So, I realize this is really hard to convey without diagrams, but I find the wikipedia representation of a hypersphere really lacking. The stereographic projection shows "infinite radius" meridians and hyper-meridian lines (because the projection space is tangential to the hypersphere?) anyway it got me thinking about the flatland analogy. What would parallels, meridians and hypermeridians look like on the 3-d intersection of a hypersphere with our space as it passes through? Imagine three scenarios: an "A" h-sphere, a "B" and a "C" with thick alternating red/blue strips painted on the single h-sphere "cell"/"surface" - in 3 different orthogonal configurations with "A" having red/blue parallel/latitudinal stripes, "B" having red/blue longitudinal meridians, and "C" having hypermeridional red/blue striping. How would an animation of each of those h-spheres passing through our space look?

I envision (and maybe all of this is wrong) all three A/B/C animations would show a tiny point appear, grow to a maximum radius sphere and shrink again until it disappears. I base this on a flatlander seeing a tiny dot grow to a maximum 2-d circle and shrink again before disappearing.

If the 2-d, 3-d relationship is described with just two "A" and "B" spheres (only latitude stripes and only meridian stripes respectively), the "A" sphere painted with latitude parallel stripes would travel through the 2-d plane and show itself growing/shrinking and and changing color all at once.

The "B" sphere painted with meridian longitude stripes would show itself growing/shrinking and stripe size/separation growing/shrinking proportionally.

When you switch your thinking to the 4-d/3-d scenario A hypersphere would travel through a 3-space and show "A" parallels as total-surface changing colors, "B" meridians as widening stripes, and "C" hypermeridians (or hyper parallels?) as .... what?.... a rotation?.... in which direction? .... meridians that flash/swap colors? ... something totally unexpected? I think my misunderstanding comes from trying to understand the "cell" analogy that assigns the name 8-cell to a hypercube (a 4-d volume with 8 different 3-d "hypersurface"/"cell"/"face"(s)). A hypertetrahedral pyramid is a 5-cell. A hypersphere is ... what? ... a 4-cell? a 1-cell? does it have one 3-dimensional "cell"/"face"?


Suppose a hypersphere were to pass through our three-space pole first. We would then see the intersection of that hypersphere with our three-space as a sequence of hyperparallels, each of which is an ordinary sphere, growing from zero radius to equatorial radius, back to zero radius as the hypersphere passed entirely through. The hypermeridians would appear as ordinary meridians of the parallels.

I'm not sure what distinction you're drawing between meridians and hypermeridians of the hypersphere, so I can't speak to that.

Next, suppose the hypersphere were to pass through our three-space equator first. We would see the hyperparallels together (although appearing and disappearing at different times), in parallel cross-sections from left to right (say), each as a circle that grows from zero radius to its own maximum and back to zero again. The equator would have the largest maximum radius, whereas the poles at either end would appear only instantaneously as a single point.

The hypermeridians would appear as a series of elliptical cross-sections, perpendicular to the hyperparallel cross-sections, each one growing from zero radius to the maximum radius of the hypersphere and back to zero again: some faster than others, but all appearing at the maximum radius at the same time. (They would be prolate spheroids if you assembled all the cross-sections, except possibly for one, if it happens to be at the point where the hypersphere first enters our space.)


In response to Brian's answer: I'm trying to avoid the mental gymnastics of having to rotate the hypersphere around some axis in the fourth dimension before shoving it through our three-space a second time, but I see how you're trying to avoid the same difficulty in thinking of painted stripes (maybe "painting a surface with stripes" doesn't carry as an analogy to higher dimensions). I guess I'm not understanding the wikipedia terminology of meridian vs hypermeridian (or your use of the term hyperparallel). If you always shove n-sphere through n-minus-1 dimensional space pole first:

A flat lander:
- sees "marked" parallels one at a time
- sees meridians as dots on the circle with increasing/decreasing separation

A 3-d person:
- sees hyper parallels as successively larger (and then smaller) spheres
- sees parallels as horizontally-oriented lines on the sphere with increasing/decreasing separation
- sees meridians as lines with increasing/decreasing separation
- sees hypermeridians as (?)

is hypermeridian synonymous with hyperparallel (seems like their should be 3 mutually orthogonal ways to "stripe" a hypersphere - no more no less)?

[hyperparallel not used here https://en.wikipedia.org/wiki/3-sphere]


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