What would a 4 dimensional object look like in 3 dimensional space? Let's leave time out of it for now.
You mean a tesseract? http://en.wikipedia.org/wiki/Tesseract The other variation is more general, it was used in the short story "He had a Crooked House" by Robert Heinlein. This is a 4D cube drawn in 3 dimensions translated into 2 dimensions. <whew> Wikipedia's article is very good on the subject, it covered everything I was going to say and more. I particularly love their animations.
Well, You kind of broach my real difficulty Bill. Certainly the tesseract is a 4D object, and it’s possibly the easiest to deal with from our (human) block headed perspective. I am familiar with all the projections and representations of it, but you’ll be hard put to convince me that any are meaningful representations of how a tesseract would look to us. Taking a 4D object and projecting into 2D is akin to taking a 3D object and projecting it into 1D. How much useful visual information could you draw about a cube from a linear projection? And you are a 3D being. Imagine how tough the visualization would be for a 2D being! Us, trying to visualize a tesseract from a 2D projection is akin to a 2D being trying to visualize a cube from a linear projection. While I find the animated projections interesting, the distortions inherent in them are horrendous and bear little resemblance to what a tesseract would look like rotating through our space. The popular Schlegel diagram, on which the animations are based, is misleading as a ‘real’ tesseract has 8 cells and none of them contain any other cells, however, the Schelgel diagram shows only 7, while an 8th that contains all the other 7 can be implied. The only solid projection of a tesseract into 3D I personally find even remotely useful is the simple cube, but still, it tells little and is also immensely misleading. Here is an example of what I mean. If I were to give you a square piece of black paper and ask you what 3D solid, convex object is this a 2D projection of, what would immediately come to mind? Probably what you are most familiar with. I think most people would choose the cube. In fact, there are at least 3 simple platonic solids that have square projections in 2D. Here they are represented as their vertices in (x,y,z) The tetrahedron: (-1,-1,0) (1,-1,0) (0,1,-1) (0,1,1) If we remove the y axis we get the square: (-1,0), (1,0), (0,-1), (0,1) The regular hexahedron (cube): (-1,-1,-1) (1,-1,-1) (-1,1,-1) (-1,-1,1) (1,1,-1) (1,-1,1) (-1,1,1) (1,1,1) If we remove any axis, and eliminate redundancies, we get the square: (-1,-1), (1,-1), (-1,1), (1,1) The octahedron: (-1,0,0) (0,-1,0) (0,0,-1) (1,0,0 (0,1,0) (0,0,1) If we remove any axis and eliminate the trivial vertices (which are occulted anyway) we get the square: (-1,0), (1,0), (0,-1), (0,1) Now, if we bring convex solids into the mix, there are an infinite number of them that could create a square projection in 2D. While I know we can create better projections of, lets say a cube in 2D, and animated ones are very revealing. Our experience with actual cubes no doubt aids in our understanding of such animations. However, a clever animation of a 2D projection of a rhombic hexahedron would look just like a cube to us. So, armed with all that: If I give you a circle, can you tell me what 3D object it represents? If I give you a cube, can you tell me what 4D object it represents? If I give you a sphere, can you tell me what 4D object it represents?
The distortions you mention are a logical consequence of trying to project a 4 dimensional shape in 3 dimensional space. It can't look right, just like a 3d cube drawn on paper doesn't look right (though it does look like the shadow of a cube would look if held a certain way). I think the tesseract is accurate as it can be as far as it goes. It can be build with straws or dowels, it is still the "shadow" of what one looks like projected on 3 dimensional space. Every edge is on the outside though. One of the reasons I like the animations is it shows what would happen if you were to walk around a tesseract in a 4 dimensional way. It is as close as I can come to being in four dimensions. I wonder if any human can truly imagine four dimensions, I can't.
I have to disagree Bill. A 3D being should be able to pass through a door in any side of any cube of a tesseract and end up in another cube. None of the projections, even the animated ones show this well. We can say the Schlegel diagram comes close, if we assume the rest of the known universe is the 8th cube. Then, the animation only shows the rotation of 7 of the cubes which sometimes become connected or disconnected from the 8th cube. That should never happen as the relationship between cubes should never change. One arrangement might be cube 1 has direct access to cubes 2, 3, 4, 5, 6, 7, but not to 8. No rotation of the tesseract should change this. I will agree that we can never imagine the shape of a 4D object in all its 4D glory, but any 4D object that shares 3D with us will have an actual shape in those 3D, not a projection. This is what I am getting at. What would be the actual shape of, a tesseract for arguments sake, be in 3D? The 'stick' projections show (or try to show) stuff we would not see of the object in 3D as they are components without extent in all 3 of our dimensions. Maybe I'm not making myself clear, but the shadow analogy does not work, as that only appears as it does to a 3D being looking at the projection onto 2D. What I'm talking about would not look that way to a 2D being. Try to imagine the result of a cube rotating randomly through a plane at various 'depths' of the cube. The inhabitants of the plane would be able to see, or more accurately, measure all sorts of cubic sections that are not represented in 3D onto 2D projections. They would be able to see any possible surface that can be created by the bisection of a cube with a plane. Triangles, lines, squares, rectangles, rhombus', kites, hexagons, and so on. Any number of cubic cross sections. See here: http://www.learner.org/courses/learningmath/geometry/session9/part_c/index.html Now take that case and try to extrapolate what we might see of a tesseract as it rotates randomly through 3D space. The shapes would be far more interesting than simple, and misleading, projections. It is what we would see in this case that I am interested in.
It's hard to answer that in words or in a 2-dimensional drawing. One can make a mapping from 4-d to 3-d and build a 3d model of the result. I've seen 3-D models of a projected 4-D cube and it looked like a cube within a cube with the corners of the outer cube connected by lines to the corners of the inner cube. This is similar to how a 3-D cube can be mapped into a 2-D plane as a square within a square with the corners of the bigger square connected by lines to the corners of the inner square. The 3-D object, that is a projection of the 4-D object, can change radically as the point of perspective in 4-D is changed. http://steve.hollasch.net/thesis/chapter4.html
The animations show all the edges being on the outside, which is why I liked them. They also show how you would walk from one cube to the other, but there does have to be an outer edge where you can walk outside the cube. If you can't see it, well that goes to my argument that we aren't wired to see it. It is outside our logic and conceptualization. I am impressed we have gone as far as we have to visualize it.
Actually they don't. If that were the case, then there are only 6 chambers shown, which is even worse. You may be thinking about them wrong. They do show chambers moving through faces. In fact, they just don't work. No, not for a 3D being. The 'outside' only exists in 4D. Just like a 2D being has no way to get outside a cube. It can only travel to other faces through the edges as it has no access to the 3rd dimensiion. In a tesseract, every face of every cubic cell has another cell attached to it. Each cell is attached to 6 others. Once a 3D being is put into a tesseract, there is no way out for it. Bill, You may still misunderstand me. I do know tessersacts and their 2D projections, and have studied other 4D shapes too. We did hundreds hours of work on projections back in the day. The more we did, the more we realized 2D projections were not a great way to visualize 4D objects. However, none of us took on the daunting task of trying to do a 3D spacial intersection with the 4D bodies. Mostly because we had no way to represent the result in 3D. We'd be back to those darn, useless 2D projections again. Perhaps when the problem of true full motion holograms is solved it will be worth the effort. But still, I'm just not interested in projections whether you might think I understand them or not. They are not what my question is about.
We did work similar to Hollasch back in the eighties. Frustrating and unrewarding as I recall. But it's still just projections of a 4D object into 2D. A 2D view of a 4D space, which is not what the question is about. I am more interested in a 3D view of a 4D object's intersection with 3D space. I am looking for a 4D/3D analogy to what I presented in the last half of post #6.
Right, and it should be. If you find this easy to think about, you are not thinking about it properly.
My post was responding along those lines, despite what you say. I specifically gave an example of what a 4D cube can look like when projected into a 3D model. And, I compared this to the 2D projection of a 3D cube. If you don't find it useful, that's fine, but don't make it sound like I'm not at least trying to address the question. As I said, "One can make a mapping from 4-d to 3-d and build a 3d model of the result. I've seen 3-D models of a projected 4-D cube and it looked like a cube within a cube with the corners of the outer cube connected by lines to the corners of the inner cube. " and I said, "The 3-D object, that is a projection of the 4-D object, can change radically as the point of perspective in 4-D is changed. " and the reference I pointed to has section 4.3 which discusses the step of projecting from 4D to 3D. Also shown there is a 2D drawing of the 3D projection of the cube I was talking about. Since I can't beam a 3D model of what I'm talking about over to your living room, I'm stuck with a 2D drawing. That's why my first comment was. "It's hard to answer that in words or in a 2-dimensional drawing"
Steve, please don't take it that way. It was not my intent. It's just that the conversation so far has been around projections. I did not mean to infer you were not trying, or that I am in any way disparaging what you said. It's actually the reason I asked the question the way I did. I personally do not find projections that useful. Rather, it's intersections I find more revealing. You had said: And I agree 100%. I wish I had a true 3D display to work with. I think running a 3D intersection of the 4D object, even one as simple as the tesseract, would be very instructive and quite captivating. Sorry for the misunderstanding.
First define what will be the fourth dimension? For example in three dimensions, we consider z-axis as third dimension's plane in addition to x and y.
Yes, it's no problem at all. I just wanted to make sure my intended message was getting through. It's a difficult question, and I'll keep thinking about it to see if I can come up with anything insightful.
The forth dimension can be any other dimension orthogonal to the 3 we are familiar with (except time), that shares the exact same mathematical properties. You can call it anything you like, but 'h' has become a popular tag. In which case you have (x, y, z, h).
It would be interesting if there were a program to draw the shadow of 4D shapes (like the sphere). I think animations tend to give the brain to grasp, but it would be very flowing to watch. I understand what you were saying about other shapes.