Tetrahedral cages for unit discs (Q2046811)

A cage is the \(1\)-skeleton of a (convex) polytope in the \(3\)-dimensional space \(\mathbb{R}^3\). A disc is a compact \(2\)-dimensional ball in \(\mathbb{R}^3\). A cage \(G\) is said to hold a compact set \(K\) disjoint from \(G\) if there is no rigid continuous motion bringing \(K\) in a position far away without meeting \(G\) on its way. The paper addresses a problem, which refers to the number of positions in which the unit disc can be held by tetrahedral cages. The main aim of the authors is to answer the question: In how many positions can unit discs be held by a tetrahedral cage? A very closed question was posed, and investigated, in [\textit{L. Yuan} and \textit{T. Zamfirescu}, Ars Math. Contemp. 17, No. 1, 255--270 (2019; Zbl 1436.52012)], where it is referred to two queries raised in [\textit{L. Montejano} and \textit{T. Zamfirescu}, Springer Proc. Math. Stat. 148, 263--264 (2016; Zbl 1373.52004)]. The problem of holding \(3\)-dimensional balls in cages seems to go back to a question of Coxeter, which was settled by \textit{A. S. Besicovitch} [Proc. Sympos. Pure Math. 7, 19--20 (1963; Zbl 0141.19903)], and \textit{O. Aberth} [Proc. Lond. Math. Soc. (3) 13, 322--336 (1963; Zbl 0113.16204)] (see also [\textit{G. Valette}, Bull. Soc. Math. Belg. 21, 124--125 (1969; Zbl 0189.52803)]). The main result in the paper is the following: Theorem 2. There are tetrahedral cages holding \(n\) unit discs for every \(n\in\{1,2,3,4,6,8,12,16\}\), and there are no such cages for any other \(n\). In [Yuan and Zamfirescu, loc. cit.], it was proven that there exist tetrahedral cages holding \(n\) discs if and only if \(n\in\{1,2,3,4,5,6,8,10,12,16\}\). Indeed, the tetrahedral cage in the mentioned result in [loc. cit.] happens to hold \(8\) unit discs. It is to notice that the cases \(n=5,10\), which are included in this result, are no longer feasible for tetrahedral cages holding unit discs (main result). In the last section of the paper, motivated by the fact that \(7\) is the smallest number for which there is no tetrahedral cage holding \(7\) discs, the authors extend Theorem 3.5 in [loc. cit.], which provides with the existence of pentahedral cage holding 7 discs. More precisely, the authors prove the following: Theorem 3. There exists a quadrilateral pyramid, such that the associated cage holds \(7\) unit discs.