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

Quantum Gravity Research, Topanga, CA 90290, USA
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Fractal Fract 2019, 3(4), 48; https://doi.org/10.3390/fractalfract3040048
Received: 12 September 2019 / Revised: 26 October 2019 / Accepted: 27 October 2019 / Published: 30 October 2019
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. View Full-Text
Keywords: tetrahedron; Golden Ratio; rotational transformation tetrahedron; Golden Ratio; rotational transformation
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Fang, F.; Irwin, K.; Kovacs, J.; Sadler, G. Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4). Fractal Fract 2019, 3, 48.

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