Finite similarity iterated function system ($\mathcal{T}(K)={\bigcup }_{i=1}^m{S}_i(K)$)

For finitely many contracting Euclidean similarities, the map $\mathcal{T}$ on nonempty compact subsets of ${\mathbb{R}}^n$ is a contraction in the Hausdorff metric, with contraction constant ${\max }_i{c}_i<1$. The contraction mapping theorem gives a unique nonempty compact invariant set. If the open set condition holds, its Hausdorff dimension is the unique $s$ satisfying ${\sum }_i{c}_i^s=1$.

The Minkowski sausage replaces a segment by eight directed unit segments whose endpoint displacement is four units. A middle straight run of length two represents two collinear unit segments. After normalizing the displacement to one, the eight contracting Euclidean similarities all have ratio $1/4$. The open diamond $\{|x-1/2|+|y|<1/2\}$ is an open set condition witness: its eight image diamonds have disjoint interiors and lie within it. Hence $8(1/4{)}^s=1$, giving Hausdorff dimension $3/2$. Treating the middle run as a single similarity of ratio $1/2$ instead defines a different iterated scheme.