Complementary Symmetry and Paired Epicycloidal Chord Families in Modular Exponential Orbits
Year: 2026
Type: article
Abstract: This paper investigates the geometric structure of modular exponential orbits through families of chords inscribed in a regular polygon and their associated epicycloidal envelopes. The study begins with the continuous family of chords joining the points determined by the angular parameters t and h·t on the unit circle. Its envelope is an epicycloid with h − 1 cusps. The cardioid appears as the special case h = 2. These classical geometric properties are used as the foundation for the modular construction developed in this work. The main arithmetic object is the family of moduli n = g^a(1 + g^g) − 1, where g is an integer greater than or equal to 2 and a is a positive integer. This family has a complementary modular structure that produces two related residue sequences. The first is generated by multiplication by g, while the second is generated by multiplication by g^g. The central result is the identification of two complementary congruences connecting these sequences. Together, they establish a relationship between the corresponding modular chord systems and their epicycloidal envelopes. In particular, the complementary residues determine reflection axes in the regular polygon, linking the arithmetic structure of the modulus to a geometric symmetry between the two chord families. The paper develops this relationship analytically and illustrates it with the example g = 2, a = 3, which gives the modulus n = 39. In this case, the construction produces a cardioidal envelope associated with the multiplier 2 and a second epicycloidal envelope associated with the multiplier 4. The complementary congruence determines the corresponding reflection axes. The general theory of chord envelopes and epicycloids has been studied in earlier mathematical literature. Accordingly, this paper does not claim the discovery of epicycloidal envelopes themselves. Its contribution is the arithmetic construction of a particular family of moduli that generates paired modular chord systems, together with the complementary congruence and reflection symmetry connecting them. The results establish an exact relationship between modular exponentiation, chord geometry, epicycloidal envelopes, and reflection symmetry. They do not imply statistical non-randomness, a reduction in computational complexity, or any cryptographic consequence. Such questions require separate investigation.
Author: Noguera, Luis
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225781066
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