arXiv subjects

Chie Nara

Publications and source records attributed to Chie Nara.

4 recordsLinked to original sources

arXiv2025

Quasigeodesics on the Cube

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le \pi$ surface to each side at every point. In contrast, a geodesic has exactly $\pi$ to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly $15$ simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail $15$ simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

math.MG↗

arXiv2024

Folding One Polyhedral Metric Graph into Another

We analyze the problem of folding one polyhedron, viewed as a metric graph of its edges, into the shape of another, similar to 1D origami. We find such foldings between all pairs of Platonic solids and prove corresponding lower bounds, establishing the optimal scale factor when restricted to integers. Further, we establish that our folding problem is also NP-hard, even if the source graph is a tree. It turns out that the problem is hard to approximate, as we obtain NP-hardness even for determining the existence of a scale factor 1.5-{\epsilon}. Finally, we prove that, in general, the optimal scale factor has to be rational. This insight then immediately results in NP membership. In turn, verifying whether a given scale factor is indeed the smallest possible, requires two independent calls to an NP oracle, rendering the problem DP-complete.

cs.CG↗

arXiv2024

Continuous Flattening and Reversing of Convex Polyhedral Linkages

We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio.

cs.CG↗

arXiv2021

Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases

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.

cs.CG↗