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

1. Introduction

Each rectangle contains at least one square with an edge h equal to the shorter edge of the rectangle. If a rectangle contains k such squares and its edges $kh+d$ and h satisfy

$$M\left(k\right):=\frac{kh+d}{h}=\frac{h}{d},$$

(1)

they satisfy a metallic ratio; the golden ratio for $k=1$, the silver ratio for $k=2$, shown in Figure 1, the bronze ratio for $k=3$, etc.

Solving the relation (1) for $M\left(k\right)$ leads to the quadratic equation

$$M{\left(k\right)}_{\pm }^2-kM{\left(k\right)}_{\pm }-1=0,$$

(2)

having roots

$$M{\left(k\right)}_{\pm }=\frac{k\pm \sqrt{k^2+4}}{2},$$

(3)

shown in Figure 2 for $k\in \mathbb{R}$.

Because the edge lengths of a metallic rectangle are assumed to be nonnegative, usually only the positive principal square root $M{\left(k\right)}_{+}$ of (2) is considered. However, distance nonnegativity (corresponding to the ontological principle of identity of indiscernibles) does not hold for the Ł–K metric [1], for example; such an axiomatization is misleading [2].

Metallic ratios (3) have interesting properties, such as

$$\begin{array}{cc}\hfill 1.& M{\left(k\right)}_{-}M{\left(k\right)}_{+}=-1,\hfill \\ \hfill 2.& M{\left(k\right)}_{-}+M{\left(k\right)}_{+}=k,\hfill \\ \hfill 3.& -M{(-k)}_{-}=M{\left(k\right)}_{+},\mathrm{or}\hfill \\ \hfill 4.& M{\left(k\right)}_{\pm }=\pm e^{\mathrm{arcsinh}(\pm k/2)}.\hfill \end{array}$$

(4)

Furthermore, as k goes to ifinity, the factor $+4$ in the square root becomes negligible, and $M{\left(k\right)}_{\pm }\approx \{k,0\}$ for large k.

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

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

(5)

and for $k\ge 3$

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

(6)

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 3, whereas for $k=\{3,4\}$ it is the angle between a hypotenuse and a shorter cathetus a ($\{M{\left(3\right)}_{+},M{\left(10\right)}_{+}\}$ and $\{M{\left(4\right)}_{+},M{\left(6\right)}_{+}\}$ are defined by the same Pythagorean triples, respectively, $(5,12,13)$ and $(3,4,5)$), and

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

(7)

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

2. Results

Theorem 1.

The metallic ratio $M{\left(k\right)}_{\pm }$ 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 (5) using half-angle formulas and substituting $\phi :=\theta /2$

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

(8)

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

Multiplying the numerator and denominator of (8) by $\sqrt{(1+\cos \theta )/2}$ and performing some basic algebraic manipulations, we arrive at the quadratic equation for $M\left(\theta \right)$

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

(9)

having roots

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

(10)

corresponding to the metallic ratios (3) for $0<{\theta }_{+}<\pi /2$. □

We can extend the domain of Theorem 1 by analytic continuation to $0<{\theta }_{+}<\pi$ as $\mathrm{sgn}\left(\sin \left({\theta }_{+}/2\right)\right)=\mathrm{sgn}\left(\cos \left({\theta }_{+}/2\right)\right)=1$ in this range. However, extending it further to $-\pi <{\theta }_{-}<0$ we note that in this range $\mathrm{sgn}\left(\sin \left({\theta }_{-}/2\right)\right)=-1$. Thus, the quadratic Equation (9) becomes

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

(11)

and its roots are

$$M{\left({\theta }_{-}\right)}_{\pm }=\frac{-\left(1+\cos \left({\theta }_{-}\right)\right)\pm \sqrt{2\left(1+\cos \left({\theta }_{-}\right)\right)}}{\sin \left({\theta }_{-}\right)}.$$

(12)

Theorem 2.

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

Proof. 

Equating relations (3) and (10) and solving for k gives

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

(13)

for $0<{\theta }_{+}<\pi$. This identity can also be obtained directly from the second property (4) applied to the ratio (10). Solving the relation (13) for ${\theta }_{+}$ yields

$$\frac{k_{+}+2i}{k_{+}-2i}=e^{i{\theta }_{+}}=\cos {\theta }_{+}+i\sin {\theta }_{+}=\frac{a}{c}+\frac{b}{c}i:=z\left(k_{+}\right),$$

(14)

Similarly, applying the second property (4) to the ratio (12) gives

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

(15)

for $-\pi <{\theta }_{-}<0$. Solving the relation (15) for ${\theta }_{-}$ yields

$$\frac{k_{-}-2i}{k_{-}+2i}=e^{-i{\theta }_{-}}=\frac{a}{c}-\frac{b}{c}i:=z\left(k_{-}\right)=\overline{z\left(k_{+}\right)},$$

(16)

as a conjugate of the relation (14). The relations (14) and (16) remove the singularity of $\theta =l\pi ,l\in \mathbb{Z}$ in the relations (13), (15), and ${lim}_{k_{+}\to \pm \infty }\arg \left(z\left(k_{+}\right)\right)={lim}_{k_{+}\to \pm \infty }\arg \left(\overline{z\left(k_{+}\right)}\right)=0$. □

Equations (14) and (16) relate $k_{\pm }\in \mathbb{R}$ which defines a metallic ratio (3) to the normalized complex number $z\left(k_{+}\right)$. The angles ${\theta }_{+}=\arg \left(z\left(k_{+}\right)\right)$ and ${\theta }_{-}=\arg \left(\overline{z\left(k_{+}\right)}\right)$ are shown in Figure 4. There are two axes of symmetry.

In summary, metallic ratios as functions of $\theta$ are

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

(17)

where the first ± defines the range of $\theta$ and the second ± corresponds to positive or negative form of the ratio. Therefore, the first and second properties (4) hold for $M{\left({\theta }_{\pm }\right)}_{-}M{\left({\theta }_{\pm }\right)}_{+}=-1$ and $M{\left({\theta }_{\pm }\right)}_{-}+M{\left({\theta }_{\pm }\right)}_{+}=k_{\pm }$ but the third property (4) holds as $-M{(-{\theta }_{\pm })}_{\pm }=M{\left({\theta }_{\pm }\right)}_{\pm }$.

Figure 5 shows metallic ratios (10) and (12) as functions of $-1.2\pi \le \theta \le 1.2\pi$, ${\theta }_{+}=\arg \left(z\left(k_{+}\right)\right)$, and ${\theta }_{-}=\arg \left(\overline{z\left(k_{+}\right)}\right)$.

Theorem 3.

For $k\ne \{0,\pm 2\},k\in \mathbb{Q}$, the triple $\{a,b,c\}$ corresponding to the angle θ (14), (16) is a Pythagorean triple.

Proof. 

Plugging rational $k:=l/m,m\ne 0,l,m\in \mathbb{Z}$ into the relation (14) gives

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

(18)

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 $k=\{0.1,0.2,\cdots ,7\}$.

3. Metallic Angles

We can extend the concept of metallic ratios (1) to angles as

$$\frac{k(2\pi -\phi )+\phi }{2\pi -\phi }=\frac{2\pi -\phi }{\phi },$$

(19)

where for $k=1$ well known golden angle $\phi {\left(1\right)}_{-}\approx 2.4$, shown in Figure 6, is obtained.

Solving the relation (19) for $\phi$ leads to the quadratic equation

$$k\phi {\left(k\right)}_{\pm }^2-2\pi (k+2)\phi {\left(k\right)}_{\pm }+4{\pi }^2=0,$$

(20)

having roots

$$\phi {\left(k\right)}_{\pm }=\frac{\pi }{k}\left(k+2\pm \sqrt{k^2+4}\right),$$

(21)

shown in Figure 7.

In this case, their products and sums

$$\begin{array}{cc}\hfill \phi {\left(k\right)}_{-}\phi {\left(k\right)}_{+}& =\frac{4{\pi }^2}{k},\hfill \\ \hfill \phi {\left(k\right)}_{-}+\phi {\left(k\right)}_{+}& =2\pi \frac{k+2}{k},\hfill \end{array}$$

(22)

are dependent on k, where ${lim}_{k\to \pm \infty }\phi {\left(k\right)}_{-}\phi {\left(k\right)}_{+}=0$ and ${lim}_{k\to \pm \infty }\phi {\left(k\right)}_{-}+\phi {\left(k\right)}_{+}=2\pi$.

The positive metallic ratios (3) are equal to the positive metallic angles (21) for

$${\widehat{k}}_1=\frac{4\pi (\pi +1)}{2\pi +1}\approx 7.1459,$$

(23)

where

$$M{\left({\widehat{k}}_1\right)}_{+}=\phi {\left({\widehat{k}}_1\right)}_{+}=2\pi +1\approx 7.2832,$$

(24)

the negative metallic ratios (3) are equal to the positive metallic angles (21) for

$${\widehat{k}}_2=\frac{-1+\sqrt{8\pi +1}}{4\pi }-\frac{\sqrt{8\pi +1}}{2}-\frac{1}{2}\approx -2.7288,$$

(25)

where

$$M{\left({\widehat{k}}_2\right)}_{-}=\phi {\left({\widehat{k}}_2\right)}_{+}=\frac{-\sqrt{8\pi +1}-1}{2}\approx -3.0560,$$

(26)

and the positive metallic ratios (3) are equal to the negative metallic angles (21) for

$${\widehat{k}}_3=\frac{-1-\sqrt{8\pi +1}}{4\pi }+\frac{\sqrt{8\pi +1}}{2}-\frac{1}{2}\approx 1.5696,$$

(27)

where

$$M{\left({\widehat{k}}_3\right)}_{+}=\phi {\left({\widehat{k}}_3\right)}_{-}=\frac{\sqrt{8\pi +1}-1}{2}\approx 2.0560.$$

(28)