Toward Unfolding Doubly Covered $n$-Stars

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.