. But, that question just does not make sense in Flatland. In Flatland, the real question is does that line run north-south or east-west?

In theory, the line should be infinitely thin, but I can't imagine an infinitely thin line. Finally, when I imagine seeing only a line, I see blackness on either side of it. I can't imagine not having that black space, or another colour.

Here's a flatland view of circle and a cube and a nonagon:

Do you have trouble imagining that this is all there is? I can see having trouble imagining that this is infinitely thin. How about one photon high? Is there really trouble imagining that there is no way to see above or below it---this is all there is in Flatland?

[pat]

Here's another Flatland visualization. This is a stereo-pair of a textured cube.

This is assuming Fred still has two eyes (which he should)

[RQ]

The first dimension is up and down(Gravity). The second is forward and backward, and the third is left and right.
Fred would see the line, just as we see the plane, yet are the third dimension, and he's in the 2nd dimension. Simple as that.

[PWrong]

The first dimension is up and down(Gravity). The second is forward and backward, and the third is left and right.

The three dimensions are arbitrary and aren't in any particular order.

Fred would see the line, just as we see the plane, yet are the third dimension, and he's in the 2nd dimension. Simple as that.

We already know that. I understand the idea of seeing only a line. I'm just saying that I can't imagine seeing only a line, without anything on either side of it, and that is neither horizontal nor vertical, because we're used to seeing a 2D image.

Maybe our brains and eyes could adapt though. The introduction says that living in the 2nd dimension is like living in a very thin trench. You can look up or down, but not left or right, and you can't turn around. What if Bob had to live in a trench like that? I don't mean actually becoming 2D, but near enough.

He'd only ever need to focus on the vertical line in front of him. Looking away would be a waste of time, so his brain would probably limit his eyes to the line.

[RQ]

Yes, I meant with respect to the observer (I always get mixed up).
Of course we couldn't visualize only a line. It would have 0 space.