Preprint Communication Version 2 Preserved in Portico This version is not peer-reviewed

Metallic Ratios are Defined by an Argument of a Normalized Complex Number

Version 1 : Received: 5 January 2024 / Approved: 5 January 2024 / Online: 8 January 2024 (06:20:05 CET)
Version 2 : Received: 9 January 2024 / Approved: 10 January 2024 / Online: 10 January 2024 (04:28:16 CET)
Version 3 : Received: 15 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (06:19:33 CET)
Version 4 : Received: 1 February 2024 / Approved: 1 February 2024 / Online: 2 February 2024 (04:33:46 CET)

A peer-reviewed article of this Preprint also exists.

Łukaszyk, S. (2024). Metallic Ratios and Angles of a Real Argument. IPI Letters, 2(1), 26–33. https://doi.org/10.59973/ipil.55 Łukaszyk, S. (2024). Metallic Ratios and Angles of a Real Argument. IPI Letters, 2(1), 26–33. https://doi.org/10.59973/ipil.55

Abstract

We show that metallic ratios for real k are defined by an argument of a normalized complex number, while for rational k ≠ {0, ±2}, they are defined by Pythagorean triples.

Keywords

metallic ratios; Pythagorean triples; emergent dimensionality

Subject

Computer Science and Mathematics, Mathematics

Comments (1)

Comment 1
Received: 10 January 2024
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author
Comment: 1. Extended validity of the θ(k) relation to 0 ≤ θ < 2π by analytic continuation.
2. Proof of conjecture about the Pythagorean triple corresponding to θ(k) for rational k.
3. The title was amended to reflect the results of the study.
4. Improvements in reasoning and clarity.
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