All Polyhedral Manifolds are Connected by a 2-Step Refolding (2412.02174v1)

Abstract: We prove that, for any two polyhedral manifolds P, Q, there is a polyhedral manifold I such that P, I share a common unfolding and I, Q share a common unfolding. In other words, we can unfold P, refold (glue) that unfolding into I, unfold I, and then refold into Q. Furthermore, if P, Q are embedded in 3D, then I can be embedded in 3D (without self-intersection). These results generalize to n given manifolds P_1, P_2, ..., P_n; they all have a common unfolding with an intermediate manifold I. Allowing more than two unfold/refold steps, we obtain stronger results for two special cases: for doubly covered convex planar polygons, we achieve that all intermediate polyhedra are planar; and for tree-shaped polycubes, we achieve that all intermediate polyhedra are tree-shaped polycubes.

• The paper introduces a formal 2-step refolding algorithm that guarantees a transformation between any two equal-area polyhedral manifolds via a common intermediate state.
• It demonstrates the method with special cases like doubly covered convex polygons and tree-shaped polycubes, achieving polynomial-time transformations.
• The study leverages nonobtuse triangulations and explicit embedding constructs to maintain 3D embeddability and manifold integrity during refolding.

Overview of the Paper "All Polyhedral Manifolds are Connected by a 2-Step Refolding"

This paper investigates the intriguing problem of transforming one polyhedral manifold into another through a specific process of unfolding and refolding, all within two steps. The authors establish a significant result demonstrating that for any two polyhedral manifolds of the same surface area, there exists a 2-step refolding pathway. This introduces an intermediate manifold, which acts as a transient state through which the transitions are orchestrated. Importantly, this pathway maintains manifold properties such as being embeddable in 3D-space and not having boundaries if the input manifolds are boundary-free.

Key Contributions and Results

1. 2-Step Refolding Framework: The main contribution is the formalization of a 2-step refolding algorithm. For any two polyhedral manifolds, they compute a common subdivision such that each piece can be isometrically laid out to permit a refold into a specified intermediate polyhedral embedding. This intermediary serves as a pivot state connecting the initial and final manifolds, while ensuring no intermediate results violate the manifold's embeddability or boundary constraints.
2. Specific Case Analysis: The paper extends the general formulation to particular instances, such as doubly covered convex polygons and tree-shaped polycubes. In each instance, the algorithmic pathways are specifically engineered to exploit the inherent structure of these cases, achieving efficient transformations that maintain or optimize on refolding steps. The complexity is generally polynomial, with the transformation steps fine-tuned to preserve or improve practical efficiency across various polyhedral classes.
3. Common Dissection to Nonobtuse Triangulation: The technique leverages the transformation of manifolds into nonobtuse triangulations, providing a pathway through various established results in geometrical dissection. The transformation aligns the surfaces of the two manifolds using these triangulations to enable direct geometrical transformations embedded within the context of 3D space.
4. Embedding Considerations: A pivotal aspect is the embedding of the intermediate manifold. While the authors initially propose an abstract theory-based embedding through the Burago and Zalgaller theorem, they also contribute a concrete, explicit embedding construction. This involves specific folding algorithms that dovetail theoretical constructs with practical realization in 3D space.

Theoretical and Practical Implications

Theoretical Implications: The proving of connectedness in the transformations of such manifolds implies a depth of structure commonality in the field of polyhedral geometry. This understanding offers profound insight into homotopies and the potential for simpler morphing processes between complex shapes.

Practical Implications: This research has implications for fields such as material science and robotic manipulation, where the translation from one physical state to another without tearing or breaking is critical. By enabling a systematic two-step refolding process, new pathways for technological advancements in shape-shifting robotics or deployable structures may be achieved.

Future Directions

The groundwork laid by this research opens several future research avenues:

• Further Generalization and Complexity Reduction: Exploring whether one-step refolding is impossible in certain scenarios could refine our understanding and potentially lead to more optimized refolding strategies.
• Convex Intermediates for Convex Manifolds: The possibility of achieving convex intermediates when transitioning between two convex polyhedra remains an open challenge.
• Non-Intersecting Structures: Extending results to more complex polyhedral structures, especially those which are not initially well-separated, could mitigate the computational complexity further.

This paper provides foundational insights into the geometry of polyhedra and establishes a structured pathway for reassessing manifold refolding processes, adding a robust framework to the transformation of complex geometrical shapes through minimal steps.