Gold, silver, and bronze ratios within their respective rectangles.
The metallic mean (also metallic ratio, metallic constant, or noble mean[1]) of a natural numbern is a positive real number, denoted here that satisfies the following equivalent characterizations:
Metallic means are (successive) derivations of the golden () and silver ratios (), and share some of their interesting properties. The term "bronze ratio" () (Cf. Golden Age and Olympic Medals) and even metals such as copper () and nickel () are occasionally found in the literature.[2][3][a]
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.
If one removes n largest possible squares from a rectangle with ratio length/width equal to the nth metallic mean, one gets a rectangle with the same ratio length/width (in the figures, n is the number of dotted lines).
Golden ratio within the pentagram (φ = red/ green = green/blue = blue/purple) and silver ratio within the octagon.
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).
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
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.
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[4]
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 Pythagorean triangles.
Relation to Pythagorean triples
Metallic Ratios in Primitive Pythagorean Triangles
For a primitive Pythagorean triple, a2 + b2 = c2, with positive integers a < b < c that are relatively prime, if the difference between the hypotenusec and longer legb 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.[5]
More precisely, for a primitive Pythagorean triple (a, b, c) with a < b < c, the smaller acute angle α satisfies When c − b ∈ {1, 2, 8}, we will always get that is an integer and that the n-th metallic mean.
The reverse direction also works. For n ≥ 5, the primitive Pythagorean triple that gives the n-th metallic mean is given by (n, n2/4 − 1, n2/4 + 1) if n is a multiple of 4, is given by (n/2, (n2 − 4)/8, (n2 + 4)/8) if n is even but not a multiple of 4, and is given by (4n, n2 − 4, n2 + 4) if n is odd. For example, the primitive Pythagorean triple (20, 21, 29) gives the 5th metallic mean; (3, 4, 5) gives the 6th metallic mean; (28, 45, 53) gives the 7th metallic mean; (8, 15, 17) gives the 8th metallic mean; and so on.
1 2 3 This name appears to have originated from de Spinadel's paper.
References
↑ 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.
↑ de Spinadel, Vera W. (1998). Williams, Kim (ed.). "The Metallic Means and Design". Nexus II: Architecture and Mathematics. Fucecchio (Florence): Edizioni dell'Erba: 141–157.
Stakhov, Alekseĭ Petrovich (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, p.228, 231. World Scientific. ISBN9789812775832.
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