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Menger Sponge

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

Menger (1926) proved that the Menger sponge is universal for all compact 1-dimensional metric spaces, meaning that any such metric space has a homeomorphic copy as a subspace of the Menger sponge (Peitgen et al. 1992, Broden et al. 2024).

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 side length of each retained cube, and V_n the fractional volume after the nth iteration, then

N_n=20^n
(1)
L_n=(1/3)^n=3^(-n)
(2)
V_n=L_n^3N_n=((20)/(27))^n.
(3)

The capacity dimension is therefore

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(4)
=log_320
(5)
=(ln20)/(ln3)
(6)
=2.726833028....
(7)

(OEIS A102447).

Broden et al. (2024) proved that every tame knot can be embedded into the Menger sponge (Barber 2024).

Menger sponge metal sculpture (Bathsheba Grossman)

The image above shows a metal print of the Menger sponge created by digital sculptor Bathsheba Grossman (https://www.bathsheba.com/).

See also

Menger Sponge Graph, Sierpiński Carpet, Tetrix

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References

Barber, G. "Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal." Quanta Mag., Nov. 26, 2024. https://www.quantamagazine.org/teen-mathematicians-tie-knots-through-a-mind-blowing-fractal-20241126/.Broden, J.; Espinosa, M.; Nazareth, N.; and Voth, N. "Knots Inside Fractals." 5 Sep 2024. https://arxiv.org/abs/2409.03639. Chung, S. and Hur, K. "Volume and Surface Area of the Menger Sponge." Wolfram Demonstrations Project, 2014. https://demonstrations.wolfram.com/VolumeAndSurfaceAreaOfTheMengerSponge/. Dickau, R. "Sierpinski-Menger Sponge Code and Graphic." https://library.wolfram.com/infocenter/MathSource/4662/.Dickau, R. M. "Menger (Sierpinski) Sponge." https://www.robertdickau.com/sponge.html.Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 101, 1988.Grossman, B. "Menger Sponge." http://www.bathsheba.com/math/menger.Kosmulski, M. "Modulus Origami--Fractals, IFS." http://hektor.umcs.lublin.pl/~mikosmul/origami/fractals.html.Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, p. 145, 1983.Menger, K. "Allgemeine Räume und Cartesische Räume. I." Comm. Amsterdam Acad. Sci., 1926.Menger, K. Dimensionstheorie. Leipzig, Germany: Teubner, 1928.Mosely, J. "Menger's Sponge (Depth 3)." https://theworld.com/~j9/sponge/.Peitgen, 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." http://www.angelfire.com/art2/stw/.

Referenced on Wolfram|Alpha

Menger Sponge

Cite this as:

Weisstein, Eric W. "Menger Sponge." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MengerSponge.html

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