Metallic Ratios are Uniquely Defined by an Acute Angle of a Right Triangle

Szymon Łukaszyk

doi:10.20944/preprints202401.0551.v1

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Communication

Metallic Ratios are Uniquely Defined by an Acute Angle of a Right Triangle

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Szymon Łukaszyk^*

This version is not peer-reviewed.

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Submitted:

05 January 2024

Posted:

08 January 2024

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Abstract

We show that metallic ratios for real k are uniquely defined by an acute angle of a right triangle, where for k ≠ {0, ±2} this angle is a logarithm of a normalized complex number, and for rational k this number is defined by a Pythagorean triple.

Keywords:

metallic ratios

;

Pythagorean triples

;

emergent dimensionality

Subject:

Computer Science and Mathematics - Mathematics

1. Introduction

Metallic ratios of

k \in 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 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 \neq {0, 2}

the metallic ratios (1) can be expressed by primitive Pythagorean triples, as

M_{k +} = cot (\frac{θ}{4}),

(3)

and for

k \geq 3

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

(4)

where

θ

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 (\frac{π - θ_{(3, 5)}}{4}) .

(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.

Metallic ratios are uniquely defined by an acute angle of a right triangle.

Proof.

We express the RHS of the equation (3) using half-angle formulae for sine and cosine, and substituting

φ : = θ / 2

\begin{matrix} cot (\frac{θ}{4}) & = cot (\frac{φ}{2}) = \frac{1 + cos φ}{sin φ} = \frac{1 + cos (\frac{θ}{2})}{sin (\frac{θ}{2})} = \\ = \frac{1 + \sqrt{\frac{1 + cos θ}{2}}}{\sqrt{\frac{1 - cos θ}{2}}} = M_{k}, \end{matrix}

(6)

since

0 < θ < π / 2

, so

sgn (sin (θ / 2) = sgn (cos (θ / 2) = 1

Multiplying the numerator and denominator of (6) by

\sqrt{(1 + cos θ) / 2}

and performing some basic algebraic manipulations, we arrive at the quadratic equation for

M_{k}

sin {(θ)}^{2} M_{k}^{2} - 2 sin (θ) [1 + cos (θ)] M_{k} - sin {(θ)}^{2} = 0,

(7)

having roots

M_{θ \pm} = \frac{1 + cos (θ) \pm \sqrt{2 (1 + cos (θ))}}{sin (θ)},

(8)

corresponding to the metallic ratios (1). □

The metallic ratios

M_{θ \pm}

are shown in Figure 3.

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

k (θ) = \frac{2 (cos (θ) + 1)}{sin (θ)},

(9)

as shown in Figure 4.

Conjecture 1.

For

k \neq {0, \pm 2}, k \in R

the angle

θ_{k}

defining the metallic ratio is the argument of a normalized complex number

z_{k} = e^{i θ_{k}}

, wherein for

k \in Z

z_{k}

is defined by a Pythagorean triple as

\begin{matrix} z_{k} & = \{\begin{matrix} + \frac{a}{c} - \frac{b}{c} i & k < - \frac{π}{2} \\ - \frac{a}{c} - \frac{b}{c} i & - \frac{π}{2} < k \leq - 1 \\ - \frac{b}{c} - \frac{a}{c} i & - 1 < k < 0 \\ - \frac{b}{c} + \frac{a}{c} i & 0 < k < 1 \\ - \frac{a}{c} + \frac{b}{c} i & 1 \leq k < \frac{π}{2} \\ + \frac{a}{c} + \frac{b}{c} i & k > - \frac{π}{2}, \end{matrix} \end{matrix}

(10)

a < b

are catheti and c is hypotenuse of the right triangle shown in Figure 2,

θ_{\pm 2} = \pm π / 2

, and

θ_{0}

is undefined.

Conjecture (1) has been numerically validated. This form of

z_{k}

does not hold for irrational k. For example

z_{\sqrt{2}} = - \frac{1}{3} + \frac{2 \sqrt{2}}{3} i, z_{π} = \frac{π + 2 i}{π - 2 i} .

(11)

Acknowledgments

I thank my wife Magdalena Bartocha for her unwavering motivation and my friend, Renata Sobajda, for her prayers.

References

Łukaszyk, S. A new concept of probability metric and its applications in approximation of scattered data sets. Computational Mechanics 2004, 33, 299–304. [Google Scholar] [CrossRef]
Łukaszyk, S.; Tomski, A. Omnidimensional Convex Polytopes. Symmetry 2023, 15, 755. [Google Scholar] [CrossRef]
Rajput, C. Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles. JOURNAL OF ADVANCES IN MATHEMATICS 2021, 20, 312–344. [Google Scholar] [CrossRef]

Figure 1. Metallic ratios: positive

M_{k +}

(red), negative

M_{k -}

(green) as continuous functions of

0 \leq k \leq 10

and Ł-K metric

D_{N N} (k)

s = \sqrt{π} / 2

(blue).

Figure 1. Metallic ratios: positive

M_{k +}

(red), negative

M_{k -}

(green) as continuous functions of

0 \leq k \leq 10

and Ł-K metric

D_{N N} (k)

s = \sqrt{π} / 2

(blue).

Figure 2. Right triangle showing a longer (b), shorter (a) hypotenuse, catheti (c) and angles

θ = θ_{(b, c)}

and

θ_{(a, c)}

Figure 2. Right triangle showing a longer (b), shorter (a) hypotenuse, catheti (c) and angles

θ = θ_{(b, c)}

and

θ_{(a, c)}

Figure 3. Metallic ratios: positive

M_{θ +}

(red), negative

M_{θ -}

(green) as a function of

0 \leq θ \leq 2 π

Figure 3. Metallic ratios: positive

M_{θ +}

(red), negative

M_{θ -}

(green) as a function of

0 \leq θ \leq 2 π

Figure 4.

k (θ)

for

0 \leq θ \leq 2 π

k (π / 2) = 2

k (π) = 0

k (3 π / 2) = - 2

Figure 4.

k (θ)

for

0 \leq θ \leq 2 π

k (π / 2) = 2

k (π) = 0

k (3 π / 2) = - 2

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Metallic Ratios are Uniquely Defined by an Acute Angle of a Right Triangle

Abstract

Keywords:

Subject:

1. Introduction

2. Results

Acknowledgments

References

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