In this paper, we consider a kind of discrete surfaces in the three-dimensional Euclidean space called a regular pentagon ring. It is a discrete surface obtained by attaching a finite number of pairwise congruent regular pentagons along their edges such that the closed polygonal line on the surface, which connects the midpoints of those edges with line segments, is a trivial knot.
In the main theorem, we will show that if a regular pentagon ring is planar, it can be folded in one regular pentagon.
Hiromi Ei. Hiroko Hayashi. Kazushi Komatsu. "On folding of planar regular pentagon rings." Nihonkai Math. J. 31 (1) 45 - 58, 2020.