Menger Sponge


The Menger sponge is a fractal which is the three-dimensional analog of the Sierpiński carpet.

The nth iteration of the Menger sponge is implemented in the Wolfram Language as MengerMesh[n, 3].

Let N_n be the number of filled boxes, L_n the length of a side of a hole, and V_n the fractional volume after the nth iteration, then


The capacity dimension is therefore


(OEIS A102447).

The Menger sponge, in addition to being a fractal, is also a super-object for all compact one-dimensional objects, i.e., the topological equivalent of all one-dimensional objects can be found in a Menger sponge (Peitgen et al. 1992).

Menger sponge metal sculpture (Bathsheba Grossman)

The image above shows a metal print of the Menger sponge created by digital sculptor Bathsheba Grossman (

See also

Menger Sponge Graph, Sierpiński Carpet, Tetrix

Explore with Wolfram|Alpha


Dickau, R. "Sierpinski-Menger Sponge Code and Graphic.", R. M. "Menger (Sierpinski) Sponge.", J. Chaos: Making a New Science. New York: Penguin Books, p. 101, 1988.Grossman, B. "Menger Sponge.", M. "Modulus Origami--Fractals, IFS.", B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, p. 145, 1983.Mosely, J. "Menger's Sponge (Depth 3).", H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.Sloane, N. J. A. Sequence A102447 in "The On-Line Encyclopedia of Integer Sequences."Werbeck, S. "A Journey into Menger's Sponge."

Referenced on Wolfram|Alpha

Menger Sponge

Cite this as:

Weisstein, Eric W. "Menger Sponge." From MathWorld--A Wolfram Web Resource.

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