A342843 a(n) is the number of edges of the regular polygon such that packing n nonoverlapping equal circles inside the regular polygon gives the highest packing density. a(n) = 0 if such a regular polygon is a circle.
0, 4, 3, 4, 5, 3, 6, 7, 4, 3, 9, 6, 10, 6, 3, 4
Offset: 1
Examples
a(1) = 0. The maximum packing density for packing 1 circle in regular m-gon is (Pi/m)*cot(Pi/m), which is an increasing function of m. Highest packing density of 1 is achieved as m tends to infinity and the regular n-gon becomes a circle.
a(2) = 4. The maximum packing density for packing 2 circles in regular polygon with odd number of edges m >= 3 is 4*Pi/(m*sin(2*Pi/m))/(sec(Pi/(2*m))+sec(Pi/m))^2, which is smaller than the packing density in regular polygon with even number of edges m >= 4, 4*Pi/(m*sin(2*Pi/m))/(1+sec(Pi/m))^2, which is a decreasing function of m with a maximum of Pi/(3+2*sqrt(2)) at m = 4.
Symmetry type (S) of the n-circle configuration achieving the highest packing density and the corresponding number of edges (N) of the regular polygon and packing density for n up to 16 are listed below.
n S N Packing density
------ ------ --- ---------------------------------------------------------
1 O(2) oo 1
2 D_{4} 4 Pi/(3+2*sqrt(2)) = 0.53901+
3 D_{6} 3 (Pi/2)/(1+2/sqrt(3)) = 0.72900+
4,9,16 D_{8} 4 Pi/4 = 0.78539+
5 D_{10} 5 (Pi/2)/(1+4/sqrt(10+2*sqrt(5))) = 0.76569+
6 D_{6} 3 6*Pi/(12+7*sqrt(3)) = 0.78134+
7 D_{12} 6 7*Pi/(12+8*sqrt(3)) = 0.85051+
8 D_{14} 7 (4*Pi/7)/(1+1/sin(2*Pi/7)) = 0.78769+
10 D_{6} 3 (5*Pi/3)/(3+2*sqrt(3)) = 0.81001+
11 D_{2} 9 (11*Pi/18)/(1+csc(2*Pi/9)) = 0.75120+
12 D_{6} 6 6*Pi/(12+7*sqrt(3)) = 0.78134+
13 D_{2} 10 (13*Pi/20)/(1+sqrt(50+10*sqrt(5))/5) = 0.75594+
14 D_{4} 6 (49*Pi/2)/(21+20*sqrt(3)+6*sqrt(7)+6*sqrt(21)) = 0.77737+
15 D_{6} 3 15*Pi/(24+19*sqrt(3)) = 0.82805+
Links
- Erich Friedman, Packing Equal Copies
- Eckard Specht, Packomania, Packings of equal and unequal circles in fixed-sized containers with maximum packing density
Comments