The metallic mean (also metallic ratio, metallic constant, or noble means[3]) of a natural numbern is a positive real number, denoted here that satisfies the following equivalent characterizations:
Golden ratio within the pentagram and silver ratio within the octagon.
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.[4][5]
As the golden ratio is connected to the pentagon (first diagonal/side), the silver ratio is connected to the octagon (second diagonal/side). As the golden ratio is connected to the Fibonacci numbers, the silver ratio is connected to the Pell numbers, and the bronze ratio is connected to OEIS:A006190. Each Fibonacci number is the sum of the previous number times one plus the number before that, each Pell number is the sum of the previous number times two and the one before that, and each "bronze Fibonacci number" is the sum of the previous number times three plus the number before that. Taking successive Fibonacci numbers as ratios, these ratios approach the golden mean, the Pell number ratios approach the silver mean, and the "bronze Fibonacci number" ratios approach the bronze mean.
Powers
This section does not cite any sources. (August 2020)
Gold, silver, and bronze ratios within their respective rectangles.
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.
This section does not cite any sources. (April 2024)
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]
The metallic mean for any given integer can be constructed geometrically in the following way. Define a right triangle with sides and having lengths of and , respectively. The th metallic mean is simply the sum of the length of and the hypotenuse, .[7]
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.
This article uses material from the Wikipedia article Bronze_ratio, and is written by contributors.
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