Metallic mean explained

The metallic mean (also metallic ratio, metallic constant, or noble means[1]) of a natural number is a positive real number, denoted here

Sn,

that satisfies the following equivalent characterizations:

x

such that x=n+\frac 1x

x2-nx-1=0

[n;n,n,n,n,...]=n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\ddots}}}}

Metallic means are generalizations of the golden ratio (

n=1

) and silver ratio (

n=2

), and share some of their interesting properties. The term "bronze ratio" (

n=3

), and terms using other metals names (such as copper or nickel), are occasionally used to name subsequent metallic means.[2] [3]

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

x2-nx-1=0

of the th metallic mean is the characteristic equation of a linear recurrence relation of the form

xk=nxk-1+xk-2.

It follows that, given such a recurrence the solution can be expressed as

xk=aS

k+b\left(-1
Sn
n

\right)k,

where

Sn

is the th metallic mean, and and are constants depending only on

x0

and

x1.

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,

Sn

is the golden ratio. If

x0=0

and

x1=1,

the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If

n=1,x0=2,x1=1

one has the Lucas numbers. If

n=2,

the metallic mean is called the silver ratio, and the elements of the sequence starting with

x0=0

and

x1=1

are called the Pell numbers. The third metallic mean is sometimes called the "bronze ratio".

Geometry

The defining equation x=n+\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 segments 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.

Powers

Denoting by

Sm

the metallic mean of m one has
n
S
m

=KnSm+Kn-1,

where the numbers

Kn

are defined recursively by the initial conditions and,and the recurrence relation

Kn=mKn-1+Kn-2.

Proof: The equality is immediately true for

n=1.

The recurrence relation implies

K2=m,

which makes the equality true for

k=2.

Supposing the equality true up to

n-1,

one has
n
\begin{align} S
m

&=

n-1
mS
m
n-2
+S
m

&&(definingequation)\\ &=m(Kn-1Sn+Kn-2)+(Kn-2Sm+Kn-3)&&(recurrencehypothesis)\\ &=(mKn-1+Kn-2)Sn+(mKn-2+Kn-3)&&(regrouping)\\ &=KnSm+Kn-1&&(recurrenceonKn). \end{align}

End of the proof.

One has also

Kn=

n+1
S-
n+1
(m-S
m)
m
\sqrt{m2+4
} .

The odd powers of a metallic mean are themselves metallic means. More precisely, if is an odd natural number, then

n=S
S
Mn

,

where

Mn

is defined by the recurrence relation

Mn=mMn-1+Mn-2

and the initial conditions

M0=2

and

M1=m.

Proof: Let

a=Sm

and

b=-1/Sm.

The definition of metallic means implies that

a+b=m

and

ab=-1.

Let
n+b
M
n=a

n.

Since

anbn=(ab)n=-1

if is odd, the power

an

is a root of

x2-Mn-1=0.

So, it remains to prove that

Mn

is an integer that satisfies the given recurrence relation. This results from the identity

\begin{align}an+bn&=(a+b)(an-1+bn-1)-ab(an-2+an-2)\\ &=m(an-1+bn-1)+(an-2+an-2). \end{align}

This completes the proof, given that the initial values are easy to verify.

In particular, one has

\begin{align}

3
S
m

&=

S
m3+3m

\\

5
S
m

&=

S
m5+5m3+5m

\\

7
S
m

&=

S
m7+7m5+14m3+7m

\\

9
S
m

&=

S
m9+9m7+27m5+30m3+9m

\\

11
S
m

&=

S
m11+11m9+44m7+77m5+55m3+11m

\end{align}

and, in general,
2n+1
S
m

=SM,

where
n
M=\sum
k=0

{{2n+1}\over{2k+1}}{{n+k}\choose{2k}}m2k+1.

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

n
{S
m

-\left\lfloor

n
S
m

\right\rfloor}=1-

-n
S
m

.

Additionally,

{1\over

4
{S
m

-\left\lfloor

4
S
m

\right\rfloor}}+\left\lfloor

4
S
m

-1\right\rfloor=

S
\left(m4+4m2+1\right)

{1\over

6
{S
m

-\left\lfloor

6
S
m

\right\rfloor}}+\left\lfloor

6
S
m

-1\right\rfloor=

S
\left(m6+6m4+9m2+1\right)

.

Generalization

One may define the metallic mean

S-n

of a negative integer as the positive solution of the equation

x2-(-n)-1.

The metallic mean of is the multiplicative inverse of the metallic mean of :

S-n=

1
Sn

.

Another generalization consists of changing the defining equation from

x2-nx-1=0

to

x2-nx-c=0

. If
R=n\pm\sqrt{n2+4c
}, is any root of the equation, one has

R-n=

c
R

.

The silver mean of m is also given by the integral

Sm=

m
\int
0

{\left({x\over{2\sqrt{x2+4}}}+{{m+2}\over{2m}}\right)}dx.

Another form of the metallic mean is

n+\sqrt{n2+4
} = e^.

Numerical values

First metallic means[4]
NRatioValueName
01
11.618033989Golden
22.414213562Silver
33.302775638Bronze
44.236067978
55.192582404
66.162277660
77.140054945
88.123105626
99.109772229
1010.099019513

See also

Further reading

External links

Notes and References

  1. M. 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.
  2. Vera W.. de Spinadel. The metallic means family and multifractal spectra. Nonlinear analysis, theory, methods and applications. 36. 6. 721–745. 1999. Elsevier Science.
  3. Vera W.. de Spinadel. The Metallic Means and Design. 141–157. Nexus II: Architecture and Mathematics. Kim. Williams. Fucecchio (Florence). Edizioni dell'Erba. 1998.
  4. "An Introduction to Continued Fractions: The Silver Means", maths.surrey.ac.uk.