Menger sponge

The Menger sponge heard as the Sierpinski triangle and the Koch curve to the objects of fractal geometry. Named after Karl Menger sponge was first published in 1926 in his work on the dimensionality of point sets. The Menger sponge is a three-dimensional analogue of the Cantor set or the Sierpinski carpet.

Formal definition

Formally, a Menger sponge M defined as follows:

Where M0 denote the unit cube and

Construction

Expands to the Sierpinski carpet on a cube, then you get a structure which resembles a sponge. In each iteration of the cube is decomposed (or any part of the cube) in partial cubes and seven of these sub-cubes are removed.

The first five iteration of a Sierpinski carpet are as follows:

Level 1

Level 2

Level 3

Level 4

Level 5

Similarly, the procedure in the construction of the Menger sponge, the construction iteratively determined by the following steps:

The successive Continue this process with each iteration leads to further erosion of the cube. Executes the process is infinitely more, there is the fractal Menger sponge.

In general, for the Menger sponge that he is after iterations of individual cubes of the corresponding iteration. In other words, one obtains 20 copies of the cube at reducing the size to one third. The side of each hollow cube is a function of the iteration. Hence the volume of the n-th cube is derived. Due to the continued erosion of the volume converges in the limit to zero, while the surface of tends to infinity. The convergence speed is relatively fast; From the 16th design step only 1 % of the volume of the unit cube M0 are available.

The exact value of the Hausdorff dimension of the Menger sponge is derived from the definition:

The "body" of Menger sponge has therefore a Hausdorff dimension is less than 3 (as opposed to non- fractal actually three -dimensional objects ), while at the same time has its surface a Hausdorff dimension is greater than 2 (in contrast to 2 - dimensional surface non- fractal body ). In other words, the Menger sponge is a structure that a " non-standard" features ( fractal ) dimension, which is between a two-dimensional plane and a three-dimensional cube.

Properties

Each face of the Menger sponge is a Sierpinski carpet; In addition, the section of the structure obtained with a diagonal or central line of the side face of the unit cube M0 the Cantor set. As an intersection of closed sets is the Menger sponge considered topologically to a closed set, and after the covering theorem of Heine - Borel is this also compact. He is also uncountable and its Lebesgue measure is 0

Menger showed in 1926 that the Lebesgue'sche covering dimension of the sponge to the corresponding curve is the same. It is thus a so-called spatial universal curve and is capable of all curves with a dimension ≥ 3 display (→ homeomorphism ). For example, can thus geometries of loop quantum gravity embedded in a Menger sponge.

The Menger sponge has a self-similar structure.

Importance in group theory

In Geometric group theory is defined for each finitely generated group is a metric on the Cayley graph and, if the group is word - hyperbolic, the "boundary at infinity " of the graph. Many properties of finitely generated infinite groups can be derived from this edge at infinity.

Dahmani, Guirardel and Przytycki have (based on results from Kapovich and Small ) proved that in a certain sense, almost all finitely generated groups as boundary the Menger sponge.

To formulate this result precisely we need first the notion of " overwhelming probability ". This is defined as follows. To a number d with 0 < d < 1, consider all groups with n generators and ( most) dL relations of length (at most) meets L. A property P ( for the selected d) with overwhelming probability if for each n: L tends to infinity, the proportion of groups with the property P goes to 100 %.

With this definition, one can then formulate and prove probability statements about groups. Gromov has proved that > 1/2 with overwhelming probability the trivial group or Z/2Z get for d (the group with 2 elements ). Therefore, we only consider the case d > 1/2 in the theory of random groups.

For d < 1/2 with overwhelming probability you get after Gromov hyperbolic group G with cd (G ) = 2 and therefore one -dimensional edge.

For 1 -dimensional edges of hyperbolic groups there are for a set of Kapovich - Small only 3 possibilities: the circle, the Sierpinski carpet or the Menger sponge. The first two possibilities come to results of Kapovich - Small and Dahmani - Guirardel - Przytycki "with overwhelming probability " not before, so why in the only interesting case d <1 /2 of the edge with an overwhelming probability is a Menger sponge.