We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small.

我们证明，如果扩展标准折叠模型以允许无数折痕出现，则可以将3D中的任何有限多面体流形连续平展为2D，同时保留固有距离并避免交叉，从而回答了19年的开放问题。先前已知可以连续展平的最一般情况是凸多面体和半正交多面体。对于不可定向的歧管，即使存在瞬时展平（展平折叠状态）也是新的结果。我们的解决方案扩展了一种用于展平半正交多面体的方法：沿平行平面切片多面体，并展平连续平面之间的多面体条。我们通过将条带展平到非正交角并沿着无数个平行平面切片，使切片密集地接近流形的每个顶点，从而使该方法适用于任意非凸多面体。我们还表明，需要支持移动折痕的多面体区域（贝娄定理对闭合多面体是必需的）可以任意减小。