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It is a 3D projection of a 4D object onto a 2D surface in 3D space. To be exact.Quote:
Originally Posted by Bonsay
Drawing a cube in 2D space would look like two squares drawn over each other to a 3D viewer, but they would appear as a strangely convex box rather than a cube, because everything you see in 2D is made up of 1D lines.
That is what this is, only for 4D.
It's a drawn 2D object (we might call it a jumble of quadrilaterals), a projected 3D object (looks like it could be two cubes, one inside the other), or a projected hypercube. There is this ambiguity which defines what it is to be a projection.Quote:
Originally Posted by Bonsay
EDIT: Roxxor doesn't know what the fuck he's talking about. He's just confusing himself and everyone else.
Yeah, I figured that a while ago.
That isn't a post, that's just a random jumble of pixels that one might interpret to be written words and a picture. If they're lucky, they'll see english and might be able to draw meaning from it.Quote:
Originally Posted by drewmandan
Yea, I thought his question was specific. You know, wanting a specific answer that was meaningful.
Ah I get it! So Xei's real life hypercube model is a "drawn" 3D object or a projected 4D object. Well in that case you couldn't draw it I guess.
Correct. I'm assuming his real life model is actually 3-dimensional. So something like that would be like looking at a hypercube from a certain 'angle' (and projecting), although the angle is now in 4-space and you can't picture it.Quote:
Originally Posted by Bonsay
Hmm... no, actually I just realised you can definitely draw a 4D object in 2D:
http://upload.wikimedia.org/wikipedi...simple.svg.png
Apparently that's a drawing of a circle. But it can also be a projection of a sphere or some 4D sphere equivalent.
If that's a 4-sphere, it's just a projection, not an actual drawing.Quote:
Originally Posted by Xei
Yeah, exactly. :PQuote:
Originally Posted by Xei
This is really a load of nonsense to be honest. To draw something is to represent something on a surface.Quote:
If that's a 4-sphere, it's just a projection, not an actual drawing.
If you were to ask an artist what drawing something is, he would not say, 'oh, but remember it can only be n+1 dimensions or fewer!'. Just why? You lose information when you represent a 3D object in 2D. That object is no longer 3D, it's a 2D representation. Here, the object is no longer 4D, it's another 2D representation too.
Seriously, unless you can find an accepted definition of drawing something to be 'representing something in 2D from no more than n+1 dimensions', I don't see what the argument is.
Incedentally, that's not a 4-sphere. When I drew it I intended it to be a 4D equivalent of a disc or a ball (a 4-ball I suppose it would be called).
Technically a circle is 1D, a sphere is 2D, and so actually the n+1 dimensional equivalent is a 3-sphere.
Representation is what mathematicians call "projection".Quote:
Originally Posted by Xei
I don't know where you're getting this "n+1" stuff. Any number of dimensions can be projected to the plane.Quote:
Originally Posted by Xei
Ok, technically the 4D sphere is "called" a 3-sphere. That's just arbitrary notation and it's not logical. But convention is convention. Without checking, I wouldn't have thought a 2D object would be called 1D and a 3D object called 2D, but apparently they are.Quote:
Originally Posted by Xei
I know. Swap all the instances of the word representation for projection in my post if you want.Quote:
Representation is what mathematicians call "projection".
I'm getting it from you, saying you can't draw an n+2 dimensional object on an n dimensional object, ie a hyperball on a plane.Quote:
I don't know where you're getting this "n+1" stuff. Any number of dimensions can be projected to the plane.
It is logical. This is maths we're talking about here, I can assure you it's logical. :PQuote:
Ok, technically the 4D sphere is "called" a 3-sphere. That's just arbitrary notation and it's not logical. But convention is convention. Without checking, I wouldn't have thought a 2D object would be called 1D and a 3D object called 2D, but apparently they are.
A circle is a 1D line (formed by the set of points which are of a fixed point from the origin) and a sphere is a 2D plane (formed in the same way). Hence the next dimensional analogue of a sphere is 3D.
A disc however is the 2D area/plane bounded by the line of a circle, and a ball is the 3D volume/space bounded by the plane of a sphere.
But...I didn't say that. I have said about a dozen times now: any higher dimensional object can be projected onto a lower dimensional object.Quote:
Originally Posted by Xei
But... you did. That's why we're talking.
Judging by your above post you've now decided that a projection can actually be a drawing, so we don't disagree any more.Quote:
not an actual drawing.
The word "drawing" was being used in a different way before. People were using it like a "true" representation of the object, and then saying that you can't "draw" a 4D object because you're "drawing" it on a 2D plane. So there was an inconsistency there, and I'm not quite certain everyone understands. So for now on, I suggest we all use the word "projection" when referring to a projection, and "sculpture" when referring to the recreation of the original object.Quote:
Originally Posted by Xei
Hence, a circle on a sheet of paper is a sculpture of a circle and a projection of either a sphere or any higher order of sphere. A sphere in 3D is a sculpture of a sphere and a projection of higher dimensional spheres. And so on.
In this case, a drawing is the expression of all the spacial dimensions of an object. So, if you're in 2D then a 3D drawn object is a 2D projection-- Still a drawing, but if you're going to call it 3D then it's a projection of a 3D object, not a "true" drawing of a 3D object, which would express Height, width and depth.
Any representation of an object in fewer dimensions is actually a representation, and not a real thing. That's how you can draw a cube in 2D, but you lose some aspect of it - see how you can't see the opposite side of the drawn cube. The drawing is an orthogonal projection of the image - just like a shadow, really.
4D can be represented into 2D, but instead of losing only one dimension, it actually loses 2, making it even less accurate a representation. It is a valid representation, nonetheless.
So, yes, that is the drawing of a tesseract.
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Funny as physics can be, you can actually substitute one of the dimensions in a projection for a dimension of time. That's what allows us to represent a perfect cube in 2D - you make it an animation (making use of the time dimension), and rotate it around. Unfortunately, we still haven't invented hologram monitors, but once we do, it'd be much easier to represent tesseracts.
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Regarding the original question - what's the use of a fourth physical dimension. Well, You know you can actually work with each dimension separately, including movement even. I don't see any other practical use for it right now, but maybe it can help in explanations for fundamental physics, the origin of the universe, etc. Finally, it could be used in art :D