View source for Circle packing in an equilateral triangle

{{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 {{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
 | 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
 }}.</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
 | 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
 | doi-access = free
 }}.</ref><ref>{{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}}.</ref>

A conjecture of [[Paul Erdős]] and Norman Oler states that, if {{mvar|n}} is a triangular number, then the optimal packings of {{math|''n'' &minus; 1}} and of {{mvar|n}} circles have the same side length: that is, according to the conjecture, an optimal packing for {{math|''n'' &minus; 1}} circles can be found by removing any single circle from the optimal hexagonal packing of {{mvar|n}} circles.<ref>{{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
 }}.</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
 | mr = 1439300
 | journal = [[Discrete Mathematics (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
 }}.</ref>

Minimum solutions for the side length of the triangle:<ref name="Melissen"/>

{{Table alignment}}
{| class="wikitable defaultright col2center"
|-
! Number<br/> of circles
! Triangle<br/> number
! Length
! Area
! Figure
|-
| 1
| Yes
| <math>2 \sqrt {3}</math> = 3.464...
|align="right"| 5.196...
|
|-
| 2
|
| <math>2 + 2 \sqrt {3}</math> = 5.464...
| 12.928...
|
|-
| 3
| Yes
| <math>2 + 2 \sqrt {3}</math> = 5.464...
| 12.928...
|
|-
| 4
|
| <math>4 \sqrt {3}</math> = 6.928...
| 20.784...
| [[Image:4 cirkloj en 60 60 60 triangulo.png|60x60px]]
|-
| 5
| 
| <math>4 + 2 \sqrt {3}</math> = 7.464...
| 24.124...
| [[Image:5 cirkloj en 60 60 60 triangulo v2.png|60x60px]]
|-
| 6
| Yes
| <math>4 + 2 \sqrt {3}</math> = 7.464...
|24.124...
|
|-
| 7
|
| <math>2 + 4 \sqrt {3}</math> = 8.928...
|34.516...
|
|-
| 8
|
| <math>2 + 2 \sqrt{3} + \tfrac {2} {3} \sqrt{33}</math> = 9.293...
|37.401...
|
|-
| 9
|
| <math>6 + 2 \sqrt {3}</math> = 9.464...
|38.784...
|
|-
| 10
| Yes
| <math>6 + 2 \sqrt {3}</math> = 9.464...
|38.784...
|
|-
| 11
|
| <math>4 + 2 \sqrt {3} + \tfrac {4} {3} \sqrt{6}</math> = 10.730...
|49.854...
|
|-
| 12
|
| <math>4 + 4 \sqrt {3}</math> = 10.928...
|51.712...
|
|-
| 13
|
| <math>4 + \tfrac {10} {3} \sqrt{3} + \tfrac {2} {3} \sqrt{6}</math> = 11.406...
|56.338...
|
|-
| 14
|
| <math>8 + 2 \sqrt {3}</math> = 11.464...
|56.908...
|
|-
| 15
| Yes
| <math>8 + 2 \sqrt {3}</math> = 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.<ref>{{citation
 | last = Nurmela | first = Kari J.
 | mr = 1780209
 | issue = 2
 | journal = Experimental Mathematics
 | pages = 241–250
 | title = Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles
 | url = http://projecteuclid.org/getRecord?id=euclid.em/1045952348
 | volume = 9
 | year = 2000
 | doi=10.1080/10586458.2000.10504649| s2cid = 45127090
 }}.</ref>

==See also==
*[[Circle packing in an isosceles right triangle]]
*[[Malfatti circles]], three circles of possibly unequal sizes packed into a triangle

==References==
<references/>

{{Packing problem}}

[[Category:Circle packing]]


{{elementary-geometry-stub}}