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

1. Introduction

Metallic ratios of $k\in {\mathbb{N}}_0$ are defined as

$$M_{k\pm :}\frac{k\pm \sqrt{k^2+4}}{2},$$

(1)

being the roots of the quadratic equation

$$M_{k\pm }^2-k{M}_{k\pm }-1=0,$$

(2)

with the property $M_{k-}=-1/M_{k+}=k-M_{k+}$. They are shown in Figure 1.

The positive metallic ratio $M_{k+}$ as a continuous function of $k\in \mathbb{R}$ has the same property as the Łukaszyk-Karmowski (Ł-K) metric [1] between two independent continuous random variables: It becomes asymptotic to $M_{k+}=k$ as k goes to ifinity, as the $+4$ factor in the square root becomes negligible and $M_{k\pm }\approx \{k,0\}$ for large k. Because the ratios (1) are usually visualized as ratios of the edge lengths of a rectangle and these are assumed to be nonnegative, usually only the positive principal square root $M_{k+}$ of (2) is considered, where for $k=1$ the golden ratio is obtained, for $k=2$ the silver ratio, for $k=3$ the bronze ratio, etc. However, distance nonnegativity does not hold for the Ł–K metric [1], for example; such axiomatization may be misleading [2].

It was shown [3] that for $k\ne \{0,2\}$ the metallic ratios (1) can be expressed by primitive Pythagorean triples, as

$$M_{k+}=\cot \left(\frac{\theta }{4}\right),$$

(3)

and for $k\ge 3$

$$k=2\sqrt{\frac{c+b}{c-b}},$$

(4)

where $\theta$ is the angle between a longer cathetus b and hypotenuse c of a right triangle defined by a Pythagorean triple, as shown in Figure 2, whereas for $k=\{3,4\}$ it is the angle between a hypotenuse and a shorter cathetus a ($\{M_{3+},M_{10+}\}$ and $\{M_{4+},M_{6+}\}$ are defined by the same Pythagorean triples, respectively, $(5,12,13)$ and $(3,4,5)$), and

$$M_{1+}=\cot \left(\frac{\pi -{\theta }_{(3,5)}}{4}\right).$$

(5)

For example the Pythagorean triple $(20,21,29)$ defines $M_{5+}$, the Pythagorean triple $(3,4,5)$ defines $M_{6+}$, the Pythagorean triple $(28,45,53)$ defines $M_{7+}$, and so on.

2. Results

Theorem 1. 

The metallic ratio of $k\in \mathbb{R}$ is defined by an acute angle of a right triangle $0<\theta <\pi /2$.

Proof. 

We express the RHS of the equation (3) using half-angle formulae for sine and cosine, and substituting $\phi :=\theta /2$

$$\begin{array}{cc}\hfill \cot \left(\frac{\theta }{4}\right)& =\cot \left(\frac{\phi }{2}\right)=\frac{1+\cos \phi }{\sin \phi }=\frac{1+\cos \left(\frac{\theta }{2}\right)}{\sin \left(\frac{\theta }{2}\right)}=\hfill \\ \hfill & =\frac{1+\sqrt{\frac{1+\cos \theta }{2}}}{\sqrt{\frac{1-\cos \theta }{2}}}=M_k,\hfill \end{array}$$

(6)

since $0<\theta <\pi /2$ (we exclude degenerated triangles), so $\mathrm{sgn}(\sin \left(\theta /2\right)=\mathrm{sgn}(\cos \left(\theta /2\right)=1$.

Multiplying the numerator and denominator of (6) by $\sqrt{(1+\cos \theta )/2}$ and performing some basic algebraic manipulations, we arrive at the quadratic equation for $M_k$

$$\sin {\left(\theta \right)}^2{M}_k^2-2\sin \left(\theta \right)\left[1+\cos \left(\theta \right)\right]M_k-\sin {\left(\theta \right)}^2=0,$$

(7)

having roots

$$M_{\theta \pm }=\frac{1+\cos \left(\theta \right)\pm \sqrt{2\left(1+\cos \left(\theta \right)\right)}}{\sin \left(\theta \right)},$$

(8)

corresponding to the metallic ratios (1) for $0<\theta <\pi /2$. □

Theorem 2. 

The metallic ratio of $k\in \mathbb{R}$ is defined by an angle $0\le \theta <2\pi$.

Proof. 

Equating relations (1) and (8) and solving for k gives

$$k=\frac{2\left(\cos \left(\theta \right)+1\right)}{\sin \left(\theta \right)},$$

(9)

valid for $0<\theta <\pi /2$. Solving the relation (9) for $\theta$ extends its validity to $0\le \theta <2\pi$ by analytic continuation, giving

$$\frac{k+2i}{k-2i}=e^{i{\theta }_k}=\cos {\theta }_k+i\sin {\theta }_k=\frac{a}{c}+\frac{b}{c}i{z}_k,$$

(10)

that relates $k\in \mathbb{R}$ with a normalized complex number $z_k$. □

The angle ${\theta }_k=\arg \left(z_k\right)$ is shown in Figure 3 for $z_k$ given by the relation (10). Figure 4 shows metallic ratios (8) as a function of this angle for $-\pi \le {\theta }_k\le \pi$ and extrapolated to $-2\pi \le \theta \le 2\pi$. $z_{k<0}$ and $z_{k>0}$ are complex conjugates of each other.

Theorem 3. 

For $k\ne \{0,\pm 2\},k\in \mathbb{Q}$, the triple $\{a,b,c\}$ corresponding to the angle ${\theta }_k$ (10) is a Pythagorean triple.

Proof. 

Plugging rational $k:=l/m,m\ne 0,l,m\in \mathbb{Z}$ into the relation (10) gives

$$\frac{l^2-4m^2}{l^2+4m^2}+\frac{4lm}{l^2+4m^2}i=\frac{a}{c}+\frac{b}{c}i,$$

(11)

and $a=l^2-4m^2,b=4lm,c=l^2+4m^2,a,b,c\in \mathbb{Z}$ is a possible solution. It is easy to see that $a^2+b^2=c^2$. $k=0$ implies $l=0$ and $a=-4m^2,b=0,c=4m^2$ valid $\forall m\in \{\mathbb{R},\mathbb{I}\}$. $k=\pm 2$ implies $l=\pm 2m$ and $a=0,b=\pm 8m^2,c=8m^2$ also valid $\forall m\in \{\mathbb{R},\mathbb{I}\}$. □

Table 1 shows the generalized Pythagorean triples that define the metallic ratios for $k=\{0.1,0.2,\dots ,7\}$.