Rationality Irrational ψ 1.4655712318767680266567312... ${\displaystyle \psi ={\frac {1}{3}}\left(1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}\right)}$ [1; 2, 6, 1, 3, 5, 4, 22, 1, 1, 4, 1, 2, 84, 1, ...]Not periodicInfinite 1.01110111001011111010... 1.772FAD1EDE80B46...

In mathematics, two quantities are in the supergolden ratio if their quotient equals the unique real solution to the equation ${\displaystyle x^{3}=x^{2}+1.}$ This solution is commonly denoted ${\displaystyle \psi .}$ The name supergolden ratio results of a analogy with the golden ratio ${\displaystyle \varphi }$, which is the positive root of the equation ${\displaystyle x^{2}=x+1.}$

Using formulas for the cubic equation, one can show that

${\displaystyle \psi ={\frac {1}{3}}\left(1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}\right),}$

or, using the hyperbolic cosine,

${\displaystyle \psi ={\frac {2}{3}}\cosh {\left({\frac {1}{3}}\cosh ^{-1}\left({\frac {29}{2}}\right)\right)}+{\frac {1}{3}}.}$

The decimal expansion of this number begins as 1.465571231876768026656731... ((sequence in the OEIS)). [1]

Properties

Many properties of the supergolden ratio are closely related to golden ratio ${\displaystyle \phi }$. For example, while we have ${\displaystyle \varphi -1=\varphi ^{-1}}$ for the golden ratio, the inverse square of the supergolden ratio obeys ${\displaystyle \psi -1=\psi ^{-2}}$. [3] Additionally, the supergolden ratio can be expressed in terms of itself as the infinite geometric series [3]

${\displaystyle \psi ^{3}=\sum _{n=0}^{\infty }\psi ^{-n},}$

in comparison to the golden ratio identity

${\displaystyle \varphi ^{2}=\sum _{n=0}^{\infty }\varphi ^{-n}.}$

The supergolden ratio is also the fourth smallest Pisot number, which means that its algebraic conjugates are both smaller than 1 in absolute value. [2]

Supergolden sequence

The supergolden sequence, also known as the Narayana's cows sequence, is a sequence where the ratio between consecutive terms approaches the supergolden ratio. [4] The first three terms are each one, and each term after that is calculated by adding the previous term and the term two places before that; that is, ${\displaystyle a_{n+1}=a_{n}+a_{n-2}}$, with ${\displaystyle a_{1}=a_{2}=a_{3}=1}$. The first values are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595… [4] [3] ((sequence in the OEIS)).

Supergolden rectangle

A supergolden rectangle is a rectangle whose side lengths are in a ψ∶1 ratio. When a square with the same side length as the shorter side of the rectangle is removed from one side of the rectangle, the sides of the resulting rectangle will be in a ψ2∶1 ratio. This rectangle can be divided into two more supergolden rectangles with opposite orientations and areas in a ψ2∶1 ratio. The larger rectangle has a diagonal of length ${\displaystyle 1/{\sqrt {\psi }}}$ times the short side of the original rectangle, and which is perpendicular to the diagonal of the original rectangle. [4] [3]

In addition, if the line segment that separates the two supergolden rectangles is extended across the square, then each diagonally opposite pair of rectangles has a combined area which is half that of the original rectangle. [1] The larger of the new rectangles is also a supergolden rectangle, with a diagonal of length ${\displaystyle {\sqrt {\psi }}}$ times the length of the short side of the original rectangle; [3] while the smaller one has sides in a ψ3∶1 ratio. [1]

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