The Translative Kissing Number of Tetrahedra Is 18

Abstract

We show that the maximum number of mutually nonoverlapping translates of any tetrahedron T which touch T is 18. Moreover, in the case of 18 touching translates the arrangement turns out to be unique. We also give a description of all possible arrangements of 17 touching translates. Finally, we apply these results to determine the minimum and maximum densities of 17 + -neighbor translative packings of tetrahedra.

Exponential Lower Bound for the Translative Kissing Numbers of d -Dimensional Convex Bodies

“…For additional related results and references on this topic, see [1], [2], [5], [7], [14], [16], and the survey paper by Fejes Tóth and Kuperberg [4].…”

Mysteries in Packing Regular Tetrahedra

“…Three years later, by modifying Zong's method, Talata [53] was able to improve (8) to τ t (T ) = 18.…”

A Lower Bound for the Translative Kissing Numbers of Simplices

“…denotes the usual Euclidean norm, and w is a vector parallel to v and having endpoint on the boundary ∂K of K. The metric derived from this norm is called the Minkowski metric determined by K. A set S is called 1-discrete in the Minkowski metric determined by K if s 1 −s 2 K ≥ 1 for any s 1 ,s 2 ∈ S. If the Minkowski metric is fixed, then we call S 1-discrete for short. By Minkowski [14] (also see [15], [20]), for a convex body K the quantity H(K) is equal to the maximum cardinality of 1-discrete subsets of ∂( 1 2 (K −K)) in the Minkowski metric determined by 1 2 (K −K). This clearly implies that H(K) = H( 1 2 (K − K)).…”

“…We note that their exact values are known only in two and three dimensions (Grünbaum [11], Talata [20]). Furthermore, we generalize the obtained lower bound for d-orthoplexes.…”