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Packing circles in simple bounded shapes is a common type of problem in recreational mathematics. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles. Specific problems of this type that have been studied include: Circle packing in a circle; Circle packing in a square
The optimal packing of 15 circles in a square Optimal solutions have been proven for n ≤ 30. Packing circles in a rectangle; Packing circles in an isosceles right triangle - good estimates are known for n < 300. Packing circles in an equilateral triangle - Optimal solutions are known for n < 13, and conjectures are available for n < 28. [14]
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the ... Only 26 optimal packings are thought to be ...
Solutions (not necessarily optimal) have been computed for every N ≤ 10,000. [2] Solutions up to N = 20 are shown below. [2] The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest square numbers), but ceases to be optimal for larger squares from 49 onwards.
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]
The golden angle plays a significant role in the theory of phyllotaxis; for example, the golden angle is the angle separating the florets on a sunflower. [2] Analysis of the pattern shows that it is highly sensitive to the angle separating the individual primordia, with the Fibonacci angle giving the parastichy with optimal packing density. [3]
But there’s something else printed on the back of most food packaging: several brightly-colored circles or squares that look like some sort of secret code. However, these shapes aren’t an ...
Circle packings, as studied in this book, are systems of circles that touch at tangent points but do not overlap, according to a combinatorial pattern of adjacencies specifying which pairs of circles should touch. The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it ...