Further stellations of the uniform polyhedra (Q2268036)

The paper deals with particular stellations of polyhedra. By definition, a polyhedron, \(P\), has a surface composed of subsets of planes; the subsets are called \textit{faces}, the planes are referred to as facial planes. The full set of facial planes of \(P\) divides space into regions, the finite regions are referred to as cells. A \textit{stellation} of \(P\) is a polyhedron, \(S\), comprising any set of these cells, which has the following properties: (i) the complement of \(S\) is a connected region; (ii) \(S\) maintains the original symmetries of \(P\) [cf. \textit{M. J. Wenninger}, Polyhedron models. Cambridge: University Press (1971; Zbl 0222.50010)]. A \textit{main-line stellation} of \(P\) is constructed by adding to \(P\) the external cells adjacent to its boundary surface; such iterative process produces a sequence of main-line stellations of \(P\). The author is interested in stellations of \textit{uniform non-convex} polyhedra; uniform means that all the faces are regular polygons (including the star polygons) and all the vertices are congruent to each other; non-convex means that the polyhedron is self-intersecting. A list of uniform non-convex polyhedra, which comprises 43 items, may be found in [loc. cit.]. In his previous paper [\textit{J. L. Hudson} and \textit{J. G. Kingston}, Math. Intell. 10, No.3, 50--61 (1988; Zbl 0645.51022)], the author described external main-line stellations of particular non-convex uniform polyhedra. The complete list of (external) main-line stellations produced for non-convex uniform polyhedra includes 366 items. Now, the author discusses original intrinsic analogues of stellations for non-convex polyhedra. Namely, an \textit{internal stellation} is produced by removing internal cells and retaining the mentioned properties (i)-(ii). An \textit{internal main-line stellation} of a non-convex uniform polyhedra is constructed by removing the internal cells adjacent to its boundary surface; this iterative process results in a unique sequence of internal main-line stellations of the polyhedron in question. The author enumerates the internal main-line stellations generated by the 43 non-convex uniform polyhedra, this list includes 104 items. The process of construction of internal main-line stellations is illustrated in detail with reference to two polyhedra, the quasi-truncated hexahedron and the quasi-truncated great stellated dodecahedron. The paper is written quite clearly, it is accessible even to non-specialists. Moreover, the text includes numerous fascinating images of stellated polyhedra. So it may be really recommended to all who is interested in the geometry of polyhedra.