Cantor set
Conservation law

It has been found that some form of conservation law is always responsible behind scaling and self-similarity. In the case of Cantor set it can be seen that the d f {\displaystyle d_{f}} th moment (where d f = ln (2) / ln (3) {\displaystyle d_{f}=\ln(2)/\ln(3)} is the fractal dimension) of all the surviving intervals at any stage of the construction process is equal to a constant which is one in the case of the Cantor set. We know that there are N = 2 n {\displaystyle N=2^{n}} intervals of size 1 / 3 n {\displaystyle 1/3^{n}} present in the system at the n {\displaystyle n} th step of its construction. Then if we label the surviving intervals as x 1, x 2, …, x 2 n {\displaystyle x_{1},x_{2},\ldots,x_{2^{n}}} then the d f {\displaystyle d_{f}} th moment is x 1 d f + x 2 d f + ⋯ + x 2 n d f = 1 {\displaystyle x_{1}^{d_{f}}+x_{2}^{d_{f}}+\cdots +x_{2^{n}}^{d_{f}}=1} since x 1 = x 2 = ⋯ = x 2 n = 1 / 3 n {\displaystyle x_{1}=x_{2}=\cdots =x_{2^{n}}=1/3^{n}}.

The Hausdorff dimension of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.
keywords:existence, punctuation:no_comma, combination:two_responses