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

Preprint

Communication

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

Altmetrics

Downloads

223

Views

Comments

A peer-reviewed article of this preprint also exists.

Szymon Łukaszyk^*

This version is not peer-reviewed

Submitted:

15 January 2024

Posted:

16 January 2024

Read the latest preprint version here

Alerts

Abstract

We show that metallic ratios for real k are defined by an argument of a normalized complex number, while for rational k ≠ {0, ±2}, they are defined by Pythagorean triples. We further extend the concept of metallic ratios to metallic angles.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

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

k h + d

and h satisfy

M (k) : = \frac{k h + 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 (k)

leads to the quadratic equation

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

(2)

having roots

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

(3)

shown in Figure 2 for

k \in R

Because the edge lengths of a metallic rectangle are assumed to be nonnegative, usually only the positive principal square root

M {(k)}_{+}

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{matrix} 1. & M {(k)}_{-} M {(k)}_{+} = - 1, \\ 2. & M {(k)}_{-} + M {(k)}_{+} = k, \\ 3. & - M {(- k)}_{-} = M {(k)}_{+}, or \\ 4. & M {(k)}_{\pm} = \pm e^{arcsinh (\pm k / 2)} . \end{matrix}

(4)

Furthermore, as k goes to ifinity, the factor

+ 4

in the square root becomes negligible, and

M {(k)}_{\pm} \approx {k, 0}

for large k.

It was shown [3] that for

k \neq {0, 2}

positive metallic ratios (3) can be expressed by primitive Pythagorean triples, as

M {(k)}_{+} = \cot (\frac{θ}{4}),

(5)

and for

k \geq 3

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

(6)

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 3, 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}) .

(7)

For example the Pythagorean triple

(20, 21, 29)

defines

M {(5)}_{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

M {(k)}_{\pm}

k \in R

is defined by an acute angle of a right triangle

0 < θ < π / 2

Proof.

We express the RHS of the Equation (5) using half-angle formulas 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 + sgn (\cos (\frac{θ}{2})) \sqrt{\frac{1 + \cos θ}{2}}}{sgn (\sin (\frac{θ}{2})) \sqrt{\frac{1 - \cos θ}{2}}} = M {(k)}_{+}, \end{matrix}

(8)

since

0 < θ < π / 2

(we exclude degenerated triangles), so

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

Multiplying the numerator and denominator of (8) by

\sqrt{(1 + \cos θ) / 2}

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

M (θ)

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

(9)

having roots

M {(θ_{+})}_{\pm} = \frac{(1 + \cos (θ_{+})) \pm \sqrt{2 (1 + \cos (θ_{+}))}}{\sin (θ_{+})},

(10)

corresponding to the metallic ratios (3) for

0 < θ_{+} < π / 2

. □

We can extend the domain of Theorem 1 by analytic continuation to

0 < θ_{+} < π

sgn (\sin (θ_{+} / 2)) = sgn (\cos (θ_{+} / 2)) = 1

in this range. However, extending it further to

- π < θ_{-} < 0

we note that in this range

sgn (\sin (θ_{-} / 2)) = - 1

. Thus, the quadratic Equation (9) becomes

\sin (θ_{-}) M {(θ_{-})}^{2} + 2 [1 + \cos (θ_{-})] M (θ_{-}) - \sin (θ_{-}) = 0,

(11)

and its roots are

M {(θ_{-})}_{\pm} = \frac{- (1 + \cos (θ_{-})) \pm \sqrt{2 (1 + \cos (θ_{-}))}}{\sin (θ_{-})} .

(12)

Theorem 2.

The metallic ratio of

k \in R

is defined by an angle

- π < θ \leq π

Proof.

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

k_{+} = \frac{2 (1 + \cos (θ_{+}))}{\sin (θ_{+})},

(13)

for

0 < θ_{+} < π

. This identity can also be obtained directly from the second property (4) applied to the ratio (10). Solving the relation (13) for

θ_{+}

yields

\frac{k_{+} + 2 i}{k_{+} - 2 i} = e^{i θ_{+}} = \cos θ_{+} + i \sin θ_{+} = \frac{a}{c} + \frac{b}{c} i : = z (k_{+}),

(14)

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

k_{-} = \frac{- 2 (1 + \cos (θ_{-}))}{\sin (θ_{-})},

(15)

for

- π < θ_{-} < 0

. Solving the relation (15) for

θ_{-}

yields

\frac{k_{-} - 2 i}{k_{-} + 2 i} = e^{- i θ_{-}} = \frac{a}{c} - \frac{b}{c} i : = z (k_{-}) = \bar{z (k_{+})},

(16)

as a conjugate of the relation (14). The relations (14) and (16) remove the singularity of

θ = l π, l \in Z

in the relations (13), (15), and

{lim}_{k_{+} \to \pm \infty} \arg (z (k_{+})) = {lim}_{k_{+} \to \pm \infty} \arg (\bar{z (k_{+})}) = 0

. □

Equations (14) and (16) relate

k_{\pm} \in R

which defines a metallic ratio (3) to the normalized complex number

z (k_{+})

. The angles

θ_{+} = \arg (z (k_{+}))

and

θ_{-} = \arg (\bar{z (k_{+})})

are shown in Figure 4. There are two axes of symmetry.

In summary, metallic ratios as functions of

θ

are

M {(θ_{\pm})}_{\pm} = \frac{\pm (1 + \cos (θ_{\pm})) \pm \sqrt{2 (1 + \cos (θ_{\pm}))}}{\sin (θ_{\pm})},

(17)

where the first ± defines the range of

θ

and the second ± corresponds to positive or negative form of the ratio. Therefore, the first and second properties (4) hold for

M {(θ_{\pm})}_{-} M {(θ_{\pm})}_{+} = - 1

and

M {(θ_{\pm})}_{-} + M {(θ_{\pm})}_{+} = k_{\pm}

but the third property (4) holds as

- M {(- θ_{\pm})}_{\pm} = M {(θ_{\pm})}_{\pm}

Figure 5 shows metallic ratios (10) and (12) as functions of

- 1.2 π \leq θ \leq 1.2 π

θ_{+} = \arg (z (k_{+}))

, and

θ_{-} = \arg (\bar{z (k_{+})})

Theorem 3.

For

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

, the triple

{a, b, c}

corresponding to the angle θ (14), (16) is a Pythagorean triple.

Proof.

Plugging rational

k : = l / m, m \neq 0, l, m \in Z

into the relation (14) gives

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

(18)

and

a = l^{2} - 4 m^{2}, b = 4 l m, c = l^{2} + 4 m^{2}, a, b, c \in Z

is a possible solution. It is easy to see that

a^{2} + b^{2} = c^{2}

k = 0

implies

l = 0

and

a = - 4 m^{2}, b = 0, c = 4 m^{2}

valid

\forall m \in {R, I}

k = \pm 2

implies

l = \pm 2 m

and

a = 0, b = \pm 8 m^{2}, c = 8 m^{2}

also valid

\forall m \in {R, I}

. □

Table 1 shows the generalized Pythagorean triples that define the metallic ratios for

k = {0.1, 0.2, \dots, 7}

3. Metallic Angles

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

\frac{k (2 π - φ) + φ}{2 π - φ} = \frac{2 π - φ}{φ},

(19)

where for

k = 1

well known golden angle

φ {(1)}_{-} \approx 2.4

, shown in Figure 6, is obtained.

Solving the relation (19) for

φ

leads to the quadratic equation

k φ {(k)}_{\pm}^{2} - 2 π (k + 2) φ {(k)}_{\pm} + 4 π^{2} = 0,

(20)

having roots

φ {(k)}_{\pm} = \frac{π}{k} (k + 2 \pm \sqrt{k^{2} + 4}),

(21)

shown in Figure 7.

In this case, their products and sums

\begin{matrix} φ {(k)}_{-} φ {(k)}_{+} & = \frac{4 π^{2}}{k}, \\ φ {(k)}_{-} + φ {(k)}_{+} & = 2 π \frac{k + 2}{k}, \end{matrix}

(22)

are dependent on k, where

{lim}_{k \to \pm \infty} φ {(k)}_{-} φ {(k)}_{+} = 0

and

{lim}_{k \to \pm \infty} φ {(k)}_{-} + φ {(k)}_{+} = 2 π

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

{\hat{k}}_{1} = \frac{4 π (π + 1)}{2 π + 1} \approx 7.1459,

(23)

where

M {({\hat{k}}_{1})}_{+} = φ {({\hat{k}}_{1})}_{+} = 2 π + 1 \approx 7.2832,

(24)

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

{\hat{k}}_{2} = \frac{- 1 + \sqrt{8 π + 1}}{4 π} - \frac{\sqrt{8 π + 1}}{2} - \frac{1}{2} \approx - 2.7288,

(25)

where

M {({\hat{k}}_{2})}_{-} = φ {({\hat{k}}_{2})}_{+} = \frac{- \sqrt{8 π + 1} - 1}{2} \approx - 3.0560,

(26)

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

{\hat{k}}_{3} = \frac{- 1 - \sqrt{8 π + 1}}{4 π} + \frac{\sqrt{8 π + 1}}{2} - \frac{1}{2} \approx 1.5696,

(27)

where

M {({\hat{k}}_{3})}_{+} = φ {({\hat{k}}_{3})}_{-} = \frac{\sqrt{8 π + 1} - 1}{2} \approx 2.0560 .

(28)

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. Silver ratio

M {(2)}_{+} = (2 h + d) / h = h / d

Figure 1. Silver ratio

M {(2)}_{+} = (2 h + d) / h = h / d

Figure 2. Metallic ratios: positive

M {(k)}_{+}

(red), negative

M {(k)}_{-}

(green) as continuous functions of

- 5 \leq k \leq 5

Figure 2. Metallic ratios: positive

M {(k)}_{+}

(red), negative

M {(k)}_{-}

(green) as continuous functions of

- 5 \leq k \leq 5

Figure 3. Right triangle showing a longer (b), shorter (a) hypotenuse, catheti (c) and angles

θ = θ_{(b, c)}

and

θ_{(a, c)}

Figure 3. Right triangle showing a longer (b), shorter (a) hypotenuse, catheti (c) and angles

θ = θ_{(b, c)}

and

θ_{(a, c)}

Figure 4. Phases of the complex number

z (k)

(red,

k_{+}

) and its conjugate

\bar{z (k)}

(green,

k_{-}

) for

- 7 \leq k \leq 7

θ (\pm 2) = \pm π / 2

θ (0) = π

Figure 4. Phases of the complex number

z (k)

(red,

k_{+}

) and its conjugate

\bar{z (k)}

(green,

k_{-}

) for

- 7 \leq k \leq 7

θ (\pm 2) = \pm π / 2

θ (0) = π

Figure 5. Metallic ratios: positive

M {(θ_{+})}_{+}

(red), negative

M {(θ_{+})}_{-}

(green) as a function of

- 1.2 π \leq θ \leq 1.2 π

(dotted), and

θ = \arg (z (k_{+}))

(solid), and

θ = \arg (\bar{z (k_{+})})

(positive: solid blue; negative: solid cyan) for

0 \leq k_{+} \leq 2.2

Figure 5. Metallic ratios: positive

M {(θ_{+})}_{+}

(red), negative

M {(θ_{+})}_{-}

(green) as a function of

- 1.2 π \leq θ \leq 1.2 π

(dotted), and

θ = \arg (z (k_{+}))

(solid), and

θ = \arg (\bar{z (k_{+})})

(positive: solid blue; negative: solid cyan) for

0 \leq k_{+} \leq 2.2

Figure 6. Golden angle

φ {(1)}_{-} = π (3 - \sqrt{5})

Figure 6. Golden angle

φ {(1)}_{-} = π (3 - \sqrt{5})

Figure 7. Metallic angles (solid) and ratios (dotted), positive (red), negative (green), as continuous functions of

- 10 \leq k \leq 10

Figure 7. Metallic angles (solid) and ratios (dotted), positive (red), negative (green), as continuous functions of

- 10 \leq k \leq 10

Table 1. Pythagorean triples associated with metallic ratios for rational

k = {0.1, 0.2, \dots, 7}

Table 1. Pythagorean triples associated with metallic ratios for rational

k = {0.1, 0.2, \dots, 7}

k	a	b	c	k	a	b	c
0.1	−399	40	401	3.6	28	45	53
0.2	−99	20	101	3.7	969	1480	1769
0.3	−391	120	409	3.8	261	380	461
0.4	−12	5	13	3.9	1121	1560	1921
0.5	−15	8	17	4	3	4	5
0.6	−91	60	109	4.1	1281	1640	2081
0.7	−351	280	449	4.2	341	420	541
0.8	−21	20	29	4.3	1449	1720	2249
0.9	−319	360	481	4.4	48	55	73
1	−3	4	5	4.5	65	72	97
1.1	−279	440	521	4.6	429	460	629
1.2	−8	15	17	4.7	1809	1880	2609
1.3	−231	520	569	4.8	119	120	169
1.4	−51	140	149	4.9	2001	1960	2801
1.5	−7	24	25	5	21	20	29
1.6	−9	40	41	5.1	2201	2040	3001
1.7	−111	680	689	5.2	72	65	97
1.8	−19	180	181	5.3	2409	2120	3209
1.9	−39	760	761	5.4	629	540	829
2				5.5	105	88	137
2.1	41	840	841	5.6	171	140	221
2.2	21	220	221	5.7	2849	2280	3649
2.3	129	920	929	5.8	741	580	941
2.4	11	60	61	5.9	3081	2360	3881
2.5	9	40	41	6	4	3	5
2.6	69	260	269	6.1	3321	2440	4121
2.7	329	1080	1129	6.2	861	620	1061
2.8	12	35	37	6.3	3569	2520	4369
2.9	441	1160	1241	6.4	231	160	281
3	5	12	13	6.5	153	104	185
3.1	561	1240	1361	6.6	989	660	1189
3.2	39	80	89	6.7	4089	2680	4889
3.3	689	1320	1489	6.8	132	85	157
3.4	189	340	389	6.9	4361	2760	5161
3.5	33	56	65	7	45	28	53

For

k = {- 7, - 6.9, \dots, - 0.1}

set

b \leftrightarrow - b

. E.g. for k = −7, {45, 28, 53} ↔ {45,−28, 53}.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

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

Abstract

1. Introduction

2. Results

3. Metallic Angles

Acknowledgments

References

MDPI Initiatives

Important Links

Subscribe