Supergolden ratio

In mathematics, the supergolden ratio is a geometrical proportion close to 85/58. Its true value is the real solution of the equation x³ = x² + 1.

Supergolden ratio

A supergolden rectangle contains three scaled copies of itself, ψ = ψ−1 + 2ψ−3 + ψ−5
Rationality	irrational algebraic
Symbol	ψ
Representations
Decimal	1.4655712318767680266567312...
Algebraic form	real root of x3 = x2 + 1
Continued fraction (linear)	[1;2,6,1,3,5,4,22,1,1,4,1,2,84,...] not periodic infinite

The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation x² = x + 1.

A triangle with side lengths ψ, 1, and 1 ∕ ψ has an angle of exactly 120 degrees.

Definition

Two quantities a > b > 0 are in the supergolden ratio-squared if

$\left({\frac {a+b}{a}}\right)^{2}={\frac {a}{b}}$.

The ratio ${\frac {a+b}{a}}$ is commonly denoted ⁠$\psi .$⁠

Based on this definition, one has

${\begin{aligned}1&=\left({\frac {a+b}{a}}\right)^{2}{\frac {b}{a}}\\&=\left({\frac {a+b}{a}}\right)^{2}\left({\frac {a+b}{a}}-1\right)\\&\implies \psi ^{2}\left(\psi -1\right)=1\end{aligned}}$

It follows that the supergolden ratio is found as the unique real solution of the cubic equation $\psi ^{3}-\psi ^{2}-1=0.$ The decimal expansion of the root begins as $1.465\,571\,231\,876\,768...$ (sequence A092526 in the OEIS).

The minimal polynomial for the reciprocal root is the depressed cubic $x^{3}+x-1$, thus the simplest solution with Cardano's formula,

$w_{1,2}=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {31}{3}}}\right)/2$

$1/\psi ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}$

or, using the hyperbolic sine,

$1/\psi ={\frac {2}{\sqrt {3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).$

⁠$1/\psi$⁠ is the superstable fixed point of the iteration $x\gets (2x^{3}+1)/(3x^{2}+1)$.

The iteration $x\gets {\sqrt[{3}]{1+x^{2}}}$ results in the continued radical

$\psi ={\sqrt[{3}]{1+{\sqrt[{3/2}]{1+{\sqrt[{3/2}]{1+\cdots }}}}}}$

Dividing the defining trinomial $x^{3}-x^{2}-1$ by ⁠$x-\psi$⁠ one obtains $x^{2}+x/\psi ^{2}+1/\psi$, and the conjugate elements of ⁠$\psi$⁠ are

$x_{1,2}=\left(-1\pm i{\sqrt {4\psi ^{2}+3}}\right)/2\psi ^{2},$

with $x_{1}+x_{2}=1-\psi \;$ and $\;x_{1}x_{2}=1/\psi .$

Properties

Rectangles in aspect ratios ψ, ψ² and ψ³ (from left to right) tile the square.

Many properties of ⁠$\psi$⁠ are related to golden ratio ⁠$\varphi$⁠. For example, the supergolden ratio can be expressed in terms of itself as the infinite geometric series 

$\psi =\sum _{n=0}^{\infty }\psi ^{-3n}$ and $\,\psi ^{2}=2\sum _{n=0}^{\infty }\psi ^{-7n},$

in comparison to the golden ratio identity

$\varphi =\sum _{n=0}^{\infty }\varphi ^{-2n}$ and vice versa.

Additionally, $1+\varphi ^{-1}+\varphi ^{-2}=2$, while $\sum _{n=0}^{7}\psi ^{-n}=3.$

For every integer $n$ one has

${\begin{aligned}\psi ^{n}&=\psi ^{n-1}+\psi ^{n-3}\\&=\psi ^{n-2}+\psi ^{n-3}+\psi ^{n-4}\\&=\psi ^{n-2}+2\psi ^{n-4}+\psi ^{n-6}.\end{aligned}}$

Argument $\;\theta =\operatorname {arcsec}(2\psi ^{4})\;$ satisfies the identity $\;\tan(\theta )-4\sin(\theta )=3{\sqrt {3}}.$

Continued fraction pattern of a few low powers

$\psi ^{-1}=[0;1,2,6,1,3,5,4,22,...]\approx 0.6823$ (13/19)

$\ \psi ^{0}=[1]$

$\ \psi ^{1}=[1;2,6,1,3,5,4,22,1,...]\approx 1.4656$ (22/15)

$\ \psi ^{2}=[2;6,1,3,5,4,22,1,1,...]\approx 2.1479$ (15/7)

$\ \psi ^{3}=[3;6,1,3,5,4,22,1,1,...]\approx 3.1479$ (22/7)

$\ \psi ^{4}=[4;1,1,1,1,2,2,1,2,2,...]\approx 4.6135$ (60/13)

$\ \psi ^{5}=[6;1,3,5,4,22,1,1,4,...]\approx 6.7614$ (115/17)

Notably, the continued fraction of ⁠$\psi ^{2}$⁠ begins as permutation of the first six natural numbers; the next term is equal to their sum + 1.

The supergolden ratio is the fourth smallest Pisot number. Because the absolute value $1/{\sqrt {\psi }}$ of the algebraic conjugates is smaller than 1, powers of ⁠$\psi$⁠ generate almost integers. For example: $\psi ^{11}=67.000222765...\approx 67+1/4489$. After eleven rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠$\pm 13\pi /22$⁠ – nearly align with the imaginary axis.

The minimal polynomial of the supergolden ratio $m(x)=x^{3}-x^{2}-1$ has discriminant $\Delta =-31$. The Hilbert class field of imaginary quadratic field $K=\mathbb {Q} ({\sqrt {\Delta }})$ can be formed by adjoining ⁠$\psi$⁠. With argument $\tau =(1+{\sqrt {\Delta }})/2\,$ a generator for the ring of integers of ⁠$K$⁠, one has the special value of Dedekind eta quotient

$\psi ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}$.

Expressed in terms of the Weber-Ramanujan class invariant G_n

$\psi ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{31}}{\sqrt[{4}]{2}}}$.

Properties of the related Klein j-invariant ⁠$j(\tau )$⁠ result in near identity $e^{\pi {\sqrt {-\Delta }}}\approx \left({\sqrt {2}}\,\psi \right)^{24}-24$. The difference is < 1/143092.

The elliptic integral singular value $k_{r}=\lambda ^{*}(r)$ for ⁠$r=31$⁠ has closed form expression

$\lambda ^{*}(31)=\sin(\arcsin \left(({\sqrt[{4}]{2}}\,\psi )^{-12}\right)/2)$

(which is less than 1/10 the eccentricity of the orbit of Venus).

Narayana sequence

A Rauzy fractal associated with the supergolden ratio-cubed. The central tile and its three subtiles have areas in the ratios ψ4 : ψ2 : ψ : 1.

A Rauzy fractal associated with the supergolden ratio-squared, with areas as above.

Narayana's cows is a recurrence sequence originating from a problem proposed by the 14th century Indian mathematician Narayana Pandita. He asked for the number of cows and calves in a herd after 20 years, beginning with one cow in the first year, where each cow gives birth to one calf each year from the age of three onwards.

The Narayana sequence has a close connection to the Fibonacci and Padovan sequences and plays an important role in data coding, cryptography and combinatorics. The number of compositions of n into parts 1 and 3 is counted by the nth Narayana number.

The Narayana sequence is defined by the third-order recurrence relation

$N_{n}=N_{n-1}+N_{n-3}$ for n > 2,

with initial values

$N_{0}=N_{1}=N_{2}=1$.

The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... (sequence A000930 in the OEIS). The limit ratio between consecutive terms is the supergolden ratio.

The first 11 indices n for which $N_{n}$ is prime are n = 3, 4, 8, 9, 11, 16, 21, 25, 81, 6241, 25747 (sequence A170954 in the OEIS). The last number has 4274 decimal digits.

The sequence can be extended to negative indices using

$N_{n}=N_{n+3}-N_{n+2}$.

The generating function of the Narayana sequence is given by

${\frac {1}{1-x-x^{3}}}=\sum _{n=0}^{\infty }N_{n}x^{n}$ for $x<1/\psi$

The Narayana numbers are related to sums of binomial coefficients by

$N_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}$.

The characteristic equation of the recurrence is $x^{3}-x^{2}-1=0$. If the three solutions are real root ⁠$\alpha$⁠ and conjugate pair ⁠$\beta$⁠ and ⁠$\gamma$⁠, the Narayana numbers can be computed with the Binet formula

$N_{n-2}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n}$, with real ⁠$a$⁠ and conjugates ⁠$b$⁠ and ⁠$c$⁠ the roots of $31x^{3}+x-1=0$.

Since $\left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}$ and $\alpha =\psi$, the number ⁠$N_{n}$⁠ is the nearest integer to $a\,\psi ^{n+2}$, with n ≥ 0 and $a=\psi /(\psi ^{2}+3)=$ 0.2846930799753185027474714...

Coefficients $a=b=c=1$ result in the Binet formula for the related sequence $A_{n}=N_{n}+2N_{n-3}$.

The first few terms are 3, 1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144,... (sequence A001609 in the OEIS).

This anonymous sequence has the Fermat property: if p is prime, $A_{p}\equiv A_{1}{\bmod {p}}$. The converse does not hold, but the small number of odd pseudoprimes $\,n\mid (A_{n}-1)$ makes the sequence special. The 8 odd composite numbers below 10⁸ to pass the test are n = 1155, 552599, 2722611, 4822081, 10479787, 10620331, 16910355, 66342673.

A supergolden Rauzy fractal of type a ↦ ab, with areas as above. The fractal boundary has box-counting dimension 1.50

The Narayana numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue ⁠$\psi$⁠

$Q={\begin{pmatrix}1&0&1\\1&0&0\\0&1&0\end{pmatrix}},$

$Q^{n}={\begin{pmatrix}N_{n}&N_{n-2}&N_{n-1}\\N_{n-1}&N_{n-3}&N_{n-2}\\N_{n-2}&N_{n-4}&N_{n-3}\end{pmatrix}}$

The trace of ⁠$Q^{n}$⁠ gives the above ⁠$A_{n}$⁠.

Alternatively, ⁠$Q$⁠ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet ⁠$\{a,b,c\}$⁠ with corresponding substitution rule

${\begin{cases}a\;\mapsto \;ab\\b\;\mapsto \;c\\c\;\mapsto \;a\end{cases}}$

and initiator ⁠$w_{0}=b$⁠. The series of words ⁠$w_{n}$⁠ produced by iterating the substitution have the property that the number of c's, b's and a's are equal to successive Narayana numbers. The lengths of these words are $l(w_{n})=N_{n}.$

Associated to this string rewriting process is a compact set composed of self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence.

Supergolden rectangle

Nested supergolden rectangles with perpendicular diagonals and side lengths in powers of ψ.

A supergolden rectangle is a rectangle whose side lengths are in a ⁠$\psi :1$⁠ ratio. Compared to the golden rectangle, the supergolden rectangle has one more degree of self-similarity.

Given a rectangle of height 1, length ⁠$\psi$⁠ and diagonal length ${\sqrt {\psi ^{3}}}$ (according to $1+\psi ^{2}=\psi ^{3}$). The triangles on the diagonal have altitudes $1/{\sqrt {\psi }}\,;$ each perpendicular foot divides the diagonal in ratio ⁠$\psi ^{2}$⁠.

On the left-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio $\psi ^{2}:1$ (according to $\psi -1=\psi ^{-2}$). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.

The rectangle below the diagonal has aspect ratio ⁠$\psi ^{3}$⁠, the other three are all supergolden rectangles, with a fourth one between the feet of the altitudes. The parent rectangle and the four scaled copies have linear sizes in the ratios $\psi ^{3}:\psi ^{2}:\psi :\psi ^{2}-1:1,$ the areas of the rectangles opposite the diagonal are both equal to ⁠$1/\psi ^{3}.$⁠

In the supergolden rectangle above the diagonal, the process is repeated at a scale of ⁠$1:\psi ^{2}$⁠.

Supergolden spiral

Supergolden spirals with different initial radii on a ψ− rectangle.

A supergolden spiral is a logarithmic spiral that gets wider by a factor of ⁠$\psi$⁠ with every quarter turn. It can be described by the polar equation $r(\theta )=a\exp(k\theta ),$ with initial radius ⁠$a$⁠ and parameter $k={\frac {2\ln(\psi )}{\pi }}.$ If drawn on a supergolden rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio ⁠$\psi$⁠ that are perpendicular-aligned and successively scaled by a factor ⁠$1/\psi .$⁠

See also

• Solutions of equations similar to $x^{3}=x^{2}+1$:
  • Golden ratio – the only positive solution of the equation $x^{2}=x+1$
  • Plastic ratio – the only real solution of the equation $x^{3}=x+1$
  • Supersilver ratio – the only real solution of the equation $x^{3}=2x^{2}+1$