page_title: Menger sponge
section_title: Properties
hierarchical_section_title: Menger sponge / Properties
caption_attribution_description: English: Temperature distribution in the Menger fractal structures following a shockwave loading&#160;
context_page_description: In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.
context_section_description: The nth stage of the Menger sponge, Mₙ, is made up of 20ⁿ smaller cubes, each with a side length of (1/3)ⁿ.  The total volume of Mₙ is thus (20/27)ⁿ.  The total surface area of Mₙ 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.
  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 , with .
  The sponge's Hausdorff dimension is  ≅ 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 curve 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.
  The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set.
  Experiments also showed that for the same material, cubes with a Menger sponger structure could dissipate shocks five times better than a cubes without any pores.
caption: Cubes with Menger fractal structures after shockwave loading. The color indicates the temperature rise associated with plastic deformation.[10]
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