ОБ ОДНОМ КЛАССЕ СИЛЬНО СИММЕТРИЧНЫХ МНОГОГРАННИКОВ

Abstract

We prove the completeness of the list of closed convex polyhedra in E 3 , that are strongly symmetric with respect to the rotation of the faces . Polyhedron is called symmetric if it has at least one non-trivial rotation axis. All axes intersect at a single point called the center of the polyhedron. All considered polyhedra are polyhedra with the center. A convex polyhedron is called a strongly symmetrical with respect to the rotation of the faces, if each of its faces F has an rotation axis L , intersects the relative interior of F , and L is the rotation axis of the polyhedron. It is obvious that the order of rotation axis of L does not necessarily coincide with the order of this axis, if the face of F regarded as a figure separated from the polyhedron. It has previously been shown, that the requirement of global symmetry of the polyhedron faces the rotation axis can be replaced by the weaker condition of symmetry of the star of each face of the polyhedron: to polyhedron was symmetrical with respect to the rotation of the faces, it is necessary and sufficient that some nontrivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. Under the star of face F is understood face itself and all faces have at least one common vertex with F . Given this condition, the definition of the polyhedron strongly symmetric with respect to the rotation of the faces is equivalent to the following: the polyhedron is called a strongly symmetrical with respect to the rotation of the faces , if some non-trivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. In the proof of the main theorem on the completeness of the list of this class of polyhedra using the result of the complete listing of the so- called polyhedra of 1st and 2nd class [1]. In this paper we show that in addition to the polyhedra of the 1st and 2nd class, listed in [1], only 8 types of polyhedra belongs to the class of polyhedra stronghly symmetric with respect to the rotation of faces. Seven of this eighteen types are not combinatorially equivalent regular or semi-regular (Archimedean). One type of eight is combinatorially equivalent Archimedean polyhedra, but does not belong to polyhedra of 1st or 2nd class. Turning to the polyhedra, dual strongly symmetrical about the rotation of faces, that is, to the polyhedra, stronghly symmetric about the rotation of polyhedral angles, we get their complete listing. It follows that there are 7 types of polyhedra, highly symmetric with respect to the rotation of polyhedral angles which are not combinatorially equivalent to Gessel bodies. Class of polyhedra stronghly symmetric with respect to the rotation of faces, as well as polyhedra 1st and 2nd class mentioned above can be viewed as a generalization of the class of regular (Platonic) polyhedra. Other generalizations of regular polyhedra can be found in [3],[4], [12]-[15].