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Menger Sponge

MengerSponge3

A fractal which is the three-dimensional analog of the Sierpiński carpet. 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.

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)

(Sloane's 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 (http://www.bathsheba.com/).

J. Mosely is leading an effort to construct a large Menger sponge out of old business cards.


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