Hyper4D

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Ball A = the w≥0 hemisphere of S³ (projected onto (x,y,z), losing w). Ball B = the w≤0 hemisphere, likewise. Their surfaces — each an ordinary S² sphere — are the same shared boundary: exactly the equatorial (w=0) cross-section from the “Cross-section” view, i.e. the sphere of maximum radius R. The yellow point marks a position on this boundary — the same position on both, since it's a single point, seen from both balls.sunt aceeași graniță comună: exact secțiunea de la ecuator (w=0) din vizualizarea „Secțiune", adică sfera de rază maximă R. Punctul galben marchează o poziție pe această graniță — aceeași poziție pe amândouă, pt că e un singur punct, văzut din ambele bile.

A second decomposition of S³, different from “2 glued balls”: two solid tori (filled donuts, D²×S¹), glued along their shared surface — an ordinary torus (the Clifford torus), not a sphere. It's genus 1 of the Heegaard splitting, versus genus 0 (the balls). Each solid torus renders normally, like an ordinary 3D donut — no special projection. The yellow point (φ,ψ) appears identically on both surfaces, because it's the same point of the shared boundary.

Like Bogdan's (double) wormy apple, generalized: each worm is a tunnel (a straight line in 4D) through the solid ball, with 2 ends on the “peel” (the boundary, the sphere of maximum radius R) — all verified exactly on the unit hypersphere. At a fixed cross-section you only see the points where the worms are right now — not the whole tunnel. With mystery mode on, the points become identical in appearance: the more worms there are, the harder it is to reconstruct connectivity just by watching — exactly the tomography problem from the start of the conversation, made worse by every extra worm.acum — nu tunelul întreg. Cu modul mister activat, punctele devin identice ca aspect: cu cât sunt mai mulți viermi, cu atât e mai greu să reconstruiești conectivitatea doar privind — exact problema tomografiei de la începutul conversației, agravată de fiecare vierme în plus.