Skip to content

Statement

Let \(M \subset [0,1]^3\) be the Menger sponge, the attractor of the iterated function system consisting of the \(N=20\) contractions \(x \mapsto \tfrac{1}{3}x + t_j\), where the translations \(t_j\) index the twenty sub-cubes of the \(3\times3\times3\) subdivision that remain after removing the central cube and the six face-centre cubes. Then the similarity (Hausdorff) dimension of \(M\) is

\[ D = \frac{\log 20}{\log 3} \approx 2.7268330278608. \]

Assumptions

  1. \(M\) is the unique nonempty compact attractor of \(N=20\) similarities, each of ratio \(r = 1/3\).
  2. Open set condition (OSC): taking \(U = (0,1)^3\), the twenty images \(\tfrac{1}{3}U + t_j\) are pairwise disjoint open cubes contained in \(U\). The OSC holds, so the similarity dimension equals the Hausdorff dimension (Moran 1946; Hutchinson 1981).
  3. Counting: the \(3\times3\times3=27\) sub-cubes minus the \(1\) central cube minus the \(6\) face-centre cubes leaves \(27 - 7 = 20\).

Proof

  1. [verified] Cube count. The subdivision produces \(27\) sub-cubes; the Menger removal deletes the body-centre cube and the six face-centre cubes, \(27 - 7 = 20\). SymPy: 27 - 7 = 20.
  2. [asserted] By Moran's/Hutchinson's theorem, an IFS of similarities with ratio \(r\) satisfying the OSC has an attractor of Hausdorff dimension solving the Moran equation \(N r^{D} = 1\), i.e. \(20\cdot(1/3)^{D} = 1\).
  3. [verified] Solving \(20\cdot 3^{-D} = 1\) gives \(3^{D} = 20\), hence \(D = \log 20/\log 3\). SymPy log(20)/log(3) evaluates to \(2.726833027860842\).
  4. [verified] Consistency check. Each face of the Menger sponge is a Sierpinski carpet, whose dimension \(\log 8/\log 3 \approx 1.8928\) is strictly less than \(D\), as required for a genuinely three-dimensional-in-detail object (\(2 < D < 3\)).

Verification

  • Cube count (step 1): integer arithmetic \(27-7=20\).
  • Dimension (step 3): SymPy solved \(20\cdot 3^{-D}=1\) to log(20)/log(3); float value \(2.726833027860842\).
  • Face consistency (step 4): \(\log 8/\log 3 = 1.8927892607143724 < D\), confirmed numerically.
  • Asserted external result: the Moran/Hutchinson OSC \(\Rightarrow\) Hausdorff-dimension identity (step 2). All quantitative outputs machine-checked.
  • Topological note (not required for the dimension): Menger proved \(M\) is a universal curve — it has topological dimension \(1\) and contains a homeomorphic copy of every compact metric space of topological dimension \(\le 1\). This is asserted from Menger's curve theory, not computed here.

Historical source

Karl Menger introduced the sponge in his dimension-theory work of the 1920s–30s (Allgemeine Räume und Cartesische Räume, 1926; Kurventheorie, 1932) as the three-dimensional universal curve, generalising Cantor's set (1D removal) and Sierpinski's carpet (2D removal). The similarity-dimension identification uses the later Moran (1946) / Hutchinson (1981) framework.

Related concepts

Citations

  1. K. Menger, "Allgemeine Räume und Cartesische Räume. Erste Mitteilung", Proc. Akad. Wetensch. Amsterdam 29 (1926), 476–482.
  2. P. A. P. Moran, "Additive functions of intervals and Hausdorff measure", Proc. Cambridge Philos. Soc. 42 (1946), 15–23.
  3. J. E. Hutchinson, "Fractals and self-similarity", Indiana Univ. Math. J. 30 (1981), 713–747.
  4. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., Wiley (2014), §9.1.