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Circle Packing
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. Densest packing In the two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them. Formal definition Formally, an intersection graph is an undirected graph formed from a family of sets : S_i, \,\,\, i = 0, 1, 2, \dots by creating one vertex for each set , and connecting two vertices and by an edge whenever the corresponding two sets have a nonempty intersection, that is, : E(G) = \. All graphs are intersection graphs Any undirected graph may be represented as an intersection graph. For each vertex of , form a set consisting of the edges incident to ; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, is the intersection graph of the sets . provide a construction that is more ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrally Symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be Centrosymmetry, centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometry, isometric Involution (mathematics), involutive affine transformation, which has exactly one Fixed point (mathematics), fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are Involution (mathematics), involutions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a bin packing problem, you are given: * A ''container'', usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem. * A set of ''objects'', some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circles Packed In Square 15
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is Constant (mathematics), constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a Degeneracy (mathematics), degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean space, Euclidean plane, except where otherwise noted. Specifically, a circle is a Simple polygon, simple closed curve that divides the plane into two Region (mathematics), regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tammes Problem
In geometry, the Tammes problem is a problem in packing a given number of circles on the surface of a sphere such that the minimum distance between circles is maximized. It is named after the Dutch botanist Pieter Merkus Lambertus Tammes (the nephew of pioneering botanist Jantina Tammes) who posed the problem in his 1930 doctoral dissertation on the distribution of pores on pollen grains.Pieter Merkus Lambertus Tammes (1930): ''On the number and arrangements of the places of exit on the surface of pollen-grains'', University of Groningen Mathematicians independent of Tammes began studying circle packing on the sphere in the early 1940s; it was not until twenty years later that the problem became associated with his name. It can be viewed as a particular special case of the generalized Thomson problem of minimizing the total Coulomb force of electrons in a spherical arrangement. Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomson Problem
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. Mathematical statement The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's Law, :U_(N)=k_\text. Here, k_\text is the Coulomb constant and r_=, \mathbf_i - \mathbf_j, is the distance between each pair of electrons located at points on the sphere defined by ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Circle Packings (Better)
Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ideas based on faith in a system of thought * ''The Fundamentals'', a set of books important to Christian fundamentalism * Any of a number of fundamental theorems identified in mathematics, such as: ** Fundamental theorem of algebra, awe theorem regarding the factorization of polynomials ** Fundamental theorem of arithmetic, a theorem regarding prime factorization * Fundamental analysis, the process of reviewing and analyzing a company's financial statements to make better economic decisions Music * Fun-Da-Mental, a rap group * Fundamental (Bonnie Raitt album), ''Fundamental'' (Bonnie Raitt album), 1998 * Fundamental (Pet Shop Boys album), ''Fundamental'' (Pet Shop Boys album) * Fundamental (Puya album), ''Fundamental'' (Puya album), 1999 * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Hexagonal Tiling
In geometry, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol ''sr''. Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille). There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry. There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".) Circle packing The snub trihexagonal tiling can be used 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 number).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E The lattice domain (red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Trihexagonal Tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''. Names Uniform colorings There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares. Related 2-uniform tilings The ''truncated trihexagonal tiling'' has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a central hexagon and surrounding triangles and square, in two different orientations. Circle packing The Truncated trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a truncated hexagon, t, or a twice-truncated triangle, tt. The internal angle at each vertex of a regular dodecagon is 150°. Area The area of a regular dodecagon of side length ''a'' is given by: :\begin A & = 3 \cot\left(\frac \right) a^2 = 3 \left(2+\sqrt \right) a^2 \\ & \simeq 11.19615242\,a^2 \end And in terms of the apothem ''r'' (see also inscribed figure), the area is: :\begin A & = 12 \tan\left(\frac\right) r^2 = 12 \left(2-\sqrt \right) r^2 \\ & \simeq 3.2153903\,r^2 \end In terms of the circumradius ''R'', the area is: :A = 6 \sin\left(\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices. A fundamental domain triangle is (''p'' ''q'' ''r''), and a right triangle (''p'' ''q'' 2), where ''p'', ''q'', ''r'' are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of ''p'', ''q'' and ''r''. There are a number of symbolic sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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