Using triangles to partition a disk (Q1814017)

The problem of imbedding sets in \(E^ 2\) using a small number of line segments is of interest while considering linear imbeddings of simplicial complexes in \(E^ 2\). It is shown that any partition of a disk into \(n\) subdisks has a shingled imbedding in \(E^ 2\); that is, an imbedding obtained by laying down \(n\) triangles, one after another. As a corollary to this, it is observed that every partition of a disk into \(n\) subdisks can be imbedded in \(E^ 2\) using \(2n+1\) line segments.