משתמש:Hexagone59/ספוג מנגר

Propriétés

L'éponge de Menger est une fractale dont la dimension de Hausdorff vaut ${\frac {\log 20}{\log 3}}$, soit à peu près 2,726 833.

Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry. The number of these hexagrams, in descending size, is given by $a_{n}=9a_{n-1}-12a_{n-2}$, with $a_{0}=1,\ a_{1}=6$.

La dimension topologique de l'éponge de Menger est égale à 1 ; elle fut d'ailleurs construite initialement par Menger pour explorer le concept de dimension topologique. Menger démontra 1926 que l'éponge est une courbe universelle, c’est-à-dire que toute courbe unidimensionnelle (au sens où sa dimension topologique est égale à 1) est homéomorphe à un sous-ensemble de l'éponge.

L'éponge de Menger est un espace fermé ; puisqu'il est également borné, le théorème de Heine-Borel stipule qu'il est également compact. L'éponge de Menger est un ensemble non-dénombrable de mesure de Lebesgue nulle.

It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set.

De manière similaire, le triangle de Sierpinski est une courbe universelle pour toute courbe d'un espace à deux dimensions. L'éponge de Menger étend cette notion aux courbes tri-dimensionnelles. Le raisonnement peut être étendu à un nombre de dimension quelconque.

Properties

True view of the cross-section of a level-4 Menger sponge through its centroid and perpendicular to a space diagonal. In this interactive SVG, the cross-sections are true-view and to scale.

The nth stage of the Menger sponge, M_n, is made up of 20ⁿ smaller cubes, each with a side length of (1/3)ⁿ. The total volume of M_n is thus (20/27)ⁿ. The total surface area of M_n is given by the expression 2(20/9)ⁿ + 4(8/9)ⁿ. Therefore the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues, so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.

The sponge's Hausdorff dimension is תבנית:Sfrac ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every תבנית:Interlanguage link multi is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions.