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Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack $n$ unit circles into the smallest possible equilateral triangle. Optimal solutions are known for $n < 13$ and for any triangular number of circles, and conjectures are available for $n < 28$ .^[1]^[2]^[3]

A conjecture of Paul Erdős and Norman Oler states that, if $n$ is a triangular number, then the optimal packings of $n - 1$ and of $n$ circles have the same side length: that is, according to the conjecture, an optimal packing for $n - 1$ circles can be found by removing any single circle from the optimal hexagonal packing of $n$ circles.^[4] This conjecture is now known to be true for $n \leq 15$ .^[5]

Minimum solutions for the side length of the triangle:^[1]

Number of circles	Triangle number	Length	Area
1	Yes	$2{\sqrt {3}}$ = 3.464...	5.196...
2		$2+2{\sqrt {3}}$ = 5.464...	12.928...
3	Yes	$2+2{\sqrt {3}}$ = 5.464...	12.928...
4		$4{\sqrt {3}}$ = 6.928...	20.784...
5		$4+2{\sqrt {3}}$ = 7.464...	24.124...
6	Yes	$4+2{\sqrt {3}}$ = 7.464...	24.124...
7		$2+4{\sqrt {3}}$ = 8.928...	34.516...
8		$2+2{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {33}}$ = 9.293...	37.401...
9		$6+2{\sqrt {3}}$ = 9.464...	38.784...
10	Yes	$6+2{\sqrt {3}}$ = 9.464...	38.784...
11		$4+2{\sqrt {3}}+{\tfrac {4}{3}}{\sqrt {6}}$ = 10.730...	49.854...
12		$4+4{\sqrt {3}}$ = 10.928...	51.712...
13		$4+{\tfrac {10}{3}}{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {6}}$ = 11.406...	56.338...
14		$8+2{\sqrt {3}}$ = 11.464...	56.908...
15	Yes	$8+2{\sqrt {3}}$ = 11.464...	56.908...

A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.^[6]

References

^ ^a ^b Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", The American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928.
^ Melissen, J. B. M.; Schuur, P. C. (1995), "Packing 16, 17 or 18 circles in an equilateral triangle", Discrete Mathematics, 145 (1–3): 333–342, doi:10.1016/0012-365X(95)90139-C, MR 1356610.
^ Graham, R. L.; Lubachevsky, B. D. (1995), "Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond", Electronic Journal of Combinatorics, 2: Article 1, approx. 39 pp. (electronic), MR 1309122.
^ Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.
^ Payan, Charles (1997), "Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler", Discrete Mathematics (in French), 165/166: 555–565, doi:10.1016/S0012-365X(96)00201-4, MR 1439300.
^ Nurmela, Kari J. (2000), "Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles", Experimental Mathematics, 9 (2): 241–250, doi:10.1080/10586458.2000.10504649, MR 1780209, S2CID 45127090.

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[Melissen-1] Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", The American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928.

[2] Melissen, J. B. M.; Schuur, P. C. (1995), "Packing 16, 17 or 18 circles in an equilateral triangle", Discrete Mathematics, 145 (1–3): 333–342, doi:10.1016/0012-365X(95)90139-C, MR 1356610.

[3] Graham, R. L.; Lubachevsky, B. D. (1995), "Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond", Electronic Journal of Combinatorics, 2: Article 1, approx. 39 pp. (electronic), MR 1309122.

[4] Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.

[5] Payan, Charles (1997), "Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler", Discrete Mathematics (in French), 165/166: 555–565, doi:10.1016/S0012-365X(96)00201-4, MR 1439300.

[6] Nurmela, Kari J. (2000), "Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles", Experimental Mathematics, 9 (2): 241–250, doi:10.1080/10586458.2000.10504649, MR 1780209, S2CID 45127090.

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@@ Line 1: / Line 1: @@
+{{short description|Two-dimensional packing problem}}
-'''Circle packing in an equilateral triangle''' is a [[packing problem]] in [[discrete mathematics]] where the objective is to pack ''n'' unit circles into the smallest possible [[equilateral triangle]].  Optimal solutions are known for ''n''&nbsp;<&nbsp;13 and for any [[triangular number]] of circles, and conjectures are available for ''n''&nbsp;<&nbsp;28.<ref name="Melissen">{{citation
+'''Circle packing in an equilateral triangle''' is a [[packing problem]] in [[discrete mathematics]] where the objective is to pack {{mvar|n}} [[unit circle]]s into the smallest possible [[equilateral triangle]].  Optimal solutions are known for {{math|''n'' < 13}} and for any [[triangular number]] of circles, and conjectures are available for {{math|''n'' < 28}}.<ref name="Melissen">{{citation
  | last = Melissen | first = Hans
  | doi = 10.2307/2324212
@@ Line 8: / Line 10: @@
  | title = Densest packings of congruent circles in an equilateral triangle
  | volume = 100
- | year = 1993}}.</ref><ref>{{citation
+ | year = 1993| jstor = 2324212
+ }}.</ref><ref>{{citation
  | last1 = Melissen | first1 = J. B. M.
  | last2 = Schuur | first2 = P. C.
  | doi = 10.1016/0012-365X(95)90139-C
  | mr = 1356610
- | issue = 1-3
+ | issue = 1–3
  | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
  | pages = 333–342
  | title = Packing 16, 17 or 18 circles in an equilateral triangle
  | volume = 145
+ | year = 1995| url = https://research.utwente.nl/en/publications/packing-16-17-of-18-circles-in-an-equilateral-triangle(b2172f19-9654-4ff1-9af4-59da1b6bef3d).html
- | year = 1995}}.</ref><ref>{{citation
+ | doi-access = free
+ }}.</ref><ref>{{citation
  | last1 = Graham | first1 = R. L. | author1-link = Ronald Graham
  | last2 = Lubachevsky | first2 = B. D.
@@ Line 33: / Line 38: @@
  | doi = 10.4153/CMB-1961-018-7
  | mr = 0133065
- | journal = Canadian Mathematical Bulletin
+ | journal = [[Canadian Mathematical Bulletin]]
  | pages = 153–155
  | title = A finite packing problem
  | volume = 4
- | year = 1961}}.</ref> This conjecture is now known to be true for {{math|''n'' ≤ 15}}.<ref>{{citation
+ | year = 1961| issue = 2
+ | doi-access = free
+ }}.</ref> This conjecture is now known to be true for {{math|''n'' ≤ 15}}.<ref>{{citation
  | last = Payan | first = Charles
  | doi = 10.1016/S0012-365X(96)00201-4
@@ Line 46: / Line 53: @@
  | title = Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler
  | volume = 165/166
- | year = 1997}}.</ref>
+ | year = 1997| doi-access = free
+ }}.</ref>
 Minimum solutions for the side length of the triangle:<ref name="Melissen"/>
+{{Table alignment}}
-{| class="wikitable"
+{| class="wikitable defaultright col2center"
 |-
-! Number of circles
+! Number<br/> of circles
+! Triangle<br/> number
 ! Length
+! Area
+! Figure
 |-
 | 1
+| Yes
-| 3.464...
+| <math>2 \sqrt {3}</math> = 3.464...
+|align="right"| 5.196...
+|[[File:CircleInEquilateralTrianglePacking(1).png|center|220x220px]]
 |-
 | 2
+|
-| 5.464...
+| <math>2 + 2 \sqrt {3}</math> = 5.464...
+| 12.928...
+|[[File:Circle_packing_in_equilateral_triangle_for_2_circles.png|center|220x220px]]
 |-
 | 3
+| Yes
-| 5.464...
+| <math>2 + 2 \sqrt {3}</math> = 5.464...
+| 12.928...
+|[[File:Circle_packing_in_equilateral_triangle_for_3_circles.png|center|220x220px]]
 |-
 | 4
+|
-| 6.928... [[Image:4 cirkloj en 60 60 60 triangulo.png|120x120px]]
+| <math>4 \sqrt {3}</math> = 6.928...
+| 20.784...
+| [[File:Circle_packing_in_equilateral_triangle_for_4_circles.png|center|220x220px]]
 |-
 | 5
+|
-| 7.464... [[Image:5 cirkloj en 60 60 60 triangulo v1.png|130x130px]] [[Image:5 cirkloj en 60 60 60 triangulo v2.png|130x130px]]
+| <math>4 + 2 \sqrt {3}</math> = 7.464...
+| 24.124...
+| [[File:Circle_packing_in_equilateral_triangle_for_5_circles.png|center|220x220px]]
 |-
 | 6
+| Yes
-| 7.464...
+| <math>4 + 2 \sqrt {3}</math> = 7.464...
+|24.124...
+|[[File:Circle_packing_in_equilateral_triangle_for_6.png|center|220x220px]]
 |-
 | 7
+|
-| 8.928...
+| <math>2 + 4 \sqrt {3}</math> = 8.928...
+|34.516...
+|[[File:Circle_packing_in_equilateral_triangle_for_7_circles.png|center|220x220px]]
 |-
 | 8
+|
-| 9.293...
+| <math>2 + 2 \sqrt{3} + \tfrac {2} {3} \sqrt{33}</math> = 9.293...
+|37.401...
+|[[File:Circle packing in equilateral triangle for 8 circles.png|center|220x220px]]
 |-
 | 9
+|
-| 9.464...
+| <math>6 + 2 \sqrt {3}</math> = 9.464...
+|38.784...
+|[[File:Circle_packing_in_equilateral_triangle_for_9_circles.png|right|220x220px]]
 |-
 | 10
+| Yes
-| 9.464...
+| <math>6 + 2 \sqrt {3}</math> = 9.464...
+|38.784...
+|[[File:Circle_packing_in_equilateral_triangle_for_10_circles.png|right|220x220px]]
 |-
 | 11
+|
-| 10.730...
+| <math>4 + 2 \sqrt {3} + \tfrac {4} {3} \sqrt{6}</math> = 10.730...
+|49.854...
+|[[File:Ircle_packing_in_equilateral_triangle_for_11_circles.png|right|220x220px]]
 |-
 | 12
+|
-| 10.928...
+| <math>4 + 4 \sqrt {3}</math> = 10.928...
+|51.712...
+|[[File:Circle_packing_in_equilateral_triangle_for_12_circles.png|right|220x220px]]
 |-
 | 13
+|
-| 11.406...
+| <math>4 + \tfrac {10} {3} \sqrt{3} + \tfrac {2} {3} \sqrt{6}</math> = 11.406...
+|56.338...
+|[[File:Circle_packing_in_equilateral_triangle_for_13_circles.png|220x220px]]
 |-
 | 14
+|
-| 11.464...
+| <math>8 + 2 \sqrt {3}</math> = 11.464...
+|56.908...
+|[[File:Circle_packing_in_equilateral_triangle_for_14_circles.png|220x220px]]
 |-
 | 15
+| Yes
-| 11.464...
+| <math>8 + 2 \sqrt {3}</math> = 11.464...
+|56.908...
+|[[File:Circle_packing_in_equilateral_triangle_for_15_circles.png|220x220px]]
 |}
-A closely related problem is to cover the equilateral triangle with a given number of circles, having as small a radius as possible.<ref>{{citation
+A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.<ref>{{citation
  | last = Nurmela | first = Kari J.
  | mr = 1780209
@@ Line 110: / Line 167: @@
  | url = http://projecteuclid.org/getRecord?id=euclid.em/1045952348
  | volume = 9
- | year = 2000}}.</ref>
+ | year = 2000
+ | doi=10.1080/10586458.2000.10504649| s2cid = 45127090
+ }}.</ref>
 ==See also==
 *[[Circle packing in an isosceles right triangle]]
-*[[Malfatti circles]], a construction giving the optimal solution for three circles in an equilateral triangle
+*[[Malfatti circles]], three circles of possibly unequal sizes packed into a triangle
 ==References==
@@ Line 123: / Line 182: @@
 [[Category:Circle packing]]
-{{geometry-stub}}
+{{elementary-geometry-stub}}

v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma