Best known packings of equal circles in an equilateral triangle

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Numbers

$n$

normalisation

$d_n$ or $S_n$

$d_n$:

comment: $d_2=1$.

$S_n$:

5.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_2=2+2\sqrt3$.

$d_n$:

comment: $d_3=1$.

$S_n$:

5.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_3=2+2\sqrt3$.

$d_n$:

0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252

comment: $d_4=\sqrt3/3$.

$S_n$:

6.928203230275509174109785366023489467771221015241522512223227917807732067635200148324584747028994303

comment: $S_4=4\sqrt3$.

$d_n$:

1/2

comment: $d_5=1/2$.

$S_n$:

7.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_5=4+2\sqrt3$.

$d_n$:

1/2

comment: $d_6=1/2$.

$S_n$:

7.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_6=4+2\sqrt3$.

$d_n$:

0.3660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $d_7=(\sqrt3-1)/2$.

$S_n$:

8.928203230275509174109785366023489467771221015241522512223227917807732067635200148324584747028994303

comment: $S_7=2+4\sqrt3$.

$d_n$:

0.3430703308172535824813264335273661647775330572478490459624846838207375901821755783783906706995985153

comment: $d_8=(\sqrt{33}-3)/8$.

$S_n$:

9.293810046163107026955300328491030946032453479609289501244865605947799848122536492180375950579022566

comment: $S_8=2+2\sqrt3+\frac{2}{3}\sqrt{33}$.

$d_n$:

1/3

comment: $d_9=1/3$.

$S_n$:

9.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_9=6+2\sqrt3$.

$d_n$:

1/3

comment: $d_{10}=1/3$.

$S_n$:

9.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $S_{10}=6+2\sqrt3$.

$d_n$:

0.2752551286084109509013579626470543040170262596716649357836537163745198112713424867300702834476798826

comment: $d_{11}=(3-\sqrt6)/2$.

$S_n$:

10.73008793884865871798460478261959992317354048182965476068853738190514653709402010954877161765401746

comment: $S_{11}=4+2\sqrt3+\frac{4}{3}\sqrt6$.

$d_n$:

0.2679491924311227064725536584941276330571947461896193719441930205480669830911999629188538132427514243

comment: $d_{12}=2-\sqrt3$.

$S_n$:

10.92820323027550917410978536602348946777122101524152251222322791780773206763520014832458474702899430

comment: $S_{12}=4+4\sqrt3$.

$d_n$:

0.2518132366530604814824063762582067648231888009114398492666195453063684651280394548070862563133657362

comment: $d_{13}=3/(6+2\sqrt3+\sqrt6)$.

$S_n$:

11.40649585375170971055634385482350215111998249980571551247448497634041697466754347463039357792725541

comment: $S_{13}=4+\frac{10}{3}\sqrt3+\frac{2}{3}\sqrt6$.

$d_n$:

1/4

comment: $d_{14}=1/4$.

$S_n$:

11.46410161513775458705489268301174473388561050762076125611161395890386603381760007416229237351449715

comment: $S_{14}=8+2\sqrt3$.

$d_n$:

1/4

comment: $d_{15}=1/4$.

$S_n$:

11.46410161513775458705489268301174473388561050762076125611161395890386603381760007416229237351449715

comment: $S_{15}=8+2\sqrt3$.

Definition

For $n\geq2$, $d_n$ is the largest possible minimum distance among $n$ points in a unit-side equilateral triangle. The side length $S_n=2\sqrt3+2/d_n$ is the smallest equilateral triangle that holds $n$ non-overlapping circles of radius 1. Each stored packing is listed in these two invertible normalisations.

Parameters

$n$

— number of circles ($n\geq2$)

normalisation

— whether the row holds the separation $d_n$ or the container side length $S_n$ (either separation or container)

Formulas

(1)

$S_n=2\sqrt3+2/d_n$, equivalently $d_n=2/(S_n-2\sqrt3)$.

(2)

If $n=k(k+1)/2$ with $k\geq2$, then $d_n=1/(k-1)$ and $S_n=2(k-1)+2\sqrt3$.

Comments

(3)

Each $d_n$ row measures points in a unit-side equilateral triangle. Each $S_n$ row gives the side length of the smallest equilateral triangle holding $n$ circles of radius 1. The table begins at $n=2$ because $d_1$ is undefined; the source's one-circle container value $S_1=2\sqrt3$ is not stored.

(4)

The repeated pairs $d_2=d_3$, $d_5=d_6$, $d_9=d_{10}$, and $d_{14}=d_{15}$ are the stored Erdős-Oler cases [4]. For these values, where $n=k(k+1)/2$, the optimal packing of $n-1$ circles has the same side length as the triangular packing of $n$ circles.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=64))

def triangular_packing(k):
    k = ZZ(k)
    if k < 2:
        raise ValueError('k must be at least 2')
    n = k * (k + 1) // 2
    d = R(1) / R(k - 1)
    S = R(2) * R(k - 1) + R(2) * R(3).sqrt()
    return n, d, S

print(triangular_packing(6))

References

[1]

H. Melissen, Densest packings of congruent circles in an equilateral triangle, The American Mathematical Monthly 100 (1993), no. 10, 916-925. (doi) (MR)

[2]

J. B. M. Melissen, Optimal packings of eleven equal circles in an equilateral triangle, Acta Mathematica Hungarica 65 (1994), no. 4, 389-393. (doi)

[3]

A. Joós, Packing 13 circles in an equilateral triangle, Aequationes Mathematicae 95 (2021), no. 1, 35-65. (doi)

[4]

C. Payan, Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler, Discrete Mathematics 165/166 (1997), 555-565. (doi) (MR)

Links

[5]

Wikipedia: Circle packing in an equilateral triangle

Similar tables

Best known packings of equal circles in a circle — the same finite equal-circle packing problem in a disk rather than in an equilateral triangle

Densities of the densest known lattice sphere packings — stores infinite lattice sphere packing densities rather than best known finite configurations in a bounded container

Kissing numbers — another finite extremal packing problem for equal spheres

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds both normalisations for every $2\leq n\leq15$, the range for which [5] gives an exact side length and optimality is proven)

How they were obtained:

Each container row is computed in Sage real ball arithmetic from the exact radical formula transcribed from [5], read on 23-Sep-2026. Each transcribed formula was checked against the decimal printed for the same row in the source table. The separation rows are computed from Formula (1), and the generator checks that converting them back gives the same $S_n$.

Optimality is proven for every stored row: [1] covers $2\leq n\leq10$, $n=12$, and the stored triangular rows; [2] proves $n=11$; [3] proves $n=13$; and [4] proves the Erdős-Oler case $n=14$.