The construction of Platonic bodies from constant width continuous strips

The construction of stellated regular polyhedra from constant width strips

“…The only element needed to construct both of them is the triangular element A of figure 2 (a). This triangle is obtained by a simple manual fold starting from one of the isosceles trapezes that are generated when a pentagonal knot [1] is unfolded.…”

“…The problem is already solved by the above mentioned trapezes marked on the constant width strip, since the major base of the trapeze is the convex pentagon diagonal or the stellated pentagon side, whereas the minor base is the convex pentagon side [1]. Therefore, to mark the pair of adjacent triangular elements A and C on the strip, we choose two adjacent trapezes (see figure 5 (a)) and we fold the second trapeze in such a way that P and Q coincide to mark the perpendicular at the middle point of segment PQ.…”

“…We have solved the five Platonic bodies construction problem starting from constant width continuous strips (see [1]), without performing measurement or using geometric instruments.…”

“…Recently, our group has paid attention to the study of some questions related to polyhedra topics [1,6]. We have solved the five Platonic bodies construction problem starting from constant width continuous strips (see [1]), without performing measurement or using geometric instruments.…”

“…This is possible because the manual performing of a simple knot in, for example, a paper strip marks isosceles trapezes on the strip (see [1]), and their bases are related in such a way that one is the golden section of the other, as is known to occur between sides of the stellated regular pentagon and sides of the circumscribed convex regular pentagon. As in our work on convex regular polyhedra [1], all the constructions will be manual in the present work, without performing measurements or using geometric instruments.…”