Metric Dimension (other)
In mathematics, metric dimension may refer to: * Metric dimension (graph theory), the minimum number of vertices of an undirected graph ''G'' in a subset ''S'' of ''G'' such that all other vertices are uniquely determined by their distances to the vertices in ''S'' * Minkowski–Bouligand dimension (also called the metric dimension), a way of determining the dimension of a fractal set in a Euclidean space by counting the number of fixed-size boxes needed to cover the set as a function of the box size * Equilateral dimension of a metric space (also called the metric dimension), the maximum number of points at equal distances from each other * Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ...
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Metric Dimension (graph Theory)
In graph theory, the metric dimension of a graph ''G'' is the minimum cardinality of a subset ''S'' of vertices such that all other vertices are uniquely determined by their distances to the vertices in ''S''. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete. Detailed definition For an ordered subset W = \ of vertices and a vertex ''v'' in a connected graph ''G'', the representation of ''v'' with respect to ''W'' is the ordered ''k''-tuple r(v, W) = (d(v,w_1), d(v,w_2),\dots,d(v,w_k)), where ''d''(''x'',''y'') represents the distance between the vertices ''x'' and ''y''. The set ''W'' is a resolving set (or locating set) for ''G'' if every two vertices of ''G'' have distinct representations. The metric dimension of ''G'' is the minimum cardinality of a resolving set for ''G''. A resolving set containing a minimum number of vertices is called a basis (or reference s ...
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Minkowski–Bouligand Dimension
450px, Estimating the box-counting dimension of the coast of Great Britain In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set ''S'' in a Euclidean space R''n'', or more generally in a metric space (''X'', ''d''). It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand. To calculate this dimension for a fractal ''S'', imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm. Suppose that ''N''(''ε'') is the number of boxes of side length ''ε'' required to cover the set. Then the box-counting dimension is defined as : \dim_\text(S) := \lim_ \frac . Roughly speaking, this means that the dimension is the exponent ' ...
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Equilateral Dimension
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a d-dimensional Euclidean space is d+1, achieved by a regular simplex, and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance (L^1 norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d, achieved by a cross polytope. Lebesgue spaces The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L^p norm \ \, x\, _p=\left(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\right)^. The equilateral dimension o ...
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