Hi, new member here.

As a favorite meditation subject of mine, I have figured out several 4-d and higher geometries. I can visualize cubes up to 5-d, and UNDERSTAND them up through 8-d (cubes are the easiest, for me). I recently managed to complete my visualization and understanding of triangles/pyramids in 4-d, and have logically progressed them through 6-d.

One shape that has eluded my grasp for a LONG time was the higher-d sphere, though. I still cannot for the life of me begin to visualize it, nor can I logically figure how to construct one. I had a recent breakthrough, when I realized (I swear, a lightbulb actually materialized over my head and blinded everyone nearby with THAT flash...) that a 2-d circle is a 1-d line curved through 2-d space to intersect all points at x distance in 2 dimensions from a given point c; a 3-d sphere is a 2-d plane curved through 3-d space to intersect all points at x distance in 3 dimensions from a given point c; thus a 4-d sphere must be a 3-d (something) curved through 4-d space to intersect all points at x distance in 4 dimensions from a given point c.

Where my new problem lies is in figuring out what the (something) is, exactly. I have a line and a plane; what is my logical procession? I am working on this exercise now. It is a tough little challenge, one I am thoroughly enjoying. Once I figure it out, I will have 3 points on my progression, and will be more easily able to extrapolate the next several.

I know that you can also view the sphere as a circle twisted a full rotation through z around the x or y bisecting axis, and a circle as a line twisted a full rotation through x around the y halfway point. But this is not the progression I am pursuing. It is not nearly as elegant. Plus it creates solid shapes, where the other model would just be the hollow skin encompassing the shape, which is far preferable for my purposes. (This was my original progression, but read on...)

In this alternate progression, the 4-d sphere would be a 3-d sphere twisted a full rotation through w around the xy, yz, or xz bisecting plane/circle. The problem is, although I can logically understand this, and probably continue the progression (5-d is 4-d twisted through v around the xyz, wxy, zwy, or wxz bisecting spheres. That would be the progression, but is it correct?) I still cannot visualize it, thus rendering myself incapable of determining of it is correct or just an example of fuzzy logic taken too far.

I wish to note, I have never looked at ANY models until I have designed my own first. I do not muddy the waters of my mind with other peoples' work; I work it out myself, then compare. It allows me to give things a fresh viewpoint, which has allowed me to come up with a few models I believe to be unique to myself. I did not have in my head already "Oh, that's what a tesseract looks like", because then any time I tried to visualize it, I'd just have seen that super-cross shape, and never been able to come up with my own versions. This methodology also allows me to more easily visualize the actual 4-d shape, rather than just the unfolded representation of each one.

Anyways, I am just excited about my progress on the hypersphere (which I have never seen a model of, yet, of course), and wished to share my thoughts.

See ya!