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ABSTRACT
Fractional geometry deals with fractional objects in metric spaces. A fractional object has an infinitely fragmented irregular spatial structure, and a topological dimension that is not necessarily an integer, unlike regular objects. A fractional object has a Hausdorff dimension greater than its topological dimension. A fractal object is a fractional object that also has a spatial structure that follows a deterministic or probabilistic rule involving internal self-similarity. Fractal geometry examines geometric fractal objects. This article looks at these geometries, and gives a list of practical applications, with a detailed case (Lorenz attractor).
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Jean-Charles PINOLI: Professor - École Nationale Supérieure des Mines de Saint-Étienne, Saint Étienne, France
INTRODUCTION
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Fractional geometry and fractal geometry
Fractional Geometry is a branch of geometry that deals with so-called fractional geometric objects, mainly in practice in n-dimensional Euclidean spaces, but the relevant general framework is that of separable complete metric spaces. The adjective fractional first appeared in geometry in the work of A. Besicovitch (1930s), although it was introduced into analysis as early as the 17th century by G. Leibniz to qualify certain exponents of functions.
The concept of dimension plays a key role in fractional geometry, with the Hausdorff and Besicovitch dimension introduced by F. Hausdorff (1919), then studied in detail by A. Besicovitch (1929, 1934, 1935, 1937). Dimension Theory is a branch of General Topology
[AF97] [AF98] Note: The term fraction, which appeared in French at the end of the 12th century, is derived from the Low Latin fractio - "action of breaking".
Fractal geometry is concerned with fractal geometric objects, i.e. fractional objects with a spatial structure that follows a deterministic or probabilistic rule involving internal self-similarity. The adjective fractal was introduced by B. Mandelbrot in 1967 , who was responsible for the widespread popularization of the concept of fractality
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KEYWORDS
Hausdorff measure | metric spaces | auto-similarity | random fractals objects
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