Menger sponge

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An illustration of M4, the sponge after four iterations of the construction process.


The Menger sponge simultaneously exhibits an infinite surface area and zero volume.[3]


A sculptural representation of iterations 0 (bottom) to 3 (top).

The construction of a Menger sponge can be described as follows:

  1. Begin with a cube (first image).


An illustration of the iterative construction of a Menger sponge up to M3, the third iteration.
Menger sponge animation through (4) recursion steps.

The following table lists the properties of sponges Mn where n is the number of iterations performed on the first cube of unit side length.[4][5]



The cross-section of a Menger sponge through its centroid and perpendicular to a space diagonal has sixfold symmetry containing hexagrams.[6]

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M0 is a Cantor set.

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. It is an uncountable set.

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 has infinite surface area but zero volume.[3]

The sponge has a Hausdorff dimension of log 20/log 3 (approximately 2.726833).

Formal definition[edit]

Formally, a Menger sponge can be defined as follows:

M := \bigcap_{n\in\mathbb{N}} M_n

where M0 is the unit cube and

M_{n+1} := \left\{\begin{matrix}
\begin{matrix}\exists i,j,k\in\{0,1,2\}: (3x-i,3y-j,3z-k)\in M_n
\\ \mbox{and at most one of }i,j,k\mbox{ is equal to 1}\end{matrix}


A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival
One of the MegaMengers, at the University of Bath

MegaMenger is a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University. Each small cube is made from 6 interlocking folded business cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are then covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing.[7] In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.[8]

Similar fractals[edit]

Sierpinski-Menger snowflake. Eight corner cubes and the one central cube are kept each time at the lower and lower recursion steps. This peculiar three dimensional fractal has the Hausdorff dimension of the natively two dimensional object like the plane i.e. log 9/log 3=2

See also[edit]

Notes and references[edit]

  1. ^ Menger, Karl (1928), Dimensionstheorie, B.G Teubner Publishers 
  2. ^ Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.", Communications to the Amsterdam Academy of Sciences . English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443 
  3. ^ a b "Menger sponge", Wolfram Alpha, retrieved 2013-12-12 
  4. ^ Wolfram Demonstrations Project, Volume and Surface Area of the Menger Sponge
  5. ^ University of British Columbia Science and Mathematics Education Research Group, Mathematics Geometry: Menger Sponge
  6. ^
  7. ^ Tim Chartier. "A Million Business Cards Present a Math Challenge". Retrieved 2015-04-07. 
  8. ^ "MegaMenger". Retrieved 2015-02-15. 
  9. ^
  10. ^
  11. ^

External links[edit]