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 = 0.0212105 square, n=16: A = 0.0205279 square, n=17: A = 0.0172617 square, n=18: A = 0.0155261 square, n=19: A = 0.0139479 square, n=20: A = 0.0129382 square, n=21: A = 0.0114738 square, n=22: A = 0.0106729 square, n=23: A = 0.00975729 square, n=24: A = 0.00908901 square, n=25: A = 0.00823007 square, n=26: A = 0.00757759 square, n=27: A = 0.0071024 square, n=28: A = 0.00690052 square, n=29: A = 0.00627594 square, n=30: A = 0.00603853 square, n=31: A = 0.00583707 square, n=32: A = 0.0058109 square, n=33: A = 0.00507134 square, n=34: A = 0.00483489 square, n=35: A = 0.00443229 square, n=36: A = 0.00418489 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 = 0.0183555 triangle, n=17: A = 0.0162423 triangle, n=18: A = 0.0148943 triangle, n=19: A = 0.0134819 triangle, n=20: A = 0.0126094 triangle, n=21: A = 0.011396 triangle, n=22: A = 0.0102661 triangle, n=23: A = 0.00926978 triangle, n=24: A = 0.00898515 triangle, n=25: A = 0.00780264 triangle, n=26: A = 0.00747157 triangle, n=27: A = 0.00702933 triangle, n=28: A = 0.00669434 triangle, n=29: A = 0.00612369 triangle, n=30: A = 0.0059574 triangle, n=31: A = 0.00516128 triangle, n=32: A = 0.00480076 triangle, n=33: A = 0.00472821 triangle, n=34: A = 0.00443091 triangle, n=35: A = 0.00403268 convex, n=3: A = 1 (proven) convex, n=4: A = 0.5 (proven) convex, n=5: A = 0.276393 (proven) convex, n=6: A = 0.166667 (proven) convex, n=7: A = 0.111111 (proven) convex, n=8: A = 0.0800001 (proven) 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 = 0.0309372 convex, n=14: A = 0.0278356 convex, n=15: A = 0.0245641 convex, n=16: A = 0.0222729 convex, n=17: A = 0.0190573 convex, n=18: A = 0.0182389 convex, n=19: A = 0.0159311 convex, n=20: A = 0.0144642 convex, n=21: A = 0.0131911 convex, n=22: A = 0.012255 convex, n=23: A = 0.0110825 convex, n=24: A = 0.0107126 convex, n=25: A = 0.00931611 convex, n=26: A = 0.00871479 convex, n=27: A = 0.00802912 convex, n=28: A = 0.00771884 convex, n=29: A = 0.00728912 convex, n=30: A = 0.00716018 convex, n=31: A = 0.00630967 convex, n=32: A = 0.00585071 convex, n=33: A = 0.00547014 convex, n=34: A = 0.00520622 convex, n=35: A = 0.00477381 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 = 4.772 square, n=16: n²A = 5.255 square, n=17: n²A = 4.989 square, n=18: n²A = 5.03 square, n=19: n²A = 5.035 square, n=20: n²A = 5.175 square, n=21: n²A = 5.06 square, n=22: n²A = 5.166 square, n=23: n²A = 5.162 square, n=24: n²A = 5.235 square, n=25: n²A = 5.144 square, n=26: n²A = 5.122 square, n=27: n²A = 5.178 square, n=28: n²A = 5.41 square, n=29: n²A = 5.278 square, n=30: n²A = 5.435 square, n=31: n²A = 5.609 square, n=32: n²A = 5.95 square, n=33: n²A = 5.523 square, n=34: n²A = 5.589 square, n=35: n²A = 5.43 square, n=36: n²A = 5.424 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 = 4.699 triangle, n=17: n²A = 4.694 triangle, n=18: n²A = 4.826 triangle, n=19: n²A = 4.867 triangle, n=20: n²A = 5.044 triangle, n=21: n²A = 5.026 triangle, n=22: n²A = 4.969 triangle, n=23: n²A = 4.904 triangle, n=24: n²A = 5.175 triangle, n=25: n²A = 4.877 triangle, n=26: n²A = 5.051 triangle, n=27: n²A = 5.124 triangle, n=28: n²A = 5.248 triangle, n=29: n²A = 5.15 triangle, n=30: n²A = 5.362 triangle, n=31: n²A = 4.96 triangle, n=32: n²A = 4.916 triangle, n=33: n²A = 5.149 triangle, n=34: n²A = 5.122 triangle, n=35: n²A = 4.94 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 = 5.508 convex, n=18: n²A = 5.909 convex, n=19: n²A = 5.751 convex, n=20: n²A = 5.786 convex, n=21: n²A = 5.817 convex, n=22: n²A = 5.931 convex, n=23: n²A = 5.863 convex, n=24: n²A = 6.17 convex, n=25: n²A = 5.823 convex, n=26: n²A = 5.891 convex, n=27: n²A = 5.853 convex, n=28: n²A = 6.052 convex, n=29: n²A = 6.13 convex, n=30: n²A = 6.444 convex, n=31: n²A = 6.064 convex, n=32: n²A = 5.991 convex, n=33: n²A = 5.957 convex, n=34: n²A = 6.018 convex, n=35: n²A = 5.848 convex square triangle Plotting the normalized product n² · A(n) highlights the gap between
asymptotic theory and finite-n behavior. In 1982, Komlós, Pintz, and Szemerédi
proved that A(n) ≥ c · (log n) / n², establishing that n² · A(n) must eventually grow
without bound. Meanwhile, the sharpest known upper bound, A(n) ≤ n−8/7 − 1/2000
(Cohen, Pohoata, and Zakharov, 2023), leaves substantial room between logarithmic growth
and polynomial decay. Within our computational window (n ≤ 36), the normalized curve
hovers in a stable band, demonstrating that asymptotic regimes remain distant at small n. 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: 25 tied minimal triangles square, n=16: 64 tied minimal triangles square, n=17: 27 tied minimal triangles square, n=18: 30 tied minimal triangles square, n=19: 31 tied minimal triangles square, n=20: 42 tied minimal triangles square, n=21: 37 tied minimal triangles square, n=22: 42 tied minimal triangles square, n=23: 40 tied minimal triangles square, n=24: 48 tied minimal triangles square, n=25: 44 tied minimal triangles square, n=26: 50 tied minimal triangles square, n=27: 47 tied minimal triangles square, n=28: 52 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: 64 tied minimal triangles square, n=33: 56 tied minimal triangles square, n=34: 62 tied minimal triangles square, n=35: 64 tied minimal triangles square, n=36: 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: 24 tied minimal triangles triangle, n=17: 28 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: 39 tied minimal triangles triangle, n=22: 39 tied minimal triangles triangle, n=23: 42 tied minimal triangles triangle, n=24: 48 tied minimal triangles triangle, n=25: 42 tied minimal triangles triangle, n=26: 48 tied minimal triangles triangle, n=27: 49 tied minimal triangles triangle, n=28: 54 tied minimal triangles triangle, n=29: 53 tied minimal triangles triangle, n=30: 57 tied minimal triangles triangle, n=31: 57 tied minimal triangles triangle, n=32: 59 tied minimal triangles triangle, n=33: 66 tied minimal triangles triangle, n=34: 60 tied minimal triangles triangle, n=35: 65 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: 9 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: 19 tied minimal triangles convex, n=14: 21 tied minimal triangles convex, n=15: 25 tied minimal triangles convex, n=16: 28 tied minimal triangles convex, n=17: 28 tied minimal triangles convex, n=18: 36 tied minimal triangles convex, n=19: 33 tied minimal triangles convex, n=20: 40 tied minimal triangles convex, n=21: 39 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: 45 tied minimal triangles convex, n=26: 46 tied minimal triangles convex, n=27: 48 tied minimal triangles convex, n=28: 48 tied minimal triangles convex, n=29: 52 tied minimal triangles convex, n=30: 55 tied minimal triangles convex, n=31: 55 tied minimal triangles convex, n=32: 56 tied minimal triangles convex, n=33: 61 tied minimal triangles convex, n=34: 62 tied minimal triangles convex, n=35: 70 tied minimal triangles convex square triangle The number of triangles that simultaneously achieve the minimal area (the critical set).
Understanding how the size of this tie set scales with n is an active open problem highlighted by
Sudermann-Merx (arXiv:2603.11107). Occasional drops below the general upward trend typically
indicate candidate configurations that have not yet fully converged to their highest-symmetry
optimal tie configurations, rather than genuine structural decreases in the mathematical optimum. State of the frontier
Proven optimal: square n ≤ 9, triangle n ≤ 8, convex n ≤ 8. Everything else is a best known configuration.