Circle packing in a square is a
packing problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few conta ...
in
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, where the aim is to pack
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
s into the smallest possible
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
. Equivalently, the problem is to arrange points in a unit square aiming to get the greatest minimal separation, , between points.
To convert between these two formulations of the problem, the square side for unit circles will be .
Solutions
Solutions (not necessarily optimal) have been computed for every .
Solutions up to are shown below.
The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s), but ceases to be optimal for larger squares from 49 onwards.
Circle packing in a rectangle
Dense packings of circles in non-square rectangles have also been the subject of many investigations.
See also
*
Square packing in a circle
Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle.
Square packing in a square
Square packing in a square is the problem of determining ...
References
Circle packing
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