Abstract
We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect (and so a doubly covered polygon is a polyhedron). A flat folding of a polyhedron is a folding by creases into a multilayered planar shape ([7], [8]). A. Cauchy [4] in 1813 proved that any convex polyhedron is rigid: precisely, if two convex polyhedra P, P ′ are combinatorially equivalent and their corresponding faces are congruent, then P and P ′ are congruent. By removing the condition of convexity, R. Connelly [5] in 1978 gave an example of a (non-convex) flexible polyhedron: precisely, there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that for every t 6= 0, the corresponding faces of P0 and Pt are congruent while polyhedra P0 and Pt are not congruent. (See also [6].) After then I. Sabitov [15] in 1998 proved that the volume of any polyhedron is invariant under flexing: precisely, if there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that, for every t, the corresponding faces of P0 and Pt are congruent, then the volumes P0 and Pt are equal for all 0 ≤ t ≤ 1. (See also [14].) A. Milka [12] in 1994 showed that any polyhedron admits a continuous (isometric) deformation by using moving edges, and that all Platonic polyhedra can be changed in
Highlights
We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect
[5] in 1978 gave an example of a flexible polyhedron: precisely, there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that for every t = 0, the corresponding faces of P0 and Pt are congruent while polyhedra P0 and Pt are not congruent. (See [6].) After I
Sabitov [15] in 1998 proved that the volume of any polyhedron is invariant under flexing: precisely, if there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that, for every t, the corresponding faces of P0 and Pt are congruent, the volumes P0 and Pt are equal for all 0 ≤ t ≤ 1. (See [14].)
Summary
We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect (and so a doubly covered polygon is a polyhedron). Milka [12] in 1994 showed that any polyhedron admits a continuous (isometric) deformation by using moving edges, and that all Platonic polyhedra can be changed in We discussed on flattening of convex polyhedra by a continuous (isometric) deformation which we call a continuous folding process (see Definition 1). For such a deformation, its faces must be changed by moving creases as its volume decreases. We give explicit formulas of continuous functions for a continuous flat folding process in the case of a regular tetrahedron We leave such calculation for other Platonic polyhedra in future work
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