The metallic mean (also metallic ratio, metallic constant, or noble means [1] ) of a natural number n is a positive real number, denoted here that satisfies the following equivalent characterizations:
Metallic means are generalizations of the golden ratio () and silver ratio (), and share some of their interesting properties. The term "bronze ratio" (), 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 and have as their norm.
The defining equation of the nth metallic mean is the characteristic equation of a linear recurrence relation of the form It follows that, given such a recurrence the solution can be expressed as
where is the nth metallic mean, and a and b are constants depending only on and Since the inverse of a metallic mean is less than 1, this formula implies that the quotient of two consecutive elements of such a sequence tends to the metallic mean, when k tends to the infinity.
For example, if is the golden ratio. If and the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If one has the Lucas numbers. If the metallic mean is called the silver ratio, and the elements of the sequence starting with and are called the Pell numbers.
The defining equation of the nth metallic mean induces the following geometrical interpretation.
Consider a rectangle such that the ratio of its length L to its width W is the nth metallic ratio. If one remove from this rectangle n squares of side length W, 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 the metallic mean of m one has
where the numbers are defined recursively by the initial conditions K0 = 0 and K1 = 1, and the recurrence relation
Proof: The equality is immediately true for The recurrence relation implies which makes the equality true for Supposing the equality true up to one has
End of the proof.
One has also [ citation needed ]
The odd powers of a metallic mean are themselves metallic means. More precisely, if n is an odd natural number, then where is defined by the recurrence relation and the initial conditions and
Proof: Let and The definition of metallic means implies that and Let Since if n is odd, the power is a root of So, it remains to prove that is an integer that satisfies the given recurrence relation. This results from the identity
This completes the proof, given that the initial values are easy to verify.
In particular, one has
and, in general,[ citation needed ]
where
For even powers, things are more complicate. If n is a positive even integer then[ citation needed ]
Additionally,[ citation needed ]
For the square of a metallic ratio we have:
where lies strictly between and . Therefore
One may define the metallic mean of a negative integer −n as the positive solution of the equation The metallic mean of −n is the multiplicative inverse of the metallic mean of n:
Another generalization consists of changing the defining equation from to . If
is any root of the equation, one has
The silver mean of m is also given by the integral[ citation needed ]
Another form of the metallic mean is[ citation needed ]
A tangent half-angle formula gives which can be rewritten as That is, for the positive value of , the metallic mean which is especially meaningful when is a positive integer, as it is with some primitive Pythagorean triangles.
Metallic means are precisely represented by some primitive Pythagorean triples, a2 + b2 = c2, with positive integers a < b < c.
In a primitive Pythagorean triple, if the difference between hypotenuse c and longer leg b is 1, 2 or 8, such Pythagorean triple accurately represents one particular metallic mean. The cotangent of the quarter of smaller acute angle of such Pythagorean triangle equals the precise value of one particular metallic mean.
Consider a primitive Pythagorean triple (a, b, c) in which a < b < c and c − b ∈ {1, 2, 8}. Such Pythagorean triangle (a, b, c) yields the precise value of a particular metallic mean as follows :
where α is the smaller acute angle of the Pythagorean triangle and the metallic mean index is
For example, the primitive Pythagorean triple 20-21-29 incorporates the 5th metallic mean. Cotangent of the quarter of smaller acute angle of the 20-21-29 Pythagorean triangle yields the precise value of the 5th metallic mean. Similarly, the Pythagorean triangle 3-4-5 represents the 6th metallic mean. Likewise, the Pythagorean triple 12-35-37 gives the 12th metallic mean, the Pythagorean triple 52-165-173 yields the 13th metallic mean, and so on. [4]
First metallic means [5] [6] | |||
---|---|---|---|
N | Ratio | Value | Name |
0 | 0 + √4/2 | 1 | |
1 | 1 + √5/2 | 1.618033989 [lower-alpha 1] | Golden |
2 | 2 + √8/2 | 2.414213562 [lower-alpha 2] | Silver |
3 | 3 + √13/2 | 3.302775638 [lower-alpha 3] | Bronze |
4 | 4 + √20/2 | 4.236067978 [lower-alpha 4] | |
5 | 5 + √29/2 | 5.192582404 [lower-alpha 5] | |
6 | 6 + √40/2 | 6.162277660 [lower-alpha 6] | |
7 | 7 + √53/2 | 7.140054945 [lower-alpha 7] | |
8 | 8 + √68/2 | 8.123105626 [lower-alpha 8] | |
9 | 9 + √85/2 | 9.109772229 [lower-alpha 9] | |
10 | 10+ √104/2 | 10.099019513 [lower-alpha 10] |
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