Trends
The record tables as data. Every point is a configuration on this site,
exactly verified; larger dots are proven optimal, and each point carries its
value on hover.
Best known A(n), unit-area container 1 0.3 0.1 0.03 0.01 0.003 5 10 15 20 25 30 35 n A(n), log scale square, n=3: A = 0.5 (proven) square, n=4: A = 0.5 (proven) square, n=5: A = 0.19245 (proven) square, n=6: A = 0.125 (proven) square, n=7: A = 0.083859 (proven) square, n=8: A = 0.0723764 (proven) square, n=9: A = 0.054876 (proven) square, n=10: A = 0.0465374 square, n=11: A = 0.037037 square, n=12: A = 0.0325989 square, n=13: A = 0.0270199 square, n=14: A = 0.024304 square, n=15: A = 0.0211062 square, n=16: A = 0.0205279 square, n=17: A = 0.0172617 square, n=18: A = 0.0144327 square, n=19: A = 0.0133862 square, n=20: A = 0.0129116 square, n=21: A = 0.0110003 square, n=22: A = 0.0104725 square, n=23: A = 0.00881275 square, n=24: A = 0.00849501 square, n=25: A = 0.00740718 square, n=26: A = 0.00682685 square, n=27: A = 0.00679035 square, n=28: A = 0.00677581 square, n=29: A = 0.00561967 square, n=30: A = 0.00544512 square, n=31: A = 0.00539037 square, n=32: A = 0.00469033 square, n=33: A = 0.00414638 square, n=34: A = 0.00400476 square, n=35: A = 0.00370249 triangle, n=3: A = 1 (proven) triangle, n=4: A = 0.333333 (proven) triangle, n=5: A = 0.171573 (proven) triangle, n=6: A = 0.125 (proven) triangle, n=7: A = 0.0972222 (proven) triangle, n=8: A = 0.0677892 (proven) triangle, n=9: A = 0.0548469 triangle, n=10: A = 0.0433767 triangle, n=11: A = 0.0365299 triangle, n=12: A = 0.0310048 triangle, n=13: A = 0.0265565 triangle, n=14: A = 0.0237758 triangle, n=15: A = 0.0210908 triangle, n=16: A = 0.0179763 triangle, n=17: A = 0.0155337 triangle, n=18: A = 0.0148943 triangle, n=19: A = 0.0117053 triangle, n=20: A = 0.0111378 triangle, n=21: A = 0.0106229 triangle, n=22: A = 0.00931323 triangle, n=23: A = 0.00909416 triangle, n=24: A = 0.00898515 triangle, n=25: A = 0.00699387 triangle, n=26: A = 0.00649165 triangle, n=27: A = 0.00636118 triangle, n=28: A = 0.00570263 triangle, n=29: A = 0.0054101 triangle, n=30: A = 0.00509805 triangle, n=31: A = 0.00475595 triangle, n=32: A = 0.00417795 triangle, n=33: A = 0.00396743 triangle, n=34: A = 0.00375818 triangle, n=35: A = 0.00360623 convex, n=3: A = 1 (proven) convex, n=4: A = 0.5 (proven) convex, n=5: A = 0.276393 convex, n=6: A = 0.166667 convex, n=7: A = 0.111111 (proven) convex, n=8: A = 0.0800001 convex, n=9: A = 0.0640648 convex, n=10: A = 0.0519931 convex, n=11: A = 0.0425532 convex, n=12: A = 0.0392157 convex, n=13: A = 0.0309369 convex, n=14: A = 0.0278356 convex, n=15: A = 0.0245641 convex, n=16: A = 0.0222729 convex, n=17: A = 0.0187315 convex, n=18: A = 0.0182389 convex, n=19: A = 0.0150027 convex, n=20: A = 0.0144642 convex, n=21: A = 0.0127252 convex, n=22: A = 0.012255 convex, n=23: A = 0.0110825 convex, n=24: A = 0.0107126 convex, n=25: A = 0.00887742 convex, n=26: A = 0.00840327 convex, n=27: A = 0.00780376 convex, n=28: A = 0.00762772 convex, n=29: A = 0.00607882 convex, n=30: A = 0.00594247 convex, n=31: A = 0.00521473 convex, n=32: A = 0.00488672 convex, n=33: A = 0.00461434 convex, n=34: A = 0.00441352 convex, n=35: A = 0.0040567 convex square triangle Best known values fall roughly like n−2 — the straight-line
decay on the log scale. The three variants order as convex > triangle ≥
square for small n and converge as n grows. n²·A(n) — the asymptotic view 0 2 4 6 8 10 5 10 15 20 25 30 35 n n²·A(n) square, n=3: n²A = 4.5 square, n=4: n²A = 8 square, n=5: n²A = 4.811 square, n=6: n²A = 4.5 square, n=7: n²A = 4.109 square, n=8: n²A = 4.632 square, n=9: n²A = 4.445 square, n=10: n²A = 4.654 square, n=11: n²A = 4.481 square, n=12: n²A = 4.694 square, n=13: n²A = 4.566 square, n=14: n²A = 4.764 square, n=15: n²A = 4.749 square, n=16: n²A = 5.255 square, n=17: n²A = 4.989 square, n=18: n²A = 4.676 square, n=19: n²A = 4.832 square, n=20: n²A = 5.165 square, n=21: n²A = 4.851 square, n=22: n²A = 5.069 square, n=23: n²A = 4.662 square, n=24: n²A = 4.893 square, n=25: n²A = 4.629 square, n=26: n²A = 4.615 square, n=27: n²A = 4.95 square, n=28: n²A = 5.312 square, n=29: n²A = 4.726 square, n=30: n²A = 4.901 square, n=31: n²A = 5.18 square, n=32: n²A = 4.803 square, n=33: n²A = 4.515 square, n=34: n²A = 4.63 square, n=35: n²A = 4.536 triangle, n=3: n²A = 9 triangle, n=4: n²A = 5.333 triangle, n=5: n²A = 4.289 triangle, n=6: n²A = 4.5 triangle, n=7: n²A = 4.764 triangle, n=8: n²A = 4.339 triangle, n=9: n²A = 4.443 triangle, n=10: n²A = 4.338 triangle, n=11: n²A = 4.42 triangle, n=12: n²A = 4.465 triangle, n=13: n²A = 4.488 triangle, n=14: n²A = 4.66 triangle, n=15: n²A = 4.745 triangle, n=16: n²A = 4.602 triangle, n=17: n²A = 4.489 triangle, n=18: n²A = 4.826 triangle, n=19: n²A = 4.226 triangle, n=20: n²A = 4.455 triangle, n=21: n²A = 4.685 triangle, n=22: n²A = 4.508 triangle, n=23: n²A = 4.811 triangle, n=24: n²A = 5.175 triangle, n=25: n²A = 4.371 triangle, n=26: n²A = 4.388 triangle, n=27: n²A = 4.637 triangle, n=28: n²A = 4.471 triangle, n=29: n²A = 4.55 triangle, n=30: n²A = 4.588 triangle, n=31: n²A = 4.57 triangle, n=32: n²A = 4.278 triangle, n=33: n²A = 4.321 triangle, n=34: n²A = 4.344 triangle, n=35: n²A = 4.418 convex, n=3: n²A = 9 convex, n=4: n²A = 8 convex, n=5: n²A = 6.91 convex, n=6: n²A = 6 convex, n=7: n²A = 5.444 convex, n=8: n²A = 5.12 convex, n=9: n²A = 5.189 convex, n=10: n²A = 5.199 convex, n=11: n²A = 5.149 convex, n=12: n²A = 5.647 convex, n=13: n²A = 5.228 convex, n=14: n²A = 5.456 convex, n=15: n²A = 5.527 convex, n=16: n²A = 5.702 convex, n=17: n²A = 5.413 convex, n=18: n²A = 5.909 convex, n=19: n²A = 5.416 convex, n=20: n²A = 5.786 convex, n=21: n²A = 5.612 convex, n=22: n²A = 5.931 convex, n=23: n²A = 5.863 convex, n=24: n²A = 6.17 convex, n=25: n²A = 5.548 convex, n=26: n²A = 5.681 convex, n=27: n²A = 5.689 convex, n=28: n²A = 5.98 convex, n=29: n²A = 5.112 convex, n=30: n²A = 5.348 convex, n=31: n²A = 5.011 convex, n=32: n²A = 5.004 convex, n=33: n²A = 5.025 convex, n=34: n²A = 5.102 convex, n=35: n²A = 4.969 convex square triangle Multiplying by n² makes the conjectured behavior visible:
Komlós–Pintz–Szemerédi proved A(n) ≥ c·log n / n², so n²·A(n) should grow
slowly without bound; the best known upper bound (Cohen–Pohoata–Zakharov 2023)
decays as n−8/7−1/2000 ·n² for large n. In this range the curve drifts in a
narrow band — the asymptotics are nowhere in sight at n ≤ 35. Tied minimal triangles per configuration 0 10 20 30 40 50 60 70 5 10 15 20 25 30 35 n count of tied minima square, n=3: 1 tied minimal triangles square, n=4: 4 tied minimal triangles square, n=5: 4 tied minimal triangles square, n=6: 6 tied minimal triangles square, n=7: 8 tied minimal triangles square, n=8: 12 tied minimal triangles square, n=9: 11 tied minimal triangles square, n=10: 16 tied minimal triangles square, n=11: 28 tied minimal triangles square, n=12: 20 tied minimal triangles square, n=13: 20 tied minimal triangles square, n=14: 26 tied minimal triangles square, n=15: 24 tied minimal triangles square, n=16: 64 tied minimal triangles square, n=17: 27 tied minimal triangles square, n=18: 29 tied minimal triangles square, n=19: 32 tied minimal triangles square, n=20: 36 tied minimal triangles square, n=21: 36 tied minimal triangles square, n=22: 40 tied minimal triangles square, n=23: 39 tied minimal triangles square, n=24: 56 tied minimal triangles square, n=25: 42 tied minimal triangles square, n=26: 39 tied minimal triangles square, n=27: 49 tied minimal triangles square, n=28: 56 tied minimal triangles square, n=29: 51 tied minimal triangles square, n=30: 54 tied minimal triangles square, n=31: 56 tied minimal triangles square, n=32: 58 tied minimal triangles square, n=33: 61 tied minimal triangles square, n=34: 62 tied minimal triangles square, n=35: 65 tied minimal triangles triangle, n=3: 1 tied minimal triangles triangle, n=4: 3 tied minimal triangles triangle, n=5: 4 tied minimal triangles triangle, n=6: 6 tied minimal triangles triangle, n=7: 9 tied minimal triangles triangle, n=8: 11 tied minimal triangles triangle, n=9: 13 tied minimal triangles triangle, n=10: 14 tied minimal triangles triangle, n=11: 17 tied minimal triangles triangle, n=12: 21 tied minimal triangles triangle, n=13: 20 tied minimal triangles triangle, n=14: 23 tied minimal triangles triangle, n=15: 15 tied minimal triangles triangle, n=16: 27 tied minimal triangles triangle, n=17: 29 tied minimal triangles triangle, n=18: 33 tied minimal triangles triangle, n=19: 30 tied minimal triangles triangle, n=20: 35 tied minimal triangles triangle, n=21: 37 tied minimal triangles triangle, n=22: 38 tied minimal triangles triangle, n=23: 41 tied minimal triangles triangle, n=24: 48 tied minimal triangles triangle, n=25: 45 tied minimal triangles triangle, n=26: 47 tied minimal triangles triangle, n=27: 47 tied minimal triangles triangle, n=28: 51 tied minimal triangles triangle, n=29: 53 tied minimal triangles triangle, n=30: 55 tied minimal triangles triangle, n=31: 57 tied minimal triangles triangle, n=32: 59 tied minimal triangles triangle, n=33: 60 tied minimal triangles triangle, n=34: 62 tied minimal triangles triangle, n=35: 64 tied minimal triangles convex, n=3: 1 tied minimal triangles convex, n=4: 4 tied minimal triangles convex, n=5: 5 tied minimal triangles convex, n=6: 6 tied minimal triangles convex, n=7: 6 tied minimal triangles convex, n=8: 10 tied minimal triangles convex, n=9: 11 tied minimal triangles convex, n=10: 16 tied minimal triangles convex, n=11: 28 tied minimal triangles convex, n=12: 36 tied minimal triangles convex, n=13: 15 tied minimal triangles convex, n=14: 16 tied minimal triangles convex, n=15: 25 tied minimal triangles convex, n=16: 28 tied minimal triangles convex, n=17: 29 tied minimal triangles convex, n=18: 36 tied minimal triangles convex, n=19: 32 tied minimal triangles convex, n=20: 40 tied minimal triangles convex, n=21: 37 tied minimal triangles convex, n=22: 38 tied minimal triangles convex, n=23: 38 tied minimal triangles convex, n=24: 48 tied minimal triangles convex, n=25: 44 tied minimal triangles convex, n=26: 45 tied minimal triangles convex, n=27: 45 tied minimal triangles convex, n=28: 49 tied minimal triangles convex, n=29: 52 tied minimal triangles convex, n=30: 54 tied minimal triangles convex, n=31: 61 tied minimal triangles convex, n=32: 59 tied minimal triangles convex, n=33: 60 tied minimal triangles convex, n=34: 61 tied minimal triangles convex, n=35: 65 tied minimal triangles square convex triangle How many triangles tie for the minimum, per configuration —
growth of the critical set is an open question posed by Sudermann-Merx
(arXiv:2603.11107); this is the complete census across all three variants.
Dips below the trend usually mean under-converged coordinates rather than
mathematics: rows whose records aren't public yet sit visibly low. State of the frontier
Proven optimal: square n ≤ 9, triangle n ≤ 8, convex n ≤ 7. Everything else is a best known configuration.
Three rows (square 18, convex 32, convex 35) have records whose coordinates and current figures are not public; the site shows the best verifiable configurations until they surface.
Four verified improvements are ahead of Friedman's pages: square 17, 21, 22 (cnemri, via AlphaEvolve) and square 25 (this site).