Well, I can elaborate, but I’m not sure how helpful it will be. “No one can be told what the matrix is” and that sort of thing. The basic idea is that it’s the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that’s not telling someone who has tried and failed anything they don’t know, I’m sure.
If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it’s easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.
Is it like having a complete n-dimensional construct in your head that you can view in its entirety?
I can visualise 4-dimensional polyhedra, in much the same way I can draw non-planar graphs on a sheet of paper, but it’s not what I imagine being able to visualise higher-dimensional objects to be like.
I used to be into Rubik’s Cube, and it’s quite easy for me to visualise all six faces of a 3D cube at once, but when visualising, say, a 4-octahedron, the graph is easy to visualise, (or draw on a piece of paper, for that matter), but I can only “see” one perspective of the convex hull at a time, with the rest of it abstracted away.
Well, I can elaborate, but I’m not sure how helpful it will be. “No one can be told what the matrix is” and that sort of thing. The basic idea is that it’s the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that’s not telling someone who has tried and failed anything they don’t know, I’m sure.
If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it’s easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.
Is it like having a complete n-dimensional construct in your head that you can view in its entirety?
I can visualise 4-dimensional polyhedra, in much the same way I can draw non-planar graphs on a sheet of paper, but it’s not what I imagine being able to visualise higher-dimensional objects to be like.
I used to be into Rubik’s Cube, and it’s quite easy for me to visualise all six faces of a 3D cube at once, but when visualising, say, a 4-octahedron, the graph is easy to visualise, (or draw on a piece of paper, for that matter), but I can only “see” one perspective of the convex hull at a time, with the rest of it abstracted away.