Packing $14$ unit circles in an equilateral triangle

The side length of the smallest equilateral triangle into which $14$ unit circles (of radius 1) can be packed.

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: $8+2\sqrt{3}$ (≈11.4641) Checked construction
  circle-in-equilateral-14-upper-8-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01
  [construction from Groemer, H. (1960). Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Z., 73, 285-294; Oler, N. (1961). A finite packing problem. Canad. Math. Bull., 4, 153-155.]

Updates from the literature

• 1961-05 Upper bound: $8+2\sqrt{3}$ (≈11.4641)
  Groemer, H. (1960). Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Z., 73, 285-294; Oler, N. (1961). A finite packing problem. Canad. Math. Bull., 4, 153-155.
  [via Circles in Triangles, Erich Friedman]
• 1997-03 Lower bound: $8+2\sqrt{3}$ (≈11.4641)
  Payan, C. (1997). Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler. Discrete Mathematics, 165-166, 555-565.
  [via Circle packing in an equilateral triangle - Wikipedia]