Supergolden Ratio

Author

Eric W. Weisstein
February 17, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/SupergoldenRatio.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SupergoldenRatio.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Value

psi=x/.ToRules[Reduce[x^3==x^2+1,x,Reals]]

N[psi,20]

ψ==(1/3) ((((1/2) (3 (Sqrt[93])+29))^(1/3))+(((1/2) (29-3 (Sqrt[93])))^(1/3))+1)

ToRadicals[psi]//Simplify//TraditionalForm

ψ==((Sqrt[3])/2)  csch((1/3) (sinh^(-1))(((3 (Sqrt[3]))/2)))

https://en.wikipedia.org/wiki/Supergolden_ratio

FullSimplify[psi-((Sqrt[3])/2)Csch[(1/3)ArcSinh[((3(Sqrt[3]))/2)]]]

ψ==(1/3)+(2/3) cos((1/3) (cos^(-1))((29/2)))

https://oeis.org/A092526

Finch 2003, p. 10

FullSimplify[psi-((2/3)Cos[(1/3)ArcCos[(29/2)]]+(1/3))]

RootReduce[FullSimplify[psi-((2/3)Cos[(1/3)ArcCos[(29/2)]]+(1/3))]]

Continued fraction

A369346

ContinuedFraction[psi,20]

Convergents

A381124/A381125

convergents=Convergents[psi,30]

N[convergents]

ListPlot[%-psi]

{num,den}=Transpose[NumeratorDenominator[convergents]]

StringLength[ToString[#]]&/@{num,den}

FindSequenceFunction[num,n]//Head//Timing

FindGeneratingFunction[num,x]//Head//Timing

FindSequenceFunction[den,n]//Head//Timing

FindGeneratingFunction[den,x]//Head//Timing

Sum

Self-sums

ψ==(∑_(n=0)^∞)(ψ^(-3 n))

psi==Sum[(psi^(-3n)),{n,0,∞}]//RootReduce

(ψ^2)==2 (∑_(n=0)^∞)(ψ^(-7 n))

(psi^2)==2 Sum[(psi^(-7n)),{n,0,∞}]//RootReduce

Finite sums

Table[Sum[(psi^(-k)),{k,0,n}],{n,50}]//RootReduce

Identities

Power as sums

(ψ^n)==(ψ^(n-1))+(ψ^(n-3))

Subtract@@{(psi^n),(psi^(n-1))+(psi^(n-3))}//FullSimplify

(ψ^n)==(ψ^(n-2))+(ψ^(n-3))+(ψ^(n-4))

Subtract@@{(psi^n),(psi^(n-2))+(psi^(n-3))+(psi^(n-4))}//FullSimplify

(ψ^n)==(ψ^(n-2))+2(ψ^(n-4))+(ψ^(n-6))

Subtract@@{(psi^n),(psi^(n-2))+2(psi^(n-4))+(psi^(n-6))}//FullSimplify

DedekindEta

bug(457430): Implement simplification of special value With[{tau = (1 + I Sqrt[31])/2}, Exp[Pi I/24] DedekindEta[tau]/(Sqrt[2] DedekindEta[2 tau])] == Root[-1 - #^2 + #^3& , 1, 0]

((exp(((π I)/24)) (DedekindEta[τ]))/((Sqrt[2]) (DedekindEta[2 τ])))==ψ/; τ=(1/2) (1+I (Sqrt[31]))

With[{τ=(1+I(Sqrt[31]))/2},
((Exp[((π I)/24)]DedekindEta[τ])/((Sqrt[2])DedekindEta[2τ]))]//FunctionExpand

FullSimplify[%]

N[%,20]

N[,20]

Almost integers

https://en.wikipedia.org/wiki/Supergolden_ratio#Properties

(ψ^11)≈67

N[(psi^11)-67,20]

((((Sqrt[2]) ψ))^24)-exp(π (Sqrt[31]))≈24

N[-Exp[π (Sqrt[31])]+((((Sqrt[2])psi))^24),20]

Sources

Ferguson (1976)

https://web.archive.org/web/20240223191949/https://www.fq.math.ca/Scanned/14-3/ferguson.pdf

Baker (2017)

https://arxiv.org/pdf/1711.10397

Pegg (2019)

https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/