On folding of planar regular pentagon rings

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.