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Drawing Paths on the Sierpinski Carpet

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Plot Type
Path
Array
Rules
Recursion
0
1
2
3
The Sierpinski carpet [1] is a well-known hierarchical decomposition of the square plane tiling associated with
2
, that is, pairs of integers. Consider the Sierpinski graph [2], which is the adjacency graph of
2
/S
(the complement of
S
in
2
), where
S
is one of the hierarchical subsets of
2
. Gray squares are used to depict the intersection of
S
with a
n
3
×
n
3
subset of
2
. 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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