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• Fractal_dimension abstract "A fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but rather is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which in this case is a number between one and two.".
• Fractal_dimension thumbnail Britain-fractal-coastline-200km.png?width=300.
• Fractal_dimension wikiPageExternalLink Fractals.htm.
• Fractal_dimension wikiPageExternalLink fracdim.
• Fractal_dimension wikiPageExternalLink www.trusoft-international.com.
• Fractal_dimension wikiPageID "285907".
• Fractal_dimension wikiPageRevisionID "606281696".
• Fractal_dimension align "right".
• Fractal_dimension alt "Coastline of Britain measured using a 100 km scale".
• Fractal_dimension alt "Coastline of Britain measured using a 200 km scale".
• Fractal_dimension alt "Coastline of Britain measured using a 50 km scale".
• Fractal_dimension caption "(11.5 x 200 = 2300 km)".
• Fractal_dimension caption "(28 x 100 = 2800 km)".
• Fractal_dimension caption "(70 x 50 = 3500 km)".
• Fractal_dimension footer "Figure 1. As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases.".
• Fractal_dimension hasPhotoCollection Fractal_dimension.
• Fractal_dimension image "britain-fractal-coastline-100km.png".
• Fractal_dimension image "britain-fractal-coastline-200km.png".
• Fractal_dimension image "britain-fractal-coastline-50km.png".
• Fractal_dimension width "80".
• Fractal_dimension subject Category:Chaos_theory.
• Fractal_dimension subject Category:Dimension_theory.
• Fractal_dimension subject Category:Dynamical_systems.
• Fractal_dimension subject Category:Fractals.
• Fractal_dimension type Abstraction100002137.
• Fractal_dimension type Attribute100024264.
• Fractal_dimension type Cognition100023271.
• Fractal_dimension type DynamicalSystem106246361.
• Fractal_dimension type DynamicalSystems.
• Fractal_dimension type Form105930736.
• Fractal_dimension type Fractal105931152.
• Fractal_dimension type Fractals.
• Fractal_dimension type PhaseSpace100029114.
• Fractal_dimension type PsychologicalFeature100023100.
• Fractal_dimension type Space100028651.
• Fractal_dimension type Structure105726345.
• Fractal_dimension comment "A fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.".
• Fractal_dimension label "Dimension fractale".
• Fractal_dimension label "Dimensione frattale".
• Fractal_dimension label "Dimensión fractal".
• Fractal_dimension label "Fractal dimension".
• Fractal_dimension label "Fractale dimensie".
• Fractal_dimension label "Fraktale Dimension".
• Fractal_dimension label "Wymiar podobieństwa".
• Fractal_dimension label "بعد كسيري".
• Fractal_dimension label "フラクタル次元".
• Fractal_dimension label "分形维数".
• Fractal_dimension sameAs Fraktale_Dimension.
• Fractal_dimension sameAs Μορφοκλασματική_διάσταση.
• Fractal_dimension sameAs Dimensión_fractal.
• Fractal_dimension sameAs Dimension_fractale.
• Fractal_dimension sameAs Dimensione_frattale.
• Fractal_dimension sameAs フラクタル次元.
• Fractal_dimension sameAs 프랙탈_차원.
• Fractal_dimension sameAs Fractale_dimensie.
• Fractal_dimension sameAs Wymiar_podobieństwa.
• Fractal_dimension sameAs m.01p_sc.
• Fractal_dimension sameAs Q1412452.
• Fractal_dimension sameAs Q1412452.
• Fractal_dimension sameAs Fractal_dimension.
• Fractal_dimension wasDerivedFrom Fractal_dimension?oldid=606281696.
• Fractal_dimension depiction Britain-fractal-coastline-200km.png.
• Fractal_dimension isPrimaryTopicOf Fractal_dimension.