Deca-Metallic Ratios: A Base-10 Analogue of Metallic Means with their Geometric Origin in Primitive Pythagorean Triples

Source abstract

This paper introduces the \emph{Deca-Metallic Ratios}, defined by \delta_n = n + \sqrt{n^2+10}, \qquad n\in\mathbb{Z}^{+},\which are the positive roots of the quadratic equation x^2-2nx-10=0. By fixing the algebraic norm to $10$, the base of the decimal numeral system, we establish a new family of metallic ratios whose algebraic properties manifest directly in ordinary decimal expansions. This reveals a remarkable form of self-similarity that is simultaneously algebraic and arithmetic. The principal result is the \emph{decimal-shift phenomenon}. For every $n\ge1$ and every integer $k\ge0$, the fractional part of $\delta_n$ satisfies \{\delta_n\}=\left\{\frac{10}{\delta_n}\right\}, and, more generally, \left\{\delta_n^{2k+1}\right\}=\left {\frac{10^{2k+1}}{\delta_n^{2k+1}} \right\},for every odd power. In contrast, the even powers satisfy the complementary identity \left\{\delta_n^{2k}\right\}+\left\{ \frac{10^{2k}}{\delta_n^{2k}}\right\}=1. These identities arise naturally from the sign of the conjugate root, providing a unified algebraic explanation for both the preservation law of odd powers and the complementarity law of even powers. Together, they constitute the characteristic arithmetic signature of the Deca-Metallic family. We further prove that every Deca-Metallic Ratio is canonically generated by a primitive Pythagorean triple $(a,b,c)$ through the parameter n=\frac{2a}{c-b}, which admits the trigonometric representation n=2\cot\left(\frac{\theta}{2}\right), and satisfies the Crown Identity \delta_n= \frac{2a}{c-b}+\sqrt{\frac{8c}{c-b}+6},which closely parallels the classical metallic-means identity\mu_n=\frac{a}{c-b}+ \sqrt{\frac{2c}{c-b}}. Using Euclid's parametrization, we obtain a complete classification of admissible primitive triples into three arithmetic families, referred to as \emph{Pythagoras}, \emph{Plato}, and \emph{Socrates}, together with an explicit bijective inverse construction. The distinguished case $n=5$ yields the \emph{Decimal Golden Ratio} \delta_5=5+\sqrt{35}, which is uniquely characterized by \lfloor\delta_5\rfloor=10, equal to the decimal base. Consequently, \frac{1}{\delta_5}=\frac{\{\delta_5\}}{10}, providing an exact decimal analogue of the classical identity \frac{1}{\phi}=\phi-1. The paper also develops continued-fraction expansions, a matrix formulation, the associated Deca-Fibonacci and Deca-Lucas integer sequences, a Cassini-type identity, and the exact transcendental representation \delta_n = \sqrt{10}\, \exp\left( \operatorname{arcsinh}\left(\frac{n {\sqrt{10}}\right) \right), which embeds the entire family as lattice points on a hyperbolic manifold. Taken together, these results establish a unified algebraic, geometric, arithmetic, and transcendental theory of the Deca-Metallic Ratios. This framework substantially extends the classical theory of metallic means while uncovering a previously unrecognized connection among right-triangle geometry, the decimal numeral system, and hyperbolic geometry.