No Through Line

Every dot is a pair (h, ℓ): h is a position on a 5-cycle wrapped around the tube, ℓ is which of s stacked copies it lives on — literal floors of a cylindrical building. Drag to spin it, scroll to zoom (mouse/trackpad for now), click a dot to select it. A dot turns red the instant it lands on the shortest path between two other dots you've picked — "shortest" is computed on the real graph, wraparound included, not by eye. When that happens a red line lights up too, straight from the two dots it's between through the one that broke it: the actual route being killed, not just a dot changing colour.

the surprise

A general position set here can never beat 2·gp(H), where gp(H) is the best you can do on the 5-cycle alone — computed above, live, by brute force over all 32 of its subsets. That ceiling does not move when you drag the slider. The bundle can be three copies deep or thirty; the theorem this page is built on says the maximum is the same number either way. Try to beat it — you can't, but finding a clean set of exactly that size is the game.

compare

No global scoreboard — type a handle you know and this reads their own saved records, straight from their repo.

Rendered on the cylinder are only the cycle's own edges (the pentagon ring at each level) and the path's edges (the vertical struts) — the strong product also has diagonal edges, and the distance function below accounts for them correctly via the exact theorem for strong products (d = max(d_H, d_path)), they're just left off the drawing so it doesn't turn into a tangle. One thing the cylinder gets for free that the old flat grid couldn't: the wraparound from position 4 back to position 0 is just where the ring closes — no dashed "this bends around" line needed, because it actually does. Stack depth is capped at 6 here to keep every dot a real thumb-sized target; the paper's result holds for any depth from 2 upward, this is just as deep as a clear picture goes. The red route-lines drawn on a violation are a straight guide between the two endpoints and the dot between them, not literal drawn graph edges — the actual shortest path may run along diagonal strong-product edges this picture doesn't render (see above), so the line is honest about showing which three points, not which edges.