Metallic mean

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Fibonacci spiral 34.svg
Silver spiral approximation.svg
Bronze spiral approximation.png
Gold, silver, and bronze ratios within their respective rectangles.

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:

Contents

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]

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.

Geometry

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). Gold, silver, and bronze rectangles.svg
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).
Pentagram-phi.svg
Silver ratio octagon.svg
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).

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 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

Generalization

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 ]

Relation to half-angle cotangent

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.

Relation to Pythagorean triples

Metallic Ratios in Primitive Pythagorean Triangles AgamRiaSanga.png
Metallic Ratios in 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 cb ∈ {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]

Numerical values

First metallic means [5] [6]
NRatioValueName
00 + 4/2 1
11 + 5/2 1.618033989 [a] Golden
22 + 8/2 2.414213562 [b] Silver
33 + 13/23.302775638 [c] Bronze
44 + 20/24.236067978 [d] Copper
55 + 29/25.192582404 [e] Nickel
66 + 40/26.162277660 [f]
77 + 53/27.140054945 [g]
88 + 68/28.123105626 [h]
99 + 85/29.109772229 [i]
1010+ 104/210.099019513 [j]

See also

Notes

  1. Sloane, N. J. A. (ed.). "SequenceA001622(Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  2. OEIS:  A014176 , Decimal expansion of the silver mean, 1+sqrt(2).
  3. OEIS:  A098316 , Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.
  4. OEIS:  A098317 , Decimal expansion of phi^3 = 2 + sqrt(5).
  5. OEIS:  A098318 , Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.
  6. OEIS:  A176398 , Decimal expansion of 3+sqrt(10).
  7. OEIS:  A176439 , Decimal expansion of (7+sqrt(53))/2.
  8. OEIS:  A176458 , Decimal expansion of 4+sqrt(17).
  9. OEIS:  A176522 , Decimal expansion of (9+sqrt(85))/2.
  10. OEIS:  A176537 , Decimal expansion of (10+sqrt(104)/2.

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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. de Spinadel, Vera W. (1999). "The metallic means family and multifractal spectra" (PDF). Nonlinear analysis, theory, methods and applications. 36 (6). Elsevier Science: 721–745.
  3. 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.
  4. Rajput, Chetansing; Manjunath, Hariprasad (2024). "Metallic means and Pythagorean triples | Notes on Number Theory and Discrete Mathematics". Bulgarian Academy of Sciences.{{cite web}}: CS1 maint: numeric names: authors list (link)
  5. Weisstein, Eric W. "Table of Silver means". MathWorld .
  6. "An Introduction to Continued Fractions: The Silver Means", maths.surrey.ac.uk.

Further reading