# Percolation properties of the classic Sierpinski carpet and sponge

## TL;DR

This paper demonstrates that the iterative construction of Sierpinski carpets and sponges exhibits critical phenomena similar to percolation, deriving critical exponents and using finite-size scaling to determine transport properties.

## Contribution

It introduces a novel analogy between fractal constructions and percolation, deriving critical exponents and applying finite-size scaling for transport properties.

## Key findings

- Critical exponents are derived for Sierpinski structures.
- Finite-size scaling accurately predicts scalar transport properties.
- The percolation analogy applies at any iteration stage.

## Abstract

Iterative construction of a Sierpinski carpet or sponge is shown to be a critical phenomenon analogous to uncorrelated percolation. Critical exponents are derived or calculated (by random walks over the carpet or sponge at infinite iteration) that are related by equations identical to those obtained from percolation theory. Finite-size scaling then gives accurate values for the scalar transport properties (e.g., effective conductivity) of the carpet or sponge at any stage of iteration.