Field of geometry closely arranging circles on a plane
This article is about the packing of circles on surfaces. For circle packing with a prescribed intersection graph, see Circle packing theorem.
The most efficient way to pack different-sized circles together is not obvious.
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.
The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
In geometry, circlepacking is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs...
Circlepacking in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If...
Circlepacking in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square...
squares can be packed into some larger shape, often a square or circle. Square packing in a square is the problem of determining the maximum number of...
The circlepacking theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose...
to CirclePacking: The Theory of Discrete Analytic Functions is a mathematical monograph concerning systems of tangent circles and the circlepacking theorem...
sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circlepacking in two dimensions...
Integral Apollonian circlepacking defined by circle curvatures of (−3, 5, 8, 8) Integral Apollonian circlepacking defined by circle curvatures of (−12...
as a circlepacking, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing...
as a circlepacking, placing equal-diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing...
Circlepacking in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest...
Casey's theorem Circle graph Circle map CirclepackingCirclepacking in a circleCirclepacking in an equilateral triangle Circlepacking in an isosceles...
is the three-dimensional equivalent of the circlepacking in a circle problem in two dimensions. Best packing of m>1 equal spheres in a sphere setting a...
as a circlepacking, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing...
Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circlepackings: number theory", Journal of Number Theory, 100 (1): 1–45, arXiv:math...
allocations is referred to as the 'circle-packing' or 'polygon-packing'. Using optimization algorithms, a circle-packing figure can be computed for any uniaxial...
the densest possible circlepacking. Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is π⁄√12 or 90...
as a circlepacking, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing...
This concept generalizes the circlepackings described by the circlepacking theorem, in which specified pairs of circles are tangent to each other. Although...
In geometry, the Soddy circles of a triangle are two circles associated with any triangle in the plane. Their centers are the Soddy centers of the triangle...
interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circlepacking theorem, first proved by Paul Koebe in...
Tangent lines to circlesCirclepacking theorem, the result that every planar graph may be realized by a system of tangent circles Hexafoil, the shape...
even when the locations are fixed. Circlepacking in a rectangle Square packing in a square De Bruijn's theorem: packing congruent rectangular bricks of...
as a circlepacking, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing...
It is also related to the densest circlepacking of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area...
set of Kleinian groups; see also Circlepacking theorem. The circles of Apollonius may also denote three special circles C 1 , C 2 , C 3 {\displaystyle...
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