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"?