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 topological spaces, meaning that any compact topological space of dimension 1 has a homeomorphic copy as a subspace of the Menger sponge (Peitgen et al. 1992, Broden et al. 2024).
The th iteration of the Menger sponge is implemented in the Wolfram Language as MengerMesh[n, 3].
Let be the number of filled boxes, the length of a side of a hole, and the fractional volume after the th iteration, then
(1) (2) (3)The capacity dimension is therefore
(4) (5) (6) (7)(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).
Broden et al. (2024) proved that all knots can be embedded into the Menger sponge (Barber 2024).
The image above shows a metal print of the Menger sponge created by digital sculptor Bathsheba Grossman (http://www.bathsheba.com/).
See alsoMenger Sponge Graph, Sierpiński Carpet, Tetrix Explore with Wolfram|AlphaMore things to try:
menger sponge 7-ary tree domain of f(x) = x/(x^2-1) ReferencesBarber, 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." http://library.wolfram.com/infocenter/MathSource/4662/.Dickau, R. M. "Menger (Sierpinski) Sponge." http://mathforum.org/advanced/robertd/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)." http://world.std.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|AlphaMenger Sponge Cite this as:Weisstein, Eric W. "Menger Sponge." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MengerSponge.html
Subject classifications