Circle packing in an equilateral triangle

{{unsolved|mathematics|What is the smallest possible equilateral triangle which an amount {{mvar|n}} of unit circles can be packed into?}}

Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack {{mvar|n}} unit circles 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}}.{{citation

| last = Melissen | first = Hans

| doi = 10.2307/2324212

| mr = 1252928

| issue = 10

| journal = The American Mathematical Monthly

| pages = 916–925

| title = Densest packings of congruent circles in an equilateral triangle

| volume = 100

| year = 1993| jstor = 2324212

}}.{{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

| 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

| doi-access = free

}}.{{citation

| last1 = Graham | first1 = R. L. | author1-link = Ronald Graham

| last2 = Lubachevsky | first2 = B. D.

| mr = 1309122

| journal = Electronic Journal of Combinatorics

| page = Article 1, approx. 39 pp. (electronic)

| title = Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond

| url = http://www.combinatorics.org/Volume_2/Abstracts/v2i1a1.html

| volume = 2

| year = 1995}}.

A conjecture of Paul Erdős and Norman Oler states that, if {{mvar|n}} is a triangular number, then the optimal packings of {{math|n − 1}} and of {{mvar|n}} circles have the same side length: that is, according to the conjecture, an optimal packing for {{math|n − 1}} circles can be found by removing any single circle from the optimal hexagonal packing of {{mvar|n}} circles.{{citation

| last = Oler | first = Norman

| doi = 10.4153/CMB-1961-018-7

| mr = 0133065

| journal = Canadian Mathematical Bulletin

| pages = 153–155

| title = A finite packing problem

| volume = 4

| year = 1961| issue = 2

| doi-access = free

}}. This conjecture is now known to be true for {{math|n ≤ 15}}.{{citation

| last = Payan | first = Charles

| doi = 10.1016/S0012-365X(96)00201-4

| mr = 1439300

| journal = Discrete Mathematics

| language = French

| pages = 555–565

| title = Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler

| volume = 165/166

| year = 1997| doi-access = free

}}.

Minimum solutions for the side length of the triangle:

class="wikitable defaultright col2center"
Number of circles ! Triangle number ! Length ! Area ! Figure
1 \| Yes \| $2 \sqrt {3}$ = 3.464... \|align="right"\| 5.196... \|File:CircleInEquilateralTrianglePacking(1).png
2 \| \| $2 + 2 \sqrt {3}$ = 5.464... \| 12.928... \|File:Circle_packing_in_equilateral_triangle_for_2_circles.png
3 \| Yes \| $2 + 2 \sqrt {3}$ = 5.464... \| 12.928... \|File:Circle_packing_in_equilateral_triangle_for_3_circles.png
4 \| \| $4 \sqrt {3}$ = 6.928... \| 20.784... \| File:Circle_packing_in_equilateral_triangle_for_4_circles.png
5 \| \| $4 + 2 \sqrt {3}$ = 7.464... \| 24.124... \| File:Circle_packing_in_equilateral_triangle_for_5_circles.png
6 \| Yes \| $4 + 2 \sqrt {3}$ = 7.464... \|24.124... \|File:Circle_packing_in_equilateral_triangle_for_6.png
7 \| \| $2 + 4 \sqrt {3}$ = 8.928... \|34.516... \|File:Circle_packing_in_equilateral_triangle_for_7_circles.png
8 \| \| $2 + 2 \sqrt{3} + \tfrac {2} {3} \sqrt{33}$ = 9.293... \|37.401... \|File:Circle packing in equilateral triangle for 8 circles.png
9 \| \| $6 + 2 \sqrt {3}$ = 9.464... \|38.784... \|File:Circle_packing_in_equilateral_triangle_for_9_circles.png
10 \| Yes \| $6 + 2 \sqrt {3}$ = 9.464... \|38.784... \|File:Circle_packing_in_equilateral_triangle_for_10_circles.png
11 \| \| $4 + 2 \sqrt {3} + \tfrac {4} {3} \sqrt{6}$ = 10.730... \|49.854... \|File:Ircle_packing_in_equilateral_triangle_for_11_circles.png
12 \| \| $4 + 4 \sqrt {3}$ = 10.928... \|51.712... \|File:Circle_packing_in_equilateral_triangle_for_12_circles.png
13 \| \| $4 + \tfrac {10} {3} \sqrt{3} + \tfrac {2} {3} \sqrt{6}$ = 11.406... \|56.338... \|220x220px
14 \| \| $8 + 2 \sqrt {3}$ = 11.464... \|56.908... \|220x220px
15 \| Yes \| $8 + 2 \sqrt {3}$ = 11.464... \|56.908... \|220x220px

class="wikitable defaultright col2center"

Number
of circles

! Triangle
number

! Length

! Area

! Figure

| Yes

| $2 \sqrt {3}$ = 3.464...

|align="right"| 5.196...

|File:CircleInEquilateralTrianglePacking(1).png

| $2 + 2 \sqrt {3}$ = 5.464...

| 12.928...