DIMENSIONS OF 4D SPHERE HOW TO
DIMENSIONS OF 4D SPHERE SKIN
DIMENSIONS OF 4D SPHERE FULL

(However, they are still fully n-1D dimensional objects). In nD, (n-1)D objects occupies relatively negligible space, thus they act as if they are flat. Remember how a line occupies a small region of a plane and how a plane is flat relative to 3D space. the equator), the figure trace out is a 3-sphere (technically a 4-ball) My visualization techniques are also followsĢD Spin the line about its midpoint, the figure it traces out is a circle (technically a disk)ģD Spin a circle about its diameter, the figure it traces out is a sphere (technically a ball)ĤD Spin a sphere about one of its great circles (largest possible circle you can draw on the sphere, e.g. See ya! lightandtime Nullonian Posts: 2 Joined: Mon 5:13 pmĪctually, the hyperspheres are also my most favorite higher dimensional object This methodology also allows me to more easily visualize the actual 4-d shape, rather than just the unfolded representation of each one.Īnyways, 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. 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. 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 do not muddy the waters of my mind with other peoples' work I work it out myself, then compare. I wish to note, I have never looked at ANY models until I have designed my own first. 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. 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.

DIMENSIONS OF 4D SPHERE FULL

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. (This was my original progression, but read on.)

DIMENSIONS OF 4D SPHERE SKIN

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. But this is not the progression I am pursuing. 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. Once I figure it out, I will have 3 points on my progression, and will be more easily able to extrapolate the next several. It is a tough little challenge, one I am thoroughly enjoying. I have a line and a plane what is my logical procession? I am working on this exercise now.

Where my new problem lies is in figuring out what the (something) is, exactly. 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.

DIMENSIONS OF 4D SPHERE HOW TO

I still cannot for the life of me begin to visualize it, nor can I logically figure how to construct one.

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

Dimensions of 4d sphere