Square, n = 16 record
Symmetry
180° Rotationally symmetric (group C2, order 2).
16 points in 8 orbits (2 + 2 + 2 + 2 + 2 + 2 + 2 + 2) — hover a point to see its orbit.
Tesseract structure
This configuration is an affine image of the 16 vertices of the 4-dimensional hypercube. The animation rotates the tesseract, freezes a projection, stretches it into the optimal Heilbronn placement, then reveals the 64 minimal triangles grouped by parallelogram face.
About the structure
The 16 points have rational coordinates (denominators 31 and 33), 180-degree rotational symmetry, and exactly 64 triangles tied at area 7/341. Eight axis-aligned tesseract faces each contain 4 minimal triangles; six diagonal parallelograms contribute the remaining 24 among the 64.
Minimal triangles
64 triangles tied (within relative 10−9) at the minimal area, in 15 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 8 | 0.3098 · 0.3395 · 0.6364 |
(0,4,15) (1,5,14) (2,9,12) (3,8,13) (4,7,15) (5,6,14) (8,10,13) (9,11,12) |
| 8 | 0.2651 · 0.4237 · 0.6774 |
(0,2,10) (1,3,11) (2,6,10) (3,7,11) (4,8,12) (5,9,13) (8,12,14) (9,13,15) |
| 8 | 0.2651 · 0.6774 · 0.9374 |
(0,2,6) (0,6,10) (1,3,7) (1,7,11) (4,8,14) (4,12,14) (5,9,15) (5,13,15) |
| 8 | 0.3098 · 0.6364 · 0.9416 |
(0,4,7) (0,7,15) (1,5,6) (1,6,14) (2,9,11) (2,11,12) (3,8,10) (3,10,13) |
| 4 | 0.3098 · 0.3344 · 0.6309 |
(6,9,14) (7,8,15) (8,10,15) (9,11,14) |
| 4 | 0.2651 · 0.4297 · 0.6837 |
(0,10,12) (1,11,13) (10,12,14) (11,13,15) |
| 4 | 0.3395 · 0.4119 · 0.7432 |
(0,8,13) (1,9,12) (2,5,14) (3,4,15) |
| 4 | 0.3098 · 0.6309 · 0.9361 |
(6,9,11) (6,11,14) (7,8,10) (7,10,15) |
| 4 | 0.2651 · 0.6837 · 0.9438 |
(0,10,14) (0,12,14) (1,11,15) (1,13,15) |
| 2 | 0.2651 · 0.5554 · 0.8134 |
(4,8,9) (5,8,9) |
| 2 | 0.4035 · 0.5302 · 0.9294 |
(6,12,13) (7,12,13) |
| 2 | 0.2651 · 0.8134 · 1.0748 |
(4,5,8) (4,5,9) |
| 2 | 0.4035 · 0.6977 · 1.0985 |
(6,12,15) (7,13,14) |
| 2 | 0.4119 · 0.7123 · 1.1217 |
(0,8,11) (1,9,10) |
| 2 | 0.4035 · 0.9294 · 1.3313 |
(6,7,12) (6,7,13) |
Provenance
- Found by Mark Beyleveld, August 2006.
- Coordinates exact, as published in:
arXiv:2603.11107— Beyleveld's all-rational configuration, value exactly 7/341. - Verified in exact arithmetic: all 560 triples enumerated, 64 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
