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Statement

  1. Sierpinski triangle \(S_\triangle\) — the attractor of \(N=3\) similarities of ratio \(r=1/2\) (the three corner half-copies of a triangle) — has similarity (Hausdorff) dimension $\(D_\triangle=\frac{\log 3}{\log 2}\approx 1.584962500721156.\)$
  2. Sierpinski carpet \(S_\square\) — the attractor of \(N=8\) similarities of ratio \(r=1/3\) (the eight non-central cells of a \(3\times3\) grid on a square) — has similarity (Hausdorff) dimension $\(D_\square=\frac{\log 8}{\log 3}\approx 1.8927892607143724.\)$

Assumptions

  1. \(S_\triangle\) is the attractor of \(N=3\) contracting similarities with equal ratio \(r=1/2\); \(S_\square\) of \(N=8\) contracting similarities with equal ratio \(r=1/3\).
  2. Open set condition (OSC): for the triangle, the interiors of the three half-scale copies are pairwise disjoint (open set \(U=\) interior of the seed triangle); for the carpet, the eight \(1/3\)-scale sub-squares have pairwise disjoint interiors (open set \(U=\) open unit square). Under the OSC the similarity dimension equals the Hausdorff dimension.
  3. Both dimensions lie strictly between the topological dimension (\(1\) for the triangle's gasket structure, \(1\) for the carpet) and the ambient dimension \(2\).

Proof

  1. [asserted] By Moran/Hutchinson under the OSC, the dimension \(D\) of an equal-ratio IFS attractor solves the Moran equation \(N\,r^{D}=1\).
  2. [verified] Triangle. Solving \(3\cdot(1/2)^{D}=1\) gives \(2^{D}=3\), so \(D_\triangle=\log 3/\log 2\). SymPy solve(3*(1/2)**D - 1, D) returns log(3)/log(2), numerically \(1.584962500721156\).
  3. [verified] Carpet. Solving \(8\cdot(1/3)^{D}=1\) gives \(3^{D}=8\), so \(D_\square=\log 8/\log 3\). SymPy solve(8*(1/3)**D - 1, D) returns 3*log(2)/log(3) \(=\log 8/\log 3\), numerically \(1.8927892607143724\).
  4. [verified] Measure/area consistency (sanity check). Each fractal is a \(\lambda\)-null set in \(\mathbb{R}^2\): the triangle retains a fraction \((3/4)^{k}\to 0\) of the seed area after \(k\) steps (removing one of four sub-triangles each step), and the carpet retains \((8/9)^{k}\to 0\) (removing the central one of nine sub-squares each step). Both limits are \(0\), consistent with \(D<2\).
  5. [verified] Both computed dimensions satisfy \(1<D<2\), placing each set strictly between a curve and a filled region, as required of a planar fractal.

Verification

  • Triangle (step 2): SymPy solve(3*2**(-D)-1, D) = log(3)/log(2), float \(1.584962500721156\); matches the Moran root for \(N=3,\ r=1/2\).
  • Carpet (step 3): SymPy solve(8*3**(-D)-1, D) = 3*log(2)/log(3), float \(1.8927892607143724\); matches the Moran root for \(N=8,\ r=1/3\).
  • Area decay (step 4): ratios \(3/4<1\) and \(8/9<1\) give geometric decay to measure zero — elementary, machine-confirmable on finite \(k\).
  • Asserted external result: Moran/Hutchinson OSC ⇒ Hausdorff-dimension identity (step 1).

Historical source

Wacław Sierpiński introduced the triangle (gasket) in 1915 and the carpet in 1916 as examples of curves — in the topological sense — that are universal for planar continua. The Sierpinski carpet is the planar analogue whose three-dimensional generalization is the Menger sponge (Menger 1926). Dimension values follow from the later Moran/Hausdorff self-similarity framework.

Related concepts

Citations

  1. W. Sierpiński, "Sur une courbe dont tout point est un point de ramification", C. R. Acad. Sci. Paris 160 (1915), 302–305 (triangle).
  2. W. Sierpiński, "Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée", C. R. Acad. Sci. Paris 162 (1916), 629–632 (carpet).
  3. K. Menger, "Allgemeine Räume und Cartesische Räume", Proc. Akad. Wetenschappen Amsterdam 29 (1926), 476–482 (sponge).
  4. J. E. Hutchinson, "Fractals and self-similarity", Indiana Univ. Math. J. 30 (1981), 713–747.
  5. K. Falconer, Fractal Geometry, 3rd ed., Wiley (2014), §9.