Cabinet of curiosities: the interesting geometry of the angle {\beta} = arccos((3{\phi} - 1)/4)

TL;DR

This paper explores geometric constructions of tetrahedral aggregates involving rotations that produce specific angular relationships related to the golden ratio, revealing interesting symmetries and properties.

Contribution

It introduces new tetrahedral arrangements with specific angular displacements linked to the golden ratio, demonstrating symmetry and face contact properties.

Findings

Angular displacement {eta} = arccos((3{} - 1)/4) observed in junctions

Reduction in the number of distinct face orientations after transformations

Construction of both periodic and aperiodic tetrahedral aggregates

Abstract

In this paper we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are "closed" (in the sense that faces of adjacent tetrahedra are brought into contact to form a "face junction") while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of {\beta} = arccos((3{\phi} - 1)/4) (or a closely related angle), where {\phi} is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several "curiosities" involving the structures discussed here with the goal of inspiring the reader's interest in constructions of this nature and their attending, interesting properties.