Convex polyhedra with deltoidal vertices

V. I. Subbotin

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 491-500.

Résumé

In paper the class of closed convex symmetric polyhedra in $E^3$ with a special structure of some vertices is introduced: the set $Star(V)$ of all faces incident to such vertices consists of equal deltoids. Such vertices are called deltoidal in the article. The deltoids here are convex quadrilaterals that have two pairs of equal adjacent sides and are different from rhombuses. It is also assumed that each deltoidal vertex $V$ of a polyhedron and each face that is not included in the star of any deltoidal vertex are locally symmetric. The local symmetry of a vertex means that the rotation axis $L_V$ of order n of the figure S= $Star (Star (V))$, where n is the number of deltoids of $Star (V)$, passes through $V$; $S$ is a set of faces consisting of the set $Star(V )$ and all faces having at least one common vertex with the set $Star(V)$. The local symmetry of the face F means that the rotation axis $L_F$ , which intersects the relative interior of F and perpendicular to $F$, is the rotation axis of the star $Star(F)$.$ DS $ — so denotes a class of polyhedra that have locally symmetric deltoidal vertices and there are faces that do not belong to any star of the deltoidal vertices; In addition, all faces that do not belong to any star of deltoidal vertices are locally symmetric.In this paper we prove a theorem on the complete enumeration of polyhedra of the class $ DS $, in which all deltoidal vertices are isolated. The isolation, or separation, of the vertice $ V $ means that its star of faces does not have common elements with star of faces of any other vertex of the polyhedron.We also consider polyhedra, through each vertex $ V $ of which the rotation axis of the star $ Star (V) $ passes, and $ V $ is not assumed to be deltoidal in advance; if such polyhedra have at least one deltoidal face, then there are only three such polyhedra.The proofs of the statements in the paper are based on the properties of the so-called textit {strongly symmetric polyhedra}. Namely, polyhedra that are strongly symmetric with respect to the rotation of the faces.

Détail

Keywords: delitoidal vertex, strongly symmetric polyhedron, locally symmetric vertex, locally symmetric face.

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