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Field of geometry closely arranging circles on a plane
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
Circle_packing
Two-dimensional packing problem
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If
Circle_packing_in_a_circle
Two-dimensional packing problem
Circle packing in a square is a packing problem in recreational mathematics where the aim is to pack n unit circles into the smallest possible square
Circle_packing_in_a_square
On tangency patterns of circles
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible patterns of tangent circles among non-overlapping
Circle_packing_theorem
Two-dimensional packing problem
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
Square_packing
Problems which attempt to find the most efficient way to pack objects into containers
distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on
Packing_problems
Two-dimensional packing problem
amount n of unit circles can be packed into? More unsolved problems in mathematics Circle packing in an equilateral triangle is a packing problem in discrete
Circle packing in an equilateral triangle
Circle_packing_in_an_equilateral_triangle
Geometrical structure
sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions
Sphere_packing
Three-dimensional packing problem
is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions. Best packing of m>1 equal spheres in a sphere setting a
Sphere_packing_in_a_sphere
Japanese art of paper folding
allocations is referred to as the 'circle-packing' or 'polygon-packing'. Using optimization algorithms, a circle-packing figure can be computed for any uniaxial
Origami
2005 mathematics text
to Circle Packing: The Theory of Discrete Analytic Functions is a mathematical monograph concerning systems of tangent circles and the circle packing theorem
Introduction to Circle Packing
Introduction_to_Circle_Packing
Several sets of circles associated with Apollonius of Perga
set of Kleinian groups; see also Circle packing theorem. The circles of Apollonius may also denote three special circles C 1 , C 2 , C 3 {\displaystyle
Circles_of_Apollonius
Regular tiling of a two-dimensional space
is p6m. If a circle is inscribed in each hexagon, the resulting figure is the densest way to arrange circles in two dimensions; its packing density is π
Hexagonal_tiling
Mathematical theorem
It is also related to the densest circle packing of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area
Honeycomb_theorem
Tiling of a plane by regular hexagons and equilateral triangles
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing
Trihexagonal_tiling
Area of discrete mathematics
intersection of a circle packing is a coin graph, where a vertex and an edge represent a circle and every pair of tangent circles; by Koebe–Andreev–Thurston
Graph_theory
Uniform tiling of the Euclidean plane
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing
Truncated_trihexagonal_tiling
Semiregular tiling of the Euclidean plane
to a circle packing, each vertex becoming the center of a circle of fixed diameter. Every circle is in contact with 5 other circles in the packing (kissing
Snub_trihexagonal_tiling
Semiregular tiling of the plane
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing
Snub_square_tiling
Semiregular tiling of a plane
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing
Truncated_hexagonal_tiling
Graph-theoretic description of polyhedra
system and lifting the result into three dimensions, or by using the circle packing theorem. Several extensions of the theorem are known, in which the polyhedron
Steinitz's_theorem
Semiregular tiling
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing
Truncated_square_tiling
Semiregular tiling of the Euclidean plane
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing
Rhombitrihexagonal_tiling
Fractal composed of tangent circles
circle packing is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles,
Apollonian_gasket
Lower bound on radii in circle packings
geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing. The lemma
Ring_lemma
Rational circle tangent to the real line
Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circle packings: number theory", Journal of Number Theory, 100 (1): 1–45, arXiv:math
Ford_circle
Semiregular tiling of the plane
as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing
Elongated_triangular_tiling
Circle packing – Field of geometry closely arranging circles on a plane Circle packing in a circle – Two-dimensional packing problem Circle packing in
List_of_circle_topics
Regular tiling of the plane
the densest possible circle packing. Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is π⁄√12 or 90
Triangular_tiling
Geometric concept
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
Soddy_circles_of_a_triangle
Numbers obtained by adding the two previous ones
numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps. The Fibonacci numbers are important in computational
Fibonacci_sequence
Concept in inversive geometry
This concept generalizes the circle packings described by the circle packing theorem, in which specified pairs of circles are tangent to each other. Although
Inversive_distance
Uniform Tiling
as a circle packing. Cyan circles are in contact with 3 other circles (2 cyan, 1 pink), corresponding to the V4.6.12 planigon, and pink circles are in
3-4-6-12_tiling
Graph that can be embedded in the plane
interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circle packing theorem, first proved by Paul Koebe in
Planar_graph
Quadrilateral symmetric across a diagonal
right kites meeting at the center of its inscribed circle. More generally, a method based on circle packing can be used to subdivide any polygon with n {\displaystyle
Kite_(geometry)
Equation for radii of tangent circles
{\displaystyle C=-2} in hyperbolic geometry. Circle packing in a circle Euler's four-square identity Malfatti circles Soddy, F. (June 1936), "The Kiss Precise"
Descartes'_theorem
Circle-packing on the surface of a sphere
number of circles of that radius can be packed disjointly on the sphere. Unsolved problem in mathematics What is the optimal packing of circles on the surface
Tammes_problem
Optimization problem in mathematics
Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon
Rectangle_packing
Circle in the arbelos congruent to the twin circles
geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed
Archimedean_circle
Uniform tiling of the plane with regular polygons
as a circle packing. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.122 planigon, and pink circles are in
3-4-3-12_tiling
Circle packing arranged in spirals
circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles.
Doyle_spiral
Graph formed by subdivision of triangles
polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction. An Apollonian network may be formed, starting from a single
Apollonian_network
Circles related to a point in the plane
Tangent lines to circles Circle packing theorem, the result that every planar graph may be realized by a system of tangent circles Feuerbach's theorem
Tangent_circles
Circle packing
loxodromic sequence of tangent circles is an infinite sequence of circles arranged so that any four consecutive circles in the sequence are pairwise mutually
Coxeter's loxodromic sequence of tangent circles
Coxeter's_loxodromic_sequence_of_tangent_circles
Branch of geometry that studies combinatorial properties and constructive methods
in the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry
Discrete_geometry
Shape with four equal sides and angles
to these problems remain unsolved; the same is true for circle packing in a square. Packing squares into other shapes can have high computational complexity:
Square
Packing problem
three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of
Sphere_packing_in_a_cube
Israeli mathematician
processes and SLE, he made fundamental contributions to several topics: The circle packing theorem and discrete conformal geometry. Embeddings of Gromov hyperbolic
Oded_Schramm
Sphere tangent to every edge of a polyhedron
the distances from its two endpoints to their corresponding circles in this circle packing. Every convex polyhedron has a combinatorially equivalent polyhedron
Midsphere
Convex polygon which can tile the plane by itself
demonstrates that all the 14 VRPs are cocyclic, as alternatively shown by circle packings. The ratio of the incircle to the circumcircle is: sin π 12 = sin
Planigon
series. A conjecture with Norman Oler on circle packing in an equilateral triangle with a number of circles one less than a triangular number. The minimum
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Set of circles related by tangency
circle. Closed Steiner chains are the systems of circles obtained as the circle packing theorem representation of a bipyramid. Annular Steiner chains n = 3
Steiner_chain
Carter, Ithiel; Rodin, Burt (December 1992). "An Inverse Problem for Circle Packing and Conformal Mapping". Transactions of the American Mathematical Society
List_of_spirals
Two-dimensional packing problem
Circle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right
Circle packing in an isosceles right triangle
Circle_packing_in_an_isosceles_right_triangle
Triangle with circular arc edges
given three points. Circle packing theorem, for which the gaps between circles are often circular triangles Hart circle, a circle associated with certain
Circular_triangle
Three tangent circles in a triangle
mathematics Does the greedy algorithm always find area-maximizing packings of more than three circles in any triangle? More unsolved problems in mathematics Gian
Malfatti_circles
Single, usually cylindrical, flexible strand or bar or rod of metal
unavoidable gaps between the strands (this is the circle packing problem for circles within a circle). A stranded wire with the same cross-section of conductor
Wire
Graph representing tangency between geometric objects
intersect each other. The circle packing theorem states that every planar graph can be represented as a contact graph of circles, known as a coin graph.
Contact_graph
Arrangement of points on a sphere
of Hardin and Saff. Notable cases include: α = ∞, the Tammes problem (packing); α = 1, the Thomson problem; α = 0, to maximize the product of distances
Thomson_problem
Ring of circles between two tangent circles
of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Given two circles CU and CV, let the inner circle CU
Pappus_chain
Shape with three equal sides
is known as Van Schooten's theorem. A packing problem asks the objective of n {\displaystyle n} circles packing into the smallest possible equilateral
Equilateral_triangle
Mathematics book
symmetries §7.1 Conjugate element, §7.7 arrangement of lines, §7.8 Circle packing 8 Colored patterns and tilings §§8.1-8.7 Dichromatic symmetry, polychromatic
Tilings_and_patterns
Math theorem about sphere packing
and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of
Kepler_conjecture
Planar graph drawn by relaxing springs
problem of determining the circle centers in a circle packing representation of a planar graph, given the radii of the circles, as a weighted Tutte embedding
Tutte_embedding
Family of digital modulation methods
phase-shift keying (APSK) Carrierless amplitude phase modulation (CAP) Circle packing § Applications Error correction code In-phase and quadrature components
Quadrature amplitude modulation
Quadrature_amplitude_modulation
Open-source visualization software package
Multi-set bar chart Stacked bar chart Beeswarm plot Box plot Bumpchart Circle packing Dendrograms: Circular dendrogram Linear dendrogram Gantt chart Horizon
RAWGraphs
American mathematician (1952–2002)
mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. Brooks was born in 1952 in Washington,
Robert_W._Brooks
Simple Proof of Thue's Theorem on Circle Packing". arXiv:1009.4322v1 [math.MG]. Hales, Thomas; Kusner, Wöden (2016). "Packings of regular pentagons in the plane"
List of shapes with known packing constant
List_of_shapes_with_known_packing_constant
Graph formed by touching unit circles
graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for
Penny_graph
American mathematician (1946–2012)
37:7 (September 1990) pp 844–850 Automatic group Cannon–Thurston map Circle packing theorem Hyperbolic volume Hyperbolic Dehn surgery Thurston boundary
William_Thurston
Collective behavior of decentralized, self-organized systems
successfully used to provide a general model for this problem, related to circle packing and set covering. It has been shown that the SDS can be applied to identify
Swarm_intelligence
Circle of immediate corresponding curvature of a curve at a point
{\displaystyle R(t)={\frac {4r}{\left|\csc \left({\frac {t}{2}}\right)\right|}}} Circle packing theorem Osculating curve Osculating sphere Ghys, Étienne; Tabachnikov
Osculating_circle
Theorem in hyperbolic geometry
Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations of triangulated planar graphs. A consequence of Mostow
Mostow_rigidity_theorem
Uniform tiling of the plane using regular polygons
as a circle packings. In the first 2-uniform tiling (whose dual resembles a key-lock pattern): cyan circles are in contact with 5 other circles (3 cyan
33344-33434_tiling
South Korean American mathematician
worked extensively on counting and equidistribution for Apollonian circle packings, Sierpinski carpets and Schottky dances. Her recent work continues
Hee_Oh
case of osculating or tangential circles which has continued to be actively studied in the theory of circle packing. Carathéodory's theorem (conformal
Planar_Riemann_surface
finite connected planar graphs (approximately computing the theoretical circle-packing given by the Koebe-Andreev-Thurston theorem). See also Fáry's theorem
List_of_algorithms
Graph representing intersections between given sets
of unit disks in the plane. A circle graph is the intersection graph of a set of chords of a circle. The circle packing theorem states that planar graphs
Intersection_graph
Metaheuristic method for optimization problems
alternates between different formulations which was investigated for circle packing problem (CPP) where stationary point for a nonlinear programming formulation
Variable_neighborhood_search
Any shape with straight sides can be cut from a single sheet of folded paper with one cut
instances of the fold-and-cut problem, based on straight skeletons and on circle packing respectively. Demaine, Erik D.; Demaine, Martin L. (2004), "Fold-and-Cut
Fold-and-cut_theorem
2007 mathematics book by Demaine and O'Rourke
folded flat), the work of Robert J. Lang using tree structures and circle packing to automate the design of origami folding patterns, the fold-and-cut
Geometric_Folding_Algorithms
Any planar graph can be subdivided by removing a few vertices
According to the Koebe–Andreev–Thurston circle-packing theorem, any planar graph may be represented by a packing of circular disks in the plane with disjoint
Planar_separator_theorem
Planar graphs have straight drawings
The Circle packing theorem states that every planar graph may be represented as the intersection graph of a collection of non-crossing circles in the
Fáry's_theorem
On line segment intersection graphs
of Ehrlich et al. It can also be viewed as a generalization of the circle packing theorem, which shows the same result when curves are allowed to intersect
Scheinerman's_conjecture
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
quasiconformal map with any given bounded measurable Beltrami coefficient. Circle packing theorem, a discrete analogue of the uniformization theorem DeTurck &
Uniformization_theorem
American mathematician
arXiv:1107.3776 [math.NT]. Kontorovich, Alex; Oh, Hee (2008). "Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds". arXiv:0811.2236
Alex_Kontorovich
Study of graphs defined by geometric means
the plane is a unit disk graph. The Circle packing theorem states that the intersection graphs of non-crossing circles are exactly the planar graphs. Scheinerman's
Geometric_graph_theory
Hungarian mathematician
following six circle conjecture of László Fejes Tóth: if in a planar circle packing each circle is tangent to at least 6 other circles, then either it
Imre_Bárány
Hungarian mathematician (1915–2005)
this compact binary circle packing was shown to be the densest possible planar packing of discs with this size ratio. A dense packing of spheres Dodecahedron
László_Fejes_Tóth
Topics referred to by the same term
2-dimensional analog of Kepler's conjecture: the regular hexagonal packing is the densest circle packing in the plane (1890). This disambiguation page lists mathematics
Thue's_theorem
1990 book by Gerhard Ringel and Nora Hartsfield
formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embedding on topological surfaces
Pearls_in_Graph_Theory
Polygon associated with a compact Riemann surface
related to results in convexity theory, the geometry of numbers and circle packing, such as the Brunn–Minkowski inequality. Two elementary proofs due to
Fundamental_polygon
Number of edge slopes in graph drawing
constant factor, and applying the circle packing theorem to represent this augmented graph as a collection of tangent circles. If the degree of the initial
Slope_number
2012 book by Irving Adler
each other, and makes connections with the mathematical theories of circle packing and space-filling curves. Subsequent papers refine this theory, make
Solving the Riddle of Phyllotaxis
Solving_the_Riddle_of_Phyllotaxis
Intersection graph for curves in the plane
planar graph. Alternatively, by the circle packing theorem, any planar graph may be represented as a collection of circles, any two of which cross if and only
String_graph
Shape with five sides
double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular
Pentagon
Shape with seven sides
Euclidean plane. The regular heptagon has a double lattice packing of the Euclidean plane of packing density approximately 0.89269. This has been conjectured
Heptagon
American mathematician (born 1941)
Thurston's conjecture about the approximation of the Riemann map by circle packings. Sullivan and Moira Chas started the field of string topology, which
Dennis_Sullivan
Chinese scientist and statesman
mathematical problems, including many complex formulas for geometry, circle packing, and chords and arcs problems employing trigonometry. Shen addressed
Shen_Kuo
Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 2–3 Circle packings, plane tessellations, and networks, pp. 34–40). Asaro, Laura; Hyde
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
CIRCLE PACKING
CIRCLE PACKING
Boy/Male
French Israeli
The circle.
Female
Yiddish
(מִירל) Yiddish form of Hebrew Miryam, MIRELE means "obstinacy, rebelliousness" or "their rebellion."Â
Male
Celtic
, sea circle.
Girl/Female
French American
The french form of the English Carol, a dimunitive of Charles meaning strong.
Surname or Lastname
English
English : variant spelling of Cordell.Possibly an Americanized spelling of German Kördel, a pet form of an old German personal name, formed with kuoni ‘daring’. Compare Conrad.
Female
Slovene
Feminine form of Slovene Ciril, CIRILA means "lord."
Boy/Male
English
Birch.
Female
French
French form of Latin Carola, CAROLE means "man."
Boy/Male
English
From the bird hill.
Girl/Female
Japanese
Ball; circle.
Girl/Female
British, English
Botanical Name; The Myrtle is a Dark Green Shrub with Pink or White Blossoms
Female
English
English name derived from the vocabulary word, from Latin miraculum, MIRACLE means "marvel, wonder."
Girl/Female
Latin
Circle of light.
Girl/Female
Latin
Circle of light.
Girl/Female
Greek Latin
A witch.
Boy/Male
Spanish Greek
noble.
Girl/Female
Latin
Circle of light.
Boy/Male
Christian, Hindu, Indian
Bright Circle
Girl/Female
Bengali, Indian
Circle; Normal
Male
Slovene
Slovene form of Greek Kyrillos, CIRIL means "lord."
CIRCLE PACKING
CIRCLE PACKING
Boy/Male
Tamil
Devotee of Lord Shiva
Biblical
master of the opening
Girl/Female
Arabic, Muslim
Flame of Fire
Girl/Female
Tamil
Agrima | அகà¯à®°à¯€à®®à®¾
Leadership
Boy/Male
Tamil
Winner in war, The brave warrior
Boy/Male
American, Anglo, British, English
From the Stag's Forest
Girl/Female
Hindu
Boy/Male
Tamil
Navakanth | நாவாகாஂத
New light
Girl/Female
Hindu
Full of desires
Girl/Female
Australian, Danish, German, Greek, Russian
Torch; Sun Ray; Shining Light; Wicker; Reed; Shoot; Basket; Most Beautiful Woman in the World; A Lady Attending on Imogen; The Bright One; Similar to Helen
CIRCLE PACKING
CIRCLE PACKING
CIRCLE PACKING
CIRCLE PACKING
CIRCLE PACKING
v. i.
To change into curd; to coagulate; as, rennet causes milk to curdle.
n.
A circle.
p. pr. & vb. n.
of Circle
v. t.
See Encircle.
n.
A little circle; esp., an ornament for the person, having the form of a circle; that which encircles, as a ring, a bracelet, or a headband.
n.
A miracle play.
n.
An imaginary circle or orbit in the heavens; one of the celestial spheres.
n.
An instrument of observation, the graduated limb of which consists of an entire circle.
v. t.
To form a circle about; to inclose within a circle or ring; to surround; as, to encircle one in the arms; the army encircled the city.
n.
A circlet.
n.
A circle; a circus; a circular erection or arrangement of objects.
n.
To encompass, as by a circle; to surround; to inclose; to encircle.
n.
Alt. of Corcule
imp. & p. p.
of Circle
a.
Having the form of a circle; round.
v. t.
To girdle; to encircle.
n.
An amphitheatrical circle for sports; a circus.
v. i.
To move circularly; to form a circle; to circulate.
n.
One entire round in a circle or a spire; as, a cycle or set of leaves.
a.
Having the nature, properties, or qualities, of an adult man; characteristic of developed manhood; hence, masterful; forceful; specifically, capable of begetting; -- opposed to womanly, feminine, and puerile; as, virile age, virile power, virile organs.