Line packing in real projective space
Choose n lines through the origin of R^d so that the minimum angle between any two is as large as possible — equivalently, minimize the largest coherence μ = max |cos ∠(v_i, v_j)|.
Formal definition
- ContainerReal d-space R^d, every line through the origin; an answer is n points of real projective space RP^{d-1}
- SubmissionExactly n nonzero vectors, d coordinates each; a vector stands for the line it spans
- ObjectiveMinimize μ² = max (v_i·v_j)²/(|v_i|²|v_j|²), compared exactly by cross-multiplication — no normalization, no square roots
- ScoringThe record is ceil(μ²·10¹⁸), rounded against the submitter; the page shows μ, rounded up at the ninth decimal
Getting a feel for it
Another way to say it
At d = 4 a normalized nonzero vector is a unit quaternion, and q and −q are the same 3D rotation — so the d = 4 sub-problems ask for n maximally separated 3D orientations, a problem robotics and rendering actually use.
Where the frontier is
Grassmannian frames drive noise- and erasure-robust data representations, wireless communication and compressed sensing. Sloane's packing table maintains the best known values for these parameters and openly invites improvement; optimality proofs in d = 3 stop at n = 8.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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