The metallic mean (also metallic ratio, metallic constant, or noble means[1]) of a natural number is a positive real number, denoted here
Sn,
x
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
n=2
n=3
In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than
1
-1
The defining equation
x2-nx-1=0
xk=nxk-1+xk-2.
xk=aS
| ||||
n |
\right)k,
Sn
x0
x1.
For example, if
n=1,
Sn
x0=0
x1=1,
n=1,x0=2,x1=1
n=2,
x0=0
x1=1
The defining equation 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.
Denoting by
Sm
n | |
S | |
m |
=KnSm+Kn-1,
where the numbers
Kn
Kn=mKn-1+Kn-2.
Proof: The equality is immediately true for
n=1.
K2=m,
k=2.
n-1,
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}
One has also
Kn=
| ||||||||||||||||
\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 |
,
Mn
Mn=mMn-1+Mn-2
M0=2
M1=m.
Proof: Let
a=Sm
b=-1/Sm.
a+b=m
ab=-1.
n+b | |
M | |
n=a |
n.
anbn=(ab)n=-1
an
x2-Mn-1=0.
Mn
\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}
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}
2n+1 | |
S | |
m |
=SM,
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) |
.
One may define the metallic mean
S-n
x2-(-n)-1.
S-n=
1 | |
Sn |
.
Another generalization consists of changing the defining equation from
x2-nx-1=0
x2-nx-c=0
R= | n\pm\sqrt{n2+4c |
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 | |
First metallic means[4] | |||
---|---|---|---|
N | Ratio | Value | Name |
0 | 1 | ||
1 | 1.618033989 | Golden | |
2 | 2.414213562 | Silver | |
3 | 3.302775638 | Bronze | |
4 | 4.236067978 | ||
5 | 5.192582404 | ||
6 | 6.162277660 | ||
7 | 7.140054945 | ||
8 | 8.123105626 | ||
9 | 9.109772229 | ||
10 | 10.099019513 |