Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve

Examples of differential and integral operations are given whose dimension is modified as a result of the introduction of new radial variables. Based on the integral measure $$x^\gamma\,dx$$ , $$\gamma>-1$$ , with a weak singularity, we introduce an operator that is interpreted as the Laplace operator in the space of functions of a fractional number of variables. The integration with respect to the measure $$x^\gamma\,dx$$ , $$\gamma>-1$$ , can also be interpreted as the integration over a domain of fractional dimension. The coefficient $$\gamma>-1$$ of hidden spherical symmetry is introduced. A formula is obtained that relates this coefficient to the Hausdorff dimension of a set in $$\mathbb{R}_n$$ and the Euclidean dimension $$n$$ . The existence of hidden spherical symmetries is verified by calculating the dimension of the $$m$$ th generation of the Koch curve for arbitrary positive integer $$m$$ .