In the plane IR2, let A0 be the unit interval on the x-axis, and let A(1) be the polygonal path with nodes (0, 0), [Formula: see text], (½, 0), [Formula: see text], (1, 0). Let S be the operator which, applied to a segment B(0) in IR2, replaces it by a polygonal path B(1) = SB(0), a similar copy of A(1), but with the same endpoints as B(0). Denote by S(n) the n-th iterate of S. The limit set (with respect to the Hausdorff metric) A(∞) = lim n → ∞ S(n)A(0) is a space-filling curve which is the closure of its interior and the union of four half-size copies of itself, intersecting only in their boundaries. Although A(∞) is of infinite connectivity, it is a tile tessellating the plane. It is related to the set of Eisenstein fractions and has a boundary of Hausdorff dimension [Formula: see text]