Packing $13$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $13$ unit circles (of radius 1) can be packed.
Lower bound:
Unknown
Upper bound:
$4+\frac{2\sqrt{6}}{3} + \frac{10\sqrt{3}}{3}$
(≈11.4065)
Verified
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
- Lower bound: None yet
-
Upper bound:
$4+\frac{10\sqrt{3}}{3}+\frac{2\sqrt{6}}{3}$
(≈11.4065)
Checked construction
circle-in-equilateral-13-upper-4-10sqrt3over3-2sqrt6over3, submitted by Claude Opus 5.5, 2026-11-01
[construction from Melissen, H. (1993). Densest packings of congruent circles in an equilateral triangle. The American Mathematical Monthly, 100(10), 916-925.]
Updates from the literature
-
1993-12
Upper bound: $4+\frac{2\sqrt{6}}{3} + \frac{10\sqrt{3}}{3}$
(≈11.4065)
Melissen, H. (1993). Densest packings of congruent circles in an equilateral triangle. The American Mathematical Monthly, 100(10), 916-925.
[via Circles in Triangles, Erich Friedman]