Square, n = 6 proven

A =

\frac{1}{8}

= 0.125000000000000000000000000000000000000000000…

6 minimal triangles in 2 congruence classes, colored by class.

The family of optima

The Dress–Yang–Zeng family: the shear x ↦ x + t(1−2y) slides the bottom pair right and the top pair left. A shear preserves every triangle's area, so all six minimal triangles stay at exactly ⅛ for the whole range t ∈ [0, ½]; the container is the only constraint. t = ½ is the 90°-rotated image of the original.

Symmetry

2-fold dihedral symmetry (group D2, order 4).

6 points in 2 orbits (4 + 2) — hover a point to see its orbit.

Minimal triangles

6 triangles tied (within relative 10⁻⁹) at the minimal area, in 2 congruence classes:

	count	side lengths	triangles (point indices)
	4	`0.5000 · 0.7071 · 1.1180`	(0,1,2) (0,3,4) (1,2,5) (3,4,5)
	2	`0.5000 · 0.5000 · 0.7071`	(0,1,4) (2,3,5)

Provenance

Proved optimal by A. Dress, L. Yang, and Z. B. Zeng, 1995.
One of an infinite family of solutions.
Coordinates from spiralulam/heilbronn (MIT): spiralulam/heilbronn config_n06.json — Dress, Yang, Zeng (1995).
Verified in exact arithmetic: all 20 triples enumerated, 6 tied at the minimum.

Downloads

points.txt coordinates, tab-separated
points.csv coordinates, CSV
points.json full record: value, provenance, verification
figure.svg this figure
family.json the optimal family: exact endpoints, 30-decimal sample members

References

A. Dress, L. Yang, Z. Zeng, Heilbronn problem for six points in a planar convex body, in Minimax and Applications, Springer (1995) 173–190