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Solutions to Apollonius's problem generally occur in pairs; for each solution circle, there is a conjugate solution circle (Figure 6). [1] One solution circle excludes the given circles that are enclosed by its conjugate solution, and vice versa.
CLP problems generally have 4 solutions. The solution of this special case is similar to that of the CPP Apollonius solution. Draw a circle centered on the given point P; since the solution circle must pass through P, inversion in this [clarification needed] circle transforms the solution circle
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible ... solutions for n = 2, 3, 4 ...
To convert between these two formulations of the problem, the square side for unit circles will be = + /. 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.
This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the original circle problem. [9] It can be intuitively understood as the question of how many trees within a distance of r are visible in the Euclid's orchard , standing in the origin.
The solutions to this problem are sometimes called the circles of Apollonius. The Apollonian gasket —one of the first fractals ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
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The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.