No-net Polyhedra

1. Introduction. It is often said that when the conditions are right, discoveries are made by different people at the same time. The discovery of the non-Euclidean geometry, and the invention of calculus are the most frequently mentioned examples. Here is another example (with an incomparably less significant subject) which I noticed recently. My note [4] described a starshaped polyhedron which admits no "net", as this term is understood by anybody building cardboard models of polyhedra. The note was motivated by the paper of Tarasov [7] published in 1999. Tarasov constructed a nonet starshaped polyhedron with convex faces, and thereby solved a problem posed by N. G. Dolbilin. (Throughout this note, we consider only starshaped polyhedra with convex faces. All nets are assumed to be obtained by cuts along the edges of the polyhedra in question.) After the publication of [4] I became aware of the paper [1], also published in 1999, in which a different nonet polyhedron is constructed. While preparing the present note, a message from Craig Kaplan led me to the more detailed account [2] of the results of [1]; this seems not to have been published in print so far, but is available on the Internet. Various nonet polyhedra are presented there, including one with only 24 faces. Another result of [2] is an affirmative resolution of Conjecture 2 of [4], establishing the existence of nonet polyhedra with only triangles as faces. The main purpose of this note is to present a nonet polyhedron with only 13 faces; this is done in Section 2. Section 3 discusses certain related questions, as well as some other remarks. 2. A small nonet polyhedron. We shall show: Theorem. There exist convex-faced, starshaped nonet polyhedra with 13 faces. Proof. A polyhedron establishing this result is shown in Figure 1(a). It is constructed as follows. We start from a sufficiently tall pyramid, with an equilateral triangle as basis. This is truncated at each of the basis vertices, to obtain a polyhedron with seven faces –– three