Metallic mean

{{multiple image | footer = Gold, silver, and bronze ratios within their respective rectangles. | direction = vertical | image1 = Fibonacci spiral 34.svg | image2 = Silver spiral approximation.svg | image3 = Bronze spiral approximation.png }}

The '''metallic mean''' (also '''metallic ratio''', '''metallic constant''', or '''noble mean'''<ref>M. Baake, U. Grimm (2013) [http://www.aperiodicorder.org/ 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.</ref>) of a natural number {{mvar|n}} is a positive real number, denoted here <math>S_n,</math> that satisfies the following equivalent characterizations: * the unique positive real number <math>x</math> such that <math display=inline>x=n+\frac 1x</math> * the positive root of the quadratic equation <math>x^2-nx-1=0</math> * the number <math display=inline>\frac{n+\sqrt{n^2+4}}2 = \frac2{\sqrt{n^2+4}-n}</math> * the number whose expression as a continued fraction is *:<math> [n;n,n,n,n,\dots] = n + \cfrac{1}{n+\cfrac{1} {n+\cfrac{1} {n+\cfrac{1} {n+\ddots\,}}}} </math>

Metallic means are (successive) derivations of the golden (<math>n=1</math>) and silver ratios (<math>n=2</math>), and share some of their interesting properties. The term "bronze ratio" (<math>n=3</math>) (Cf. Golden Age and Olympic Medals) and even metals such as copper (<math>n=4</math>) and nickel (<math>n=5</math>) are occasionally found in the literature.<ref>{{Cite journal|first=Vera W.|last=de Spinadel|title=The metallic means family and multifractal spectra|journal=Nonlinear Analysis, Theory, Methods and Applications|volume=36| number=6| pages=721–745| year=1999| publisher=Elsevier Science|url=https://d1wqtxts1xzle7.cloudfront.net/46781059/s0362-546x_2898_2900123-020160625-18115-1p722ic-libre.pdf?1466861385=&response-content-disposition=inline%3B+filename%3DThe_metallic_means_family_and_multifract.pdf&Expires=1713018340&Signature=AwpnybCBpuKjVLWlX6dvmK6DJLsDt1h0Z952HVahJ762TfsPMojBwo4tcJuOJ4dt-jVBqd7SZw1DGTj14VTHFI3KNGTg-IJKMbo5b6RIj3nY9SHG2yA0ecYyUfXMP2UP7Ua~uvOikAT0jyG87zgf53jT9l~7OZrQVaV-iZCjrW2Nf1n6jEmEWF9Q2htPl2QDErI5Y1X0-PCd94ETo3PavCH~Bq-GtT8iy-kSbxPslBpaoiCf1C-wGe3J0bhcB4MjFzSMxs57AwNXo2wXSyV~5zNbGYRYxNjdZ4AzurPagabaSzKixUkyfEtOzf5tm0B25bbKAENva3I72cTYokYHxg__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA}}</ref><ref>{{Cite journal|first=Vera W.|last=de Spinadel|title=The Metallic Means and Design|pages=141–157|url=https://www.nexusjournal.com/the-nexus-conferences/nexus-1998/119-n1998-spinadel.html|journal=Nexus II: Architecture and Mathematics|editor-first=Kim|editor-last=Williams|location=Fucecchio (Florence)|publisher=Edizioni dell'Erba|date=1998}}</ref><ref group=lower-alpha name=Note01>This name appears to have originated from de Spinadel's paper.</ref>

In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than <math>1</math> and have <math>-1</math> as their norm.

The defining equation <math>x^2-nx-1=0</math> of the {{mvar|n}}th metallic mean is the characteristic equation of a linear recurrence relation of the form <math>x_k=nx_{k-1}+x_{k-2}.</math> It follows that, given such a recurrence the solution can be expressed as :<math>x_k=aS_n^k+b\left(\frac{-1}{S_n}\right)^k,</math> where <math>S_n</math> is the {{mvar|n}}th metallic mean, and {{mvar|a}} and {{mvar|b}} are constants depending only on <math>x_0</math> and <math>x_1.</math> Since the inverse of a metallic mean is less than {{math|1}}, this formula implies that the quotient of two consecutive elements of such a sequence tends to the metallic mean, when {{mvar|k}} tends to the infinity.

For example, if <math>n=1,</math> <math>S_n</math> is the golden ratio. If <math>x_0=0</math> and <math>x_1=1,</math> the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If <math>n=1, x_0=2, x_1=1</math> one has the Lucas numbers. If <math>n=2,</math> the metallic mean is called the silver ratio, and the elements of the sequence starting with <math>x_0=0</math> and <math>x_1=1</math> are called the Pell numbers.

==Geometry== thumb|If one removes {{mvar|n}} largest possible squares from a rectangle with ratio length/width equal to the {{mvar|n}}th metallic mean, one gets a rectangle with the same ratio length/width (in the figures, {{mvar|n}} is the number of dotted lines). {{multiple image |direction = vertical | footer = Golden ratio within the pentagram (''φ'' = red/ green = green/blue = blue/purple) and silver ratio within the octagon. | image1 = Pentagram-phi.svg | image2 = Silver ratio octagon.svg }}

The defining equation <math display=inline>x=n+\frac 1x</math> of the {{mvar|n}}th metallic mean induces the following geometrical interpretation.

Consider a rectangle such that the ratio of its length {{mvar|L}} to its width {{mvar|W}} is the {{mvar|n}}th metallic ratio. If one remove from this rectangle {{mvar|n}} squares of side length {{mvar|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== {{Unreferenced section|date=August 2020}}

Denoting by <math>S_m</math> the metallic mean of ''m'' one has :<math> S_{m}^n = K_n S_m + K_{n-1} ,</math>

where the numbers <math>K_n</math> are defined recursively by the initial conditions {{math|''K''<sub>0</sub> {{=}} 0}} and {{math|''K''<sub>1</sub> {{=}} 1}}, and the recurrence relation :<math> K_n = mK_{n-1} + K_{n-2}. </math>

''Proof:'' The equality is immediately true for <math>n=1.</math> The recurrence relation implies <math>K_2=m,</math> which makes the equality true for <math>k=2.</math> Supposing the equality true up to <math>n-1,</math> one has :<math>\begin{align} S_m^n & = mS_m^{n-1}+S_m^{n-2} &&\text {(defining equation)}\\ & = m(K_{n-1}S_n + K_{n-2})+ (K_{n-2}S_m+K_{n-3}) &&\text{(recurrence hypothesis)}\\ & = (mK_{n-1}+K_{n-2})S_n +(mK_{n-2}+K_{n-3}) &&\text{(regrouping)}\\ & = K_nS_m+K_{n-1} &&\text{(recurrence on }K_n). \end{align}</math> ''End of the proof.''

One has also {{cn|date=April 2024}} :<math> K_n = \frac{S_m^{n+1} - (m-S_m)^{n+1}}{\sqrt{m^2 + 4}} . </math>

The odd powers of a metallic mean are themselves metallic means. More precisely, if {{mvar|n}} is an odd natural number, then <math>S_m^n=S_{M_n},</math> where <math>M_n</math> is defined by the recurrence relation <math>M_n=mM_{n-1}+M_{n-2}</math> and the initial conditions <math>M_0=2</math> and <math>M_1=m.</math>

''Proof:'' Let <math>a=S_m</math> and <math>b=-1/S_m.</math> The definition of metallic means implies that <math>a+b=m</math> and <math>ab=-1.</math> Let <math>M_n=a^n+b^n.</math> Since <math>a^nb^n =(ab)^n=-1</math> if {{mvar|n}} is odd, the power <math>a^n</math> is a root of <math>x^2- M_n-1=0.</math> So, it remains to prove that <math>M_n</math> is an integer that satisfies the given recurrence relation. This results from the identity :<math>\begin{align}a^n+b^n &= (a+b)(a^{n-1}+b^{n-1})-ab(a^{n-2}+a^{n-2})\\ &= m(a^{n-1}+b^{n-1})+(a^{n-2}+a^{n-2}). \end{align}</math> ''This completes the proof,'' given that the initial values are easy to verify.

In particular, one has :<math>\begin{align} S_m^3 &= S_{m^3 + 3m} \\ S_m^5 &= S_{m^5 + 5m^3 + 5m} \\ S_m^7 &= S_{m^7 + 7m^5 + 14m^3 + 7m} \\ S_m^9 &= S_{m^9 + 9m^7 + 27m^5 + 30m^3 + 9m} \\ S_m^{11} &= S_{m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m} \end{align}</math> and, in general,{{cn|date=April 2024}} :<math> S_m^{2n+1} = S_M,</math> where :<math>M=\sum_{k=0}^n {{2n+1} \over {2k+1}} {{n+k} \choose {2k}} m^{2k+1}.</math>

For even powers, things are more complicated. If {{math|''n''}} is a positive even integer then{{cn|date=April 2024}} :<math> {S_m^n - \left\lfloor S_m^n \right\rfloor} = 1 - S_m^{-n}. </math>

Additionally,{{cn|date=April 2024}} :<math> {1 \over {S_m^4 - \left\lfloor S_m^4 \right\rfloor}} + \left\lfloor S_m^4 - 1 \right\rfloor = S_{\left(m^4 + 4m^2 + 1\right)} </math> :<math> {1 \over {S_m^6 - \left\lfloor S_m^6 \right\rfloor }} + \left\lfloor S_m^6 - 1 \right\rfloor = S_{\left(m^6 + 6m^4 + 9m^2 +1\right)}. </math>

For the square of a metallic ratio we have:<math>S_m^2=[m\sqrt{m^2+4}+(m+2)]/2=(p+\sqrt{p^2+4})/2</math>

where <math>p=m\sqrt{m^2+4}</math> lies strictly between <math>m^2+1</math> and <math>m^2+2</math>. Therefore

==Generalization== One may define the metallic mean <math>S_{-n}</math> of a negative integer {{math|−''n''}} as the positive solution of the equation <math>x^2-(-n)x-1.</math> The metallic mean of {{math|−''n''}} is the multiplicative inverse of the metallic mean of {{math|''n''}}: :<math> S_{-n}=\frac{1}{S_n}.</math>

Another generalization consists of changing the defining equation from <math>x^2-nx-1 =0 </math> to <math>x^2-nx-c =0 </math>. If :<math> R=\frac{n\pm\sqrt{n^2+4c}}{2}, </math> is any root of the equation, one has :<math> R - n= \frac{c}{R}. </math>

The silver mean of ''m'' is also given by the integral<ref name=":0">{{Cite web |title=Metallic means - OeisWiki |url=https://oeis.org/wiki/Metallic_means#cite_ref-2 |access-date=2025-07-31 |website=oeis.org}}</ref> :<math> S_m = \int_0^m {\left( {x \over {2\sqrt{x^2+4}}} + {{m+2} \over {2m}} \right)} \, dx. </math>

Another form of the metallic mean is<ref name=":0" /> :<math> \frac{n+\sqrt{n^2+4}}{2} = e^{\operatorname{arsinh(n/2)}}. </math>

==Relation to half-angle cotangent== A tangent half-angle formula gives <math display=block>\cot\theta = \frac{\cot^2\frac\theta2 - 1}{2\cot\frac\theta2}</math> which can be rewritten as <math display=block>\cot^2\frac\theta2 - (2\cot\theta) \cot\frac\theta2 - 1 = 0\,.</math> That is, for the positive value of <math display=inline>\cot\frac\theta2</math>, the metallic mean <math display=block>S_{2\cot\theta} = \cot\frac\theta2\,,</math> which is especially meaningful when <math display=inline>2\cot\theta</math> is a positive integer, as it is with some Pythagorean triangles.

==Relation to Pythagorean triples== thumb|Metallic Ratios in Primitive Pythagorean Triangles For a primitive Pythagorean triple, {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = ''c''{{sup|2}}}}, with positive integers {{math|''a'' < ''b'' < ''c''}} that are relatively prime, if the difference between the hypotenuse {{mvar|c}} and longer leg {{mvar|b}} 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.<ref>{{cite web |last1=Rajput |first1=Chetansing |last2=Manjunath |first2=Hariprasad (2024) |title=Metallic means and Pythagorean triples {{!}} Notes on Number Theory and Discrete Mathematics |url=https://nntdm.net/volume-30-2024/number-1/184-194/ |publisher=Bulgarian Academy of Sciences}}</ref>

More precisely, for a primitive Pythagorean triple {{math|(''a'', ''b'', ''c'')}} with {{math|''a'' < ''b'' < ''c''}}, the smaller acute angle {{mvar|α}} satisfies <math display=block>\tan \frac{\alpha}{2} = \frac{c-b}{a}\,.</math> When {{math|''c'' − ''b'' ∈ {{mset|1, 2, 8}}}}, we will always get that <math display=block>n=2\cot\frac\alpha2 = \frac{2a}{c-b}</math> is an integer and that <math display=block>\cot\frac\alpha4 = S_n\,,</math> the {{mvar|n}}-th metallic mean.

The reverse direction also works. For {{math|''n'' ≥ 5}}, the primitive Pythagorean triple that gives the {{mvar|n}}-th metallic mean is given by {{math|(''n'', ''n''{{sup|2}}/4 − 1, ''n''{{sup|2}}/4 + 1)}} if {{mvar|n}} is a multiple of 4, is given by {{math|(''n''/2, (''n''{{sup|2}} − 4)/8, (''n''{{sup|2}} + 4)/8)}} if {{mvar|n}} is even but not a multiple of 4, and is given by {{math|(4''n'', ''n''{{sup|2}} − 4, ''n''{{sup|2}} + 4)}} if {{mvar|n}} is odd. For example, the primitive Pythagorean triple {{math|(20, 21, 29)}} gives the 5th metallic mean; {{math|(3, 4, 5)}} gives the 6th metallic mean; {{math|(28, 45, 53)}} gives the 7th metallic mean; {{math|(8, 15, 17)}} gives the 8th metallic mean; and so on.

==Numerical values== {| class="wikitable" |- ! colspan="4" align="center" | First metallic means<ref>{{MathWorld |title=Table of Silver means |id=SilverRatio}}</ref><ref>"[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#silver An Introduction to Continued Fractions: The Silver Means]", ''maths.surrey.ac.uk''.</ref> |- !{{mvar|n}} !Ratio !Value !Name |- | '''<math>n </math>'''||<math>\frac{n + \sqrt{4 + n^2}}{2} = \frac{n}{2} + \sqrt{1 + \left(\frac{n}{2}\right)^2}</math>|| | |- | '''0'''||<math>\frac{0 + \sqrt{4}}{2} = 0 + \sqrt{1}</math>||1 | |- | '''1'''||<math>\frac{1 + \sqrt{5}}{2}</math>||{{val|1.618033988|end=...}}<ref>{{Cite OEIS|sequencenumber=A001622 |name=Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2}}</ref> |Golden |- | '''2'''||<math>\frac{2 + \sqrt{8}}{2} = 1 + \sqrt{2}</math>||{{val|2.414213562|end=...}}<ref>{{OEIS2C|A014176}}, Decimal expansion of the silver mean, 1+sqrt(2).</ref> |Silver |- | '''3'''||<math>\frac{3 + \sqrt{13}}{2}</math>||{{val|3.302775637|end=...}}<ref>{{OEIS2C|A098316}}, Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.</ref> |Bronze<ref name=spinadel>{{cite web | url=https://www.mi.sanu.ac.rs/vismath/spinadel/ | title=The Family of Metallic Means }}</ref> |- | '''4'''||<math>\frac{4 + \sqrt{20}}{2} = 2 + \sqrt{5}</math>||{{val|4.236067977|end=...}}<ref>{{OEIS2C|A098317}}, Decimal expansion of phi^3 = 2 + sqrt(5).</ref> | |- | '''5'''||<math>\frac{5 + \sqrt{29}}{2}</math>||{{val|5.192582403|end=...}}<ref>{{OEIS2C|A098318}}, Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.</ref> | |- | '''6'''||<math>\frac{6 + \sqrt{40}}{2} = 3 + \sqrt{10}</math>||{{val|6.162277660|end=...}}<ref>{{OEIS2C|A176398}}, Decimal expansion of 3+sqrt(10).</ref> | |- | '''7'''||<math>\frac{7 + \sqrt{53}}{2}</math>||{{val|7.140054944|end=...}}<ref>{{OEIS2C|A176439}}, Decimal expansion of (7+sqrt(53))/2.</ref> | |- | '''8'''||<math>\frac{8 + \sqrt{68}}{2} = 4 + \sqrt{17}</math>||{{val|8.123105625|end=...}}<ref>{{OEIS2C|A176458}}, Decimal expansion of 4+sqrt(17).</ref> | |- | '''9'''||<math>\frac{9 + \sqrt{85}}{2}</math>||{{val|9.109772228|end=...}}<ref>{{OEIS2C|A176522}}, Decimal expansion of (9+sqrt(85))/2.</ref> | |- | '''10'''||<math>\frac{10 + \sqrt{104}}{2} = 5 + \sqrt{26}</math>||{{val|10.099019513|end=...}}<ref>{{OEIS2C|A176537}}, Decimal expansion of 5 + sqrt(26).</ref> | |}

== Relation to Aperiodic Order == The <math>k</math>-th metallic mean serves as the inflation ratio for one-dimensional substitution tilings, such as <math>a \to a^k b</math> and <math>b \to a</math>. These sequences exhibit long-range aperiodic order. By applying an interval removal process to these tilings, one can construct self-similar Cantor sets where the Hausdorff dimension is determined by the metallic mean scaling factor.<ref>{{Cite journal |last=Hutchinson |first=John |date=1981 |title=Fractals and self similarity |url=https://www.iumj.indiana.edu/docs/30055/30055.asp |journal=Indiana University Mathematics Journal |language=en |volume=30 |issue=5 |pages=713 |doi=10.1512/iumj.1981.30.30055 |issn=0022-2518}}</ref>

==See also== * Constant * Mean * Ratio * Plastic ratio

==Notes== {{notelist}}

==References== {{reflist}}

==Further reading== * Stakhov, Alekseĭ Petrovich (2009). ''The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science'', p. 228, 231. World Scientific. {{ISBN|9789812775832}}.

==External links== * Cristina-Elena Hrețcanu and Mircea Crasmareanu (2013). "[https://inmabb.criba.edu.ar/revuma/pdf/v54n2/v54n2a02.pdf Metallic Structures on Riemannian Manifolds]", ''Revista de la Unión Matemática Argentina''. * Rakočević, Miloje M. "[https://arxiv.org/abs/math/0611095 Further Generalization of Golden Mean in Relation to Euler's 'Divine' Equation]", ''Arxiv.org''.

Category:Metallic means