Origami embedding of piecewise-linear two-manifolds (Q627524)

Let \(P_i\) be an open polygon in the Euclidean plane identified with the plane \(z=0\) in \(\mathbb{E}^3\). Let \(\hat{P}_i\) be the closed polygon on the plane \(z=i\epsilon\) projecting onto the closure of \(P_i\). This induces an ordering on any family of polygons such that if \(i<j\), then \(\hat{P}_i\) lies below \(\hat{P}_j\). Moreover, given \(P_i\) and \(P_k\) sharing an edge, let us denote by \(\hat{P}_i\cdot \hat{P}_{k}\) the polygon surface consisting in \(\hat{P}_i\), \(\hat{P}_k\) and the vertical wall projecting onto the common edge. A flat-folding consists of a finite set of open polygons \(P_1,\dots, P_n\) in the Euclidean plane, along with joins between pairs of polygons sharing an edge. Two polygons \(P_i\) and \(P_k\), \(i<k\), may be joined at an edge \(e\) if: {\parindent6.5mm \begin{itemize}\item[(1)] \(e\) is a boundary edge of each of them. \item[(2)] No \(P_j\) with \(i<j<k\) intersects \(e\). \item[(3)] No \(P_j\) with \(i<j<k\) is joined to a polygon \(P_l\), with \(1\leq l\leq n\), such that \(\hat{P}_i\cdot \hat{P}_{k}\) crosses \(\hat{P}_j\cdot \hat{P}_l\). \end{itemize}} A metric piecewise-linear (PL) 2-manifold is a finite complex of Euclidean polygons with the topology of a 2-manifold (possibly with boundary). A metric PL 2-manifold \(H\) is said to have a flat-folded realization if there is a subdivision of \(H\), with faces subdivided by extra vertices and edges (``creases''), and a continuous, bijective mapping from \(H\) to a flat folding, taking each subface of \(H\) to an isometric copy in the flat folding. In this setting, the main result of the paper is: ``Every compact, orientable, metric PL 2-manifold has a flat-folded realization''. In other words, any ``reasonable'' polyhedron can be folded into a finite set of planar faces in the plane, ``plus layers''.