Triangle, n = 16 record
A = 0.018355546355403…
full precision
Exact value of the published coordinate literals:
0.0183555463554034501359529610459
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
24 triangles tied (within relative 10−9) at the minimal area, in 24 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1362 · 0.2699 · 0.2949 |
(3,4,7) |
| 1 | 0.2622 · 0.2689 · 0.5113 |
(0,9,14) |
| 1 | 0.2715 · 0.3524 · 0.6121 |
(0,10,12) |
| 1 | 0.2847 · 0.4302 · 0.7070 |
(10,14,15) |
| 1 | 0.3148 · 0.4024 · 0.7096 |
(0,8,15) |
| 1 | 0.2847 · 0.4756 · 0.7536 |
(9,10,15) |
| 1 | 0.2689 · 0.5091 · 0.7715 |
(0,5,14) |
| 1 | 0.2894 · 0.5113 · 0.7949 |
(0,9,13) |
| 1 | 0.2699 · 0.5794 · 0.8442 |
(2,3,4) |
| 1 | 0.2749 · 0.5794 · 0.8493 |
(2,4,5) |
| 1 | 0.1362 · 0.7547 · 0.8833 |
(3,7,12) |
| 1 | 0.2894 · 0.6055 · 0.8906 |
(9,12,13) |
| 1 | 0.2622 · 0.7181 · 0.9767 |
(7,9,14) |
| 1 | 0.2624 · 0.7332 · 0.9921 |
(8,11,14) |
| 1 | 0.4024 · 0.6652 · 1.0652 |
(0,3,15) |
| 1 | 0.4856 · 0.5939 · 1.0773 |
(5,7,11) |
| 1 | 0.3816 · 0.7086 · 1.0879 |
(5,8,9) |
| 1 | 0.3210 · 0.7715 · 1.0901 |
(0,5,6) |
| 1 | 0.2847 · 0.8367 · 1.1188 |
(2,10,15) |
| 1 | 0.4756 · 0.6519 · 1.1256 |
(6,9,15) |
| 1 | 0.3816 · 0.7536 · 1.1332 |
(5,9,10) |
| 1 | 0.6364 · 0.6393 · 1.2745 |
(3,11,13) |
| 1 | 0.4146 · 0.9074 · 1.3206 |
(6,8,13) |
| 1 | 0.5587 · 0.8442 · 1.4018 |
(2,3,6) |
Provenance
- Found by Fable 5.1 with Shengtong Zhang, September 2026.
- Coordinates by an external contributor, re-verified here:
Fable51 search campaign, September 2026: symmetry-reduced basin hopping (orbit parametrization, greedy ruin-and-recreate, soft-min warm-up, trust-region SLP) and insertion ladders from neighbouring records; tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— not symmetric; 24 tied minimal triangles. - Verified in exact arithmetic: all 560 triples enumerated, 24 tied at the minimum.
Record history
- 2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the smallest positive root of 1276816311639009084301936704A²⁰ − 3515727588188083487082083808A¹⁹ + 4504481674986298309572338856A¹⁸ − 3544794434601911074726581732A¹⁷ + 1897632078973176461579058411A¹⁶ − 720220168753873110033931350A¹⁵ + 194300881473767212177429833A¹⁴ − 35459425164744613336616820A¹³ + 3505481002013476904379234A¹² + 121742597444124026011824A¹¹ − 100110321847615601443203A¹⁰ + 14462072044899214707693A⁹ − 513488859485215523454A⁸ − 134719466800524854175A⁷ + 23564054226435925338A⁶ − 1686606722394967080A⁵ + 46806275724363621A⁴ + 1113735737767624A³ − 113864012350998A² + 2761817732010A − 21754092988 = 0.017976275987235… (this site, from the symmetry-reduced tie system)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
