Metallic mean

The metallic mean (also metallic ratio, metallic constant, or noble meanM. Baake, U. Grimm (2013) Aperiodic order. Vol. 1. A mathematical invitation. With a foreword by Roger Penrose. Encyclopedia of Mathematics and its Applications, 149. Cambridge University Press, Cambridge, ISBN 978-0-521-86991-1.) of a natural number is a positive real number, denoted here $S\_n,$ that satisfies the following equivalent characterizations:

• the unique positive real number $x$ such that $x=n+\backslash frac\; 1x$
• the positive root of the quadratic equation $x^2-nx-1=0$
• the number $\backslash frac\{n+\backslash sqrt\{n^2+4\}\}2\; =\; \backslash frac2\{\backslash sqrt\{n^2+4\}-n\}$
• the number whose expression as a continued fraction is
• :$$

Metallic means are (successive) derivations of the golden ratio ($n=1$) and ($n=2$), and share some of their interesting properties. The term "bronze ratio" ($n=3$) (Cf. Golden Age and Olympic Medals) and even metals such as copper ($n=4$) and nickel ($n=5$) are occasionally found in the literature.This name appears to have originated from de Spinadel's paper.

In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than $1$ and have $-1$ as their norm.

The defining equation $x^2-nx-1=0$ of the th metallic mean is the characteristic equation of a linear recurrence relation of the form $x\_k=nx\_\{k-1\}+x\_\{k-2\}.$ It follows that, given such a recurrence the solution can be expressed as

$x\_k=aS\_n^k+b\backslash left(\backslash frac\{-1\}\{S\_n\}\backslash right)^k,$

where $S\_n$ is the th metallic mean, and and are constants depending only on $x\_0$ and $x\_1.$ Since the inverse of a metallic mean is less than , this formula implies that the quotient of two consecutive elements of such a sequence tends to the metallic mean, when tends to the infinity.

For example, if $n=1,$ $S\_n$ is the golden ratio. If $x\_0=0$ and $x\_1=1,$ the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If $n=1,\; x\_0=2,\; x\_1=1$ one has the . If $n=2,$ the metallic mean is called the silver ratio, and the elements of the sequence starting with $x\_0=0$ and $x\_1=1$ are called the .

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Geometry The defining equation $x=n+\backslash frac\; 1x$ of the th metallic mean induces the following geometrical interpretation.

Consider a rectangle such that the ratio of its length to its width is the th metallic ratio. If one remove from this rectangle squares of side length , one gets a rectangle similar to the original rectangle; that is, a rectangle with the same ratio of the length to the width (see figures).

Some metallic means appear as line segment in the figure formed by a regular polygon and its diagonals. This is in particular the case for the golden ratio and the pentagon, and for the silver ratio and the octagon; see figures.

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Powers Denoting by $S\_m$ the metallic mean of m one has

$S\_\{m\}^n\; =\; K\_n\; S\_m\; +\; K\_\{n-1\}\; ,$

where the numbers $K\_n$ are defined recursively by the initial conditions and , and the recurrence relation

$K\_n\; =\; mK\_\{n-1\}\; +\; K\_\{n-2\}.$

Proof: The equality is immediately true for $n=1.$ The recurrence relation implies $K\_2=m,$ which makes the equality true for $k=2.$ Supposing the equality true up to $n-1,$ one has

$\backslash begin\{align\}$

S_m^n & = mS_m^{n-1}+S_m^{n-2} &&\text {(defining equation)}\\ & = m(K_{n-1}S_n + K_{n-2})+ (K_{n-2}S_m+K_{n-3}) &&\text{(recurrence hypothesis)}\\ & = (mK_{n-1}+K_{n-2})S_n +(mK_{n-2}+K_{n-3}) &&\text{(regrouping)}\\ & = K_nS_m+K_{n-1} &&\text{(recurrence on }K_n). \end{align} End of the proof.

One has also

$K\_n\; =\; \backslash frac\{S\_m^\{n+1\}\; -\; (m-S\_m)^\{n+1\}\}\{\backslash sqrt\{m^2\; +\; 4\}\}\; .$

The odd powers of a metallic mean are themselves metallic means. More precisely, if is an odd natural number, then $S\_m^n=S\_\{M\_n\},$ where $M\_n$ is defined by the recurrence relation $M\_n=mM\_\{n-1\}+M\_\{n-2\}$ and the initial conditions $M\_0=2$ and $M\_1=m.$

Proof: Let $a=S\_m$ and $b=-1/S\_m.$ The definition of metallic means implies that $a+b=m$ and $ab=-1.$ Let $M\_n=a^n+b^n.$ Since $a^nb^n\; =(ab)^n=-1$ if is odd, the power $a^n$ is a root of $x^2-\; M\_n-1=0.$ So, it remains to prove that $M\_n$ is an integer that satisfies the given recurrence relation. This results from the identity

&= m(a^{n-1}+b^{n-1})+(a^{n-2}+a^{n-2}). \end{align} This completes the proof, given that the initial values are easy to verify.

In particular, one has

$\backslash begin\{align\}$

S_m^3 &= S_{m^3 + 3m} \\
S_m^5 &= S_{m^5 + 5m^3 + 5m} \\
S_m^7 &= S_{m^7 + 7m^5 + 14m^3 + 7m} \\
S_m^9 &= S_{m^9 + 9m^7 + 27m^5 + 30m^3 + 9m} \\
S_m^{11} &= S_{m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m}
     

\end{align} and, in general,

$S\_m^\{2n+1\}\; =\; S\_M,$

where

$M=\backslash sum\_\{k=0\}^n\; m^\{2k+1\}.$

For even powers, things are more complicated. If is a positive even integer then

$\{S\_m^n\; -\; \backslash left\backslash lfloor\; S\_m^n\; \backslash right\backslash rfloor\}\; =\; 1\; -\; S\_m^\{-n\}.$

Additionally,

$\{1\; \backslash over\; \{S\_m^4\; -\; \backslash left\backslash lfloor\; S\_m^4\; \backslash right\backslash rfloor\}\}\; +\; \backslash left\backslash lfloor\; S\_m^4\; -\; 1\; \backslash right\backslash rfloor\; =\; S\_\{\backslash left(m^4\; +\; 4m^2\; +\; 1\backslash right)\}$

$\{1\; \backslash over\; \{S\_m^6\; -\; \backslash left\backslash lfloor\; S\_m^6\; \backslash right\backslash rfloor\; \}\}\; +\; \backslash left\backslash lfloor\; S\_m^6\; -\; 1\; \backslash right\backslash rfloor\; =\; S\_\{\backslash left(m^6\; +\; 6m^4\; +\; 9m^2\; +1\backslash right)\}.$

For the square of a metallic ratio we have:$S\_m^2=m\backslash sqrt\{m^2+4\}+(m+2)/2=(p+\backslash sqrt\{p^2+4\})/2$

where $p=m\backslash sqrt\{m^2+4\}$ lies strictly between $m^2+1$ and $m^2+2$. Therefore

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Generalization One may define the metallic mean $S\_\{-n\}$ of a negative integer as the positive solution of the equation $x^2-(-n)x-1.$ The metallic mean of is the multiplicative inverse of the metallic mean of :

$S\_\{-n\}=\backslash frac\{1\}\{S\_n\}.$

Another generalization consists of changing the defining equation from $x^2-nx-1\; =0$ to $x^2-nx-c\; =0$. If

$R=\backslash frac\{n\backslash pm\backslash sqrt\{n^2+4c\}\}\{2\},$

is any root of the equation, one has

$R\; -\; n=\; \backslash frac\{c\}\{R\}.$

The silver mean of m is also given by the integral

$S\_m\; =\; \backslash int\_0^m\; \{\backslash left(\; \{x\; \backslash over\; \{2\backslash sqrt\{x^2+4\}\}\}\; +\; \backslash right)\}\; \backslash ,\; dx.$

Another form of the metallic mean is

$\backslash frac\{n+\backslash sqrt\{n^2+4\}\}\{2\}\; =\; e^\{\backslash operatorname\{arsinh(n/2)\}\}.$

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Relation to half-angle cotangent A tangent half-angle formula gives $$\backslash cot\backslash theta\; =\; \backslash frac\{\backslash cot^2\backslash frac\backslash theta2\; -\; 1\}\{2\backslash cot\backslash frac\backslash theta2\}$$ which can be rewritten as $$\backslash cot^2\backslash frac\backslash theta2\; -\; (2\backslash cot\backslash theta)\; \backslash cot\backslash frac\backslash theta2\; -\; 1\; =\; 0\backslash ,.$$ That is, for the positive value of $\backslash cot\backslash frac\backslash theta2$, the metallic mean $$S\_\{2\backslash cot\backslash theta\}\; =\; \backslash cot\backslash frac\backslash theta2\backslash ,,$$ which is especially meaningful when $2\backslash cot\backslash theta$ is a positive integer, as it is with some Pythagorean triangles.

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Relation to Pythagorean triples For a primitive Pythagorean triple, , with positive integers that are relatively prime, if the difference between the hypotenuse and cathetus is 1, 2 or 8 then the Pythagorean triangle exhibits a metallic mean. Specifically, the cotangent of one quarter of the smaller acute angle of the Pythagorean triangle is a metallic mean.

More precisely, for a primitive Pythagorean triple with , the smaller acute angle satisfies $$\backslash tan\; \backslash frac\{\backslash alpha\}\{2\}\; =\; \backslash frac\{c-b\}\{a\}\backslash ,.$$ When , we will always get that $$n=2\backslash cot\backslash frac\backslash alpha2\; =\; \backslash frac\{2a\}\{c-b\}$$ is an integer and that $$\backslash cot\backslash frac\backslash alpha4\; =\; S\_n\backslash ,,$$ the -th metallic mean.

The reverse direction also works. For , the primitive Pythagorean triple that gives the -th metallic mean is given by if is a multiple of 4, is given by if is even but not a multiple of 4, and is given by if is odd. For example, the primitive Pythagorean triple gives the 5th metallic mean; gives the 6th metallic mean; gives the 7th metallic mean; gives the 8th metallic mean; and so on.

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Numerical values

1
golden ratio	Golden ratio
silver ratio, Decimal expansion of the silver mean, 1+sqrt(2).	Silver ratio
, Decimal expansion of 3, = (3 + sqrt(13))/2.	Bronze
, Decimal expansion of phi^3 = 2 + sqrt(5).	CopperThis name appears to have originated from de Spinadel's paper.
, Decimal expansion of 5, = (5 + sqrt(29))/2.	NickelThis name appears to have originated from de Spinadel's paper.
, Decimal expansion of 3+sqrt(10).
, Decimal expansion of (7+sqrt(53))/2.
, Decimal expansion of 4+sqrt(17).
, Decimal expansion of (9+sqrt(85))/2.
, Decimal expansion of 5 + sqrt(26).

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See also

• Constant
• Mean
• Ratio
• Plastic ratio

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Notes

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Further reading

• Stakhov, Alekseĭ Petrovich (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, p. 228, 231. World Scientific. .

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External links

• Cristina-Elena Hrețcanu and Mircea Crasmareanu (2013). " Metallic Structures on Riemannian Manifolds", Revista de la Unión Matemática Argentina.
• Rakočević, Miloje M. " Further Generalization of Golden Mean in Relation to Euler's 'Divine' Equation", Arxiv.org.

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