Deca-Metallic Ratios: A Base-10 Analogue of Metallic Means

**Version 2 (Unified Manuscript, April 2026) ** This paper introduces the **Deca-Metallic Ratios**, an infinite parametric family of quadratic irrationals δₙ = n + √ (n² + 10) for n ∈ ℤ⁺, defined as the positive roots of the equation x² − 2n x − 10 = 0. By anchoring the algebraic norm at 10 — the base of the decimal numeral system — this family extends the classical metallic means into a framework that exhibits profound self-similarity precisely within human arithmetic. A fundamental property is the reciprocal identity 1/δₙ = (δₙ − 2n) /10, which exactly preserves the sequence of fractional digits under decimal shifts. For odd powers, the fractional part of δₙ^2k+1 reappears unchanged in 1/δₙ^2k+1 after a precise 2k+1-place shift. The special case n = 5 yields the **Decimal Golden Ratio** δ₅ = 5 + √35, whose continued fraction, square law, and reciprocal structure align perfectly with base-10 scaling. This unified version establishes a deep geometric realization: every Deca-Metallic Ratio arises canonically from a primitive Pythagorean triple (a, b, c). Defining the governing parameter n = 2a/ (c − b) leads to the trigonometric representation n = 2 cot (θ/2) (where θ is the acute angle opposite the smaller leg) and the compact half-angle form δₙ = 2 cot (θ/2) + √ (cot² (θ/2) + 5/2). Via the Euclid parametrization, all admissible n are completely classified into three arithmetic families of primitive Pythagorean triples (Pythagoras, Plato, and Socrates) and admit a bijective inverse construction. Finally, the ratios possess the exact transcendental representation δₙ = √10 exp (sinh⁻¹ (n/√10) ), embedding them as integer lattice points on a hyperbolic manifold. Together, these results forge a **unified algebraic–decimal–geometric–hyperbolic framework** that substantially extends the classical theory of metallic means and reveals a deep connection between right-triangle geometry, human arithmetic, and hyperbolic space. **Keywords**: Deca-Metallic Ratios, metallic means, Decimal Golden Ratio, base-10 self-similarity, primitive Pythagorean triples, decadic metallic means, hyperbolic continued fractions, transcendental representation