Abstract
We present nonoverlapping general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n-pointed star polygon whose triangular points have base angle \(\alpha \). Specifically, we construct general unfoldings when \(n \in \{3,4,5,6,8,9,10,12\}\) (no matter the value of \(\alpha \)), and we construct general unfoldings when \(\alpha < 60^\circ (1 + 1/n)\) (i.e., when the points are shorter than equilateral, no matter the value of n, or slightly larger than equilateral, especially when n is small). Whether all doubly covered star polygons, or more broadly arbitrary nonconvex polyhedra, have general unfoldings remains open.
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Akitaya, H.A. et al. (2021). Toward Unfolding Doubly Covered n-Stars. In: Akiyama, J., Marcelo, R.M., Ruiz, MJ.P., Uno, Y. (eds) Discrete and Computational Geometry, Graphs, and Games. JCDCGGG 2018. Lecture Notes in Computer Science(), vol 13034. Springer, Cham. https://doi.org/10.1007/978-3-030-90048-9_10
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