Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

Fang Fang,Klee Irwin,Julio Kovacs,Garrett Sadler

doi:10.3390/fractalfract3040048

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 β = arccos 3 ϕ − 1 / 4 (or a closely related angle), where ϕ = 1 + 5 / 2 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.

Highlights

T√he present document introduces the reader to the angle β = arccos ((3φ − 1) /4), where φ = 1 + 5 /2 is the golden ratio, and its involvement, most notably, in the construction of several interesting aggregates of regular tetrahedra
We will perform geometric rotations on tetrahedra arranged about a common central point, common vertex, common edge, as well as those of a linear, helical arrangement known as the Boerdijk–Coxeter helix [1,2]
After performing the rotations described below, one observes a reduction in the total number of plane classes, defined as the total number of distinct facial or planar orientations in a given aggregation of polyhedra

Summary

Introduction

We will perform geometric rotations on tetrahedra arranged about a common central point, common vertex, common edge, as well as those of a linear, helical arrangement known as the Boerdijk–Coxeter helix (tetrahelix) [1,2] In each of these transformations, the angle β above appears in the projections of coincident tetrahedral faces. After performing the rotations described below, one observes a reduction in the total number of plane classes, defined as the total number of distinct facial or planar orientations in a given aggregation of polyhedra. These aggregates of tetrahedra might give clues to periodic tetrahedral packing in quasicrystals [4]

Aggregates of Tetrahedra

Aggregates about a Common Edge

Aggregates about a Common Vertex

Curiosities

Conclusions