Supergolden ratio

Template:Short description Template:Infobox non-integer number In mathematics, the supergolden ratio is a geometrical proportion close to Template:Math. Its true value is the real solution of the equation Template:Math

The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation Template:Math

A triangle with side lengths Template:Math and Template:Math has an angle of exactly 120 degrees.

Definition

Two quantities Template:Math 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 Template:Tmath

Based on this definition, one has $$\begin{array}{cc}1& ={\left(\frac{a+b}{a}\right)}^2\frac{b}{a}\\ & ={\left(\frac{a+b}{a}\right)}^2(\frac{a+b}{a}-1)\\ & ⟹{\psi }^2(\psi -1)=1\end{array}$$

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 $1.465571231876768...$ Template:OEIS.

The minimal polynomial for the reciprocal root is the depressed cubic $x^3+x-1$, thus the simplest solution with Cardano's formula, $$\begin{array}{cc}{w}_{1,2}& =(1\pm \frac{1}{3}\sqrt{\frac{31}{3}})/2\\ 1/\psi & =\sqrt[3]{w_1}+\sqrt[3]{w_2}\end{array}$$ or, using the hyperbolic sine,

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

Template:Tmath is the superstable fixed point of the iteration $x\leftarrow (2x^3+1)/(3x^2+1)$.

The iteration $x\leftarrow \sqrt[3]{1+x^2}$ 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 Template:Tmath one obtains $x^2+x/{\psi }^2+1/\psi$, and the conjugate elements of Template:Tmath are $$x_{1,2}=(-1\pm i\sqrt{4{\psi }^2+3})/2{\psi }^2,$$ with $x_1+x_2=1-\psi$ and $x_1{x}_2=1/\psi .$

Properties

Rectangles in aspect ratios Template:Math and Template:Math (from left to right) tile the square.

Many properties of Template:Tmath are related to golden ratio Template:Tmath. For example, the supergolden ratio can be expressed in terms of itself as the infinite geometric series 

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

in comparison to the golden ratio identity

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

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

For every integer Template:Tmath one has $$\begin{array}{cc}{\psi }^n& ={\psi }^{n-1}+{\psi }^{n-3}\\ & ={\psi }^{n-2}+{\psi }^{n-3}+{\psi }^{n-4}\\ & ={\psi }^{n-2}+2{\psi }^{n-4}+{\psi }^{n-6}\end{array}$$ From this an infinite number of further relations can be found.

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

Continued fraction pattern of a few low powers $$\begin{array}{cc}{\psi }^{-1}& =[0;1,2,6,1,3,5,4,22,...]\approx 0.6823(13/19)\\ {\psi }^0& =[1]\\ {\psi }^1& =[1;2,6,1,3,5,4,22,1,...]\approx 1.4656(22/15)\\ {\psi }^2& =[2;6,1,3,5,4,22,1,1,...]\approx 2.1479(15/7)\\ {\psi }^3& =[3;6,1,3,5,4,22,1,1,...]\approx 3.1479(22/7)\\ {\psi }^4& =[4;1,1,1,1,2,2,1,2,2,...]\approx 4.6135(60/13)\\ {\psi }^5& =[6;1,3,5,4,22,1,1,4,...]\approx 6.7614(115/17)\end{array}$$

Notably, the continued fraction of Template:Tmath begins as permutation of the first six natural numbers; the next term is equal to their Template:Nowrap

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 Template:Tmath 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 Template:Tmath – nearly align with the imaginary axis.

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

$\psi =\frac{𝔣(\sqrt{\mathrm{\Delta }})}{\sqrt{2}}=\frac{G_{31}}{\sqrt[4]{2}}.$Template:Efn

Properties of the related Klein j-invariant Template:Tmath result in near identity $e^{\pi \sqrt{-\mathrm{\Delta }}}\approx {\left(\sqrt{2}\psi \right)}^{24}-24$. The difference is Template:Math.

The elliptic integral singular value $k_r={\lambda }^{*}(r)$ for Template:Tmath has closed form expression

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

(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 Template:Math

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 $$N_n=N_{n-1}+N_{n-3}\text{ for }n>2,$$ 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,... Template: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 Template: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

$\frac{1}{1-x-x^3}=\sum _{n=0}^{\mathrm{\infty }}{N}_n{x}^n\text{ for }x<1/\psi .$

The Narayana numbers are related to sums of binomial coefficients by

$N_n=\sum _{k=0}^{\lfloor n/3\rfloor }(\genfrac{}{}{0ex}{}{n-2k}{k})$.

The characteristic equation of the recurrence is $x^3-x^2-1=0$. If the three solutions are real root Template:Tmath and conjugate pair Template:Tmath and Template:Tmath, the Narayana numbers can be computed with the Binet formula

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

Since $|b{\beta }^n+c{\gamma }^n|<1/{\alpha }^{n/2}$ and $\alpha =\psi$, the number Template:Tmath is the nearest integer to $a{\psi }^{n+2}$, with Template:Math and $a=\psi /({\psi }^2+3)=$ Template:Gaps

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,... Template:OEIS.

This anonymous sequence has the Fermat property: if p is prime, $A_p\equiv A_{1}modp$. 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 Template:Math 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 Template:Math of a matrix with real eigenvalue Template:Tmath $$Q=\left(\begin{array}{ccc}1& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),$$

$$Q^n=\left(\begin{array}{ccc}{N}_n& N_{n-2}& N_{n-1}\\ N_{n-1}& N_{n-3}& N_{n-2}\\ N_{n-2}& N_{n-4}& N_{n-3}\end{array}\right)$$

The trace of Template:Tmath gives the above Template:Tmath.

Alternatively, Template:Tmath can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet Template:Tmath with corresponding substitution rule $$\{\begin{array}{c}a\mapsto ab\\ b\mapsto c\\ c\mapsto a\end{array}$$ and initiator Template:Tmath. The series of words Template:Tmath produced by iterating the substitution have the property that the number of Template:Math and Template:Math 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 Template:Math.

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

Given a rectangle of height Template:Math, length Template:Tmath 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 Template:Tmath.

On the left-hand side, cut off a square of side length Template:Math 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 Template:Tmath, 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 ${\psi }^3:{\psi }^2:\psi :{\psi }^2-1:1.$ It follows from the theorem of the gnomon that the areas of the two rectangles opposite the diagonal are equal.

In the supergolden rectangle above the diagonal, the process is repeated at a scale of Template:Tmath.

Supergolden spiral

Supergolden spirals with different initial radii on a Template:Math− rectangle.

A supergolden spiral is a logarithmic spiral that gets wider by a factor of Template:Tmath for every quarter turn. It is described by the polar equation $r(\theta )=a\exp (k\theta ),$ with initial radius Template:Tmath 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 Template:Tmath which are perpendicularly aligned and successively scaled by a factor Template:Tmath

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 $x^3=2x^2+1$