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.What would a 4 dimensional object look like in 3 dimensional space? Let's leave time out of it for now.
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.The animations show all the edges being on the outside, which is why I liked them.
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.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.
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.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.
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.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
Right, and it should be. If you find this easy to think about, you are not thinking about it properly.My head is spinning...in more dimensions never possible.
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.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.
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.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.
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.Sorry for the misunderstanding.
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).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.
by Aaron Carman
by Aaron Carman