Problem of the Month
(February 2019)
What is the smallest square that can contain n non-overlapping unit golden rectangles (1×ϕ, where ϕ=(1+√5)/2)? What is the smallest golden rectangle that can contain these squares? What is the smallest golden rectangle that can contain n unit squares?
How full can we pack a unit square with golden rectangles of any size? Similarly, how full can we pack a unit golden rectangle with squares of any size? If we use the greedy algorithm, every golden rectangle or square used fills a factor of ϕ-1 of the area. Can we improve upon this?
ANSWERS
Maurizio Morandi, Sigvart Brendberg, and Yanis Gomes sent solutions.
Best-Known Packings of Golden Rectangles in Squares
1
s = ϕ = 1.618+
| 2
s = 2
| 3-4
s = 1 + ϕ = 2.618+
| 5-6
s = 2ϕ = 3.236+
| 7-8
s = 2 + ϕ = 3.618+
|
9-10
s = 1 + 2ϕ = 4.236+
| 11-12
s = 3 + ϕ = 4.618+
| 13
s = 3ϕ = 4.854+
| 14-15
s = 5
| 16
s = 2 + 2ϕ = 5.236+
|
17-18
s = 4 + ϕ = 5.618+
| 19-20
s = 1 + 3ϕ = 5.854+
| 21
s = 6
| 22-24
s = 3 + 2ϕ = 6.236+
| 25-26
s = 5 + ϕ = 6.618+
|
27-28
s = 2 + 3ϕ = 6.854+
| 29
s = 7
| 30-32
s = 4 + 2ϕ = 7.236+
| 33-35
s = 6 + ϕ = 7.618+
| 36-37
s = 3 + 3ϕ = 7.854+ (MM)
|
38
s = 8 (MM)
| 39-40
s = 5ϕ = 8.090+
| 41
s = 5+2ϕ = 8.236+ (YG)
| 42
s = 2+4ϕ = 8.472+ (YG)
| 43
s = 8.618+ (YG)
|
44
s = 7+ϕ = 8.618+ (YG)
| 45-46
s = 4+3ϕ = 8.854+ (YG)
| 47
s = 9 (YG)
| 48-49
s = 1+5ϕ = 9.090+ (YG)
| 50
s = 9.236+ (YG)
|
Best-Known Packings of Golden Rectangles in Golden Rectangles
1
s = 1
| 2
s = ϕ = 1.618+
| 3
s = 3/ϕ = 1.854+
| 4
s = 2
| 5-6
s = 1 + ϕ = 2.618+
|
7
s = 1 + 3/ϕ = 2.854+
| 8-9
s = 3
| 10
s = 2ϕ = 3.236+
| 11
s = 1 + 4/ϕ = 3.472+
| 12
s = 2 + ϕ = 3.618+
|
13
s = 6/ϕ = 3.708+
| 14-16
s = 4
| 17
s = 1 + 2ϕ = 4.236+
| 18
s = 7/ϕ = 4.326+
| 19-20
s = 3 + ϕ = 4.618+
|
21
s = 1 + 6/ϕ = 4.708+
| 22
s = 3ϕ = 4.854+
| 23-24
s = 8/ϕ = 4.944+
| 25
s = 5
| 26
s = 2 + 2ϕ = 5.236+
|
27
s = 1 + 7/ϕ = 5.326+
| 28
s = 3 + 4/ϕ = 5.472+
| 29-31
s = 4 + ϕ = 5.618+
| 32-33
s = 1 + 3ϕ = 5.854+
| 34
s = 1 + 8/ϕ = 5.944+
|
35-36
s = 6
| 37-38
s = 3+ϕ = 6.236+ (YG)
| 39-40
s = 4ϕ = 6.472+ (YG)
| 41-42
s = 9ϕ–8 = 6.562+ (YG)
| 43
s = 5+ϕ = 6.618+ (YG)
|
44
s = 11/ϕ = 6.798+
| 45
s = 6.854+ (YG)
| 46
s = 8ϕ–6 = 6.944+ (YG)
| 47-49
s = 7 (YG)
| 50
s = 4+2ϕ = 7.236+ (YG)
|
Best-Known Packings of Squares in Golden Rectangles
1
s = 1
| 2
s = 2/ϕ = 1.236+
| 3
s = 3/ϕ = 1.854+
| 4-6
s = 2
| 7
s = 2.450+ (MM)
|
8
s = 4/ϕ = 2.472+
| 9
s = 2.830+ (MM)
| 10
s = 2.945+ (MM)
| 11-12
s = 3
| 13-15
s = 5/ϕ = 3.090+
|
16
s = 3.589+ (MM)
| 17-18
s = 6/ϕ = 3.708+
| 19
s = 3.86900+ (YG)
| 20
s = 3.93177+ (YG)
| 21-24
s = 4
|
25
s = 4.294+ (MM)
| 26-28
s = 7/ϕ = 4.326+
| 29
s = 4.67244+ (YG)
| 30
s = 4.741+ (MM)
| 31
s = 4.82772+ (YG)
|
32
s = 4.86222+ (YG)
| 33
s = 8/ϕ = 4.944+ (MM)
| 34
s = 4.969+ (MM)
| 35-40
s = 5
| 41
s = 5.432+ (YG)
|
42
s = 5.485+ (YG)
| 43-45
s = 9/ϕ = 5.562+ (YG)
| 46
s = 5.774+ (YG)
| 47
s = 5.838+ (YG)
| 48-50
s = 6 (YG)
|
If you can extend any of these results, please
e-mail me.
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