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 (3φ − 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 interesting properties.
fractal and fractional
Article
Cabinet of Curiosities: The Interesting Geometry of
the Angle β = arccos ((3φ − 1) /4)
Fang Fang, Klee Irwin *, Julio Kovacs and Garrett Sadler
Quantum Gravity Research, Topanga, CA 90290, USA; Fang@quantumgravityresearch.org (F.F.);
jak3377@gmail.com (J.K.); imgarypenn@gmail.com (G.S.)
* Correspondence: klee@quantumgravityresearch.org
Received: 12 September 2019; Accepted: 27 October 2019; Published: 30 October 2019
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 (3φ− 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.
Keywords: tetrahedron; Golden Ratio; rotational transformation
1. Introduction
The 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. In the sections below, 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