The Menger sponge is a fractal which is the three-dimensional
analog of the Sierpiński carpet .
The Wolfram Language as MengerMesh n ,
3].
Let volume
after the
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).
The image above shows a metal print of the Menger sponge created by digital sculptor
Bathsheba Grossman (http://www.bathsheba.com/ ).
See also Menger Sponge Graph ,
Sierpiński
Carpet ,
Tetrix
Explore with Wolfram|Alpha
References 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. 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. 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. 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 https://mathworld.wolfram.com/MengerSponge.html
Subject classifications More... Less...
th iteration of the Menger sponge is implemented
in the Wolfram Language as MengerMesh[n,
3].
be the number of filled boxes, 
the length of a side of a hole, and
the fractional volume
after the 
th
iteration, then
Dickau, R. "Sierpinski-Menger Sponge Code
and Graphic."