# Drawing Paths on the Sierpinski Carpet

Drawing Paths on the Sierpinski Carpet

The Sierpinski carpet [1] is a well-known hierarchical decomposition of the square plane tiling associated with , that is, pairs of integers. Consider the Sierpinski graph [2], which is the adjacency graph of (the complement of in ), where is one of the hierarchical subsets of . Gray squares are used to depict the intersection of with a subset of . The replacement rule construction affirmatively answers the question, is it possible to draw a Hamiltonian cycle on the Sierpinski graph? [3]. The combinatorial ambiguity of the replacement rules defined here suggests the existence of a rich collection of valid Hamiltonian cycles, as yet unexplored (see Details). The challenge is to find your own coloring of the replacement rules that may ultimately determine a unique traversal of the Sierpinski carpet.

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