Plastic ratio

In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x + 1. Its decimal expansion begins with 1.324717957244746... (sequence A060006 in the OEIS). It is the smallest Pisot number.

The adjective plastic does not refer to the artificial material, but to the formative and sculptural qualities of this ratio, as in plastic arts.

Definition

Three quantities a > b > c > 0 are in the plastic ratio if

b

c

=

a

b

=

b

+

c

a

{\displaystyle {\frac {b}{c}}={\frac {a}{b}}={\frac {b+c}{a}}}

This ratio is commonly denoted ⁠

ρ

{\displaystyle \rho }

⁠ (rho).

Substituting

b

=

ρ

c

{\displaystyle b=\rho c\,}

and

a

=

ρ

b

=

ρ

2

c

{\displaystyle a=\rho b=\rho ^{2}c\,}

in the last fraction,

ρ

=

c

(

ρ

+

1

)

ρ

2

c

.

{\displaystyle \rho ={\frac {c(\rho +1)}{\rho ^{2}c}}.}

It follows that the plastic ratio is the unique real solution of the cubic equation

ρ

3

−

ρ

−

1

=

0.

{\displaystyle \rho ^{3}-\rho -1=0.}

Solving with Cardano's formula,

w

1

,

2

=

1

2

(

1

±

1

3

23

3

)

ρ

=

w

1

3

+

w

2

3

{\displaystyle {\begin{aligned}w_{1,2}&={\frac {1}{2}}\left(1\pm {\frac {1}{3}}{\sqrt {\frac {23}{3}}}\right)\\\rho &={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}\end{aligned}}}

or, using the hyperbolic cosine,

ρ

=

2

3

cosh

⁡

(

1

3

arcosh

⁡

(

3

3

2

)

)

.

{\displaystyle \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}

⁠

ρ

{\displaystyle \rho }

⁠ is the superstable fixed point of the iteration

x

←

(

2

x

3

+

1

)

/

(

3

x

2

−

1

)

,

{\displaystyle x\gets (2x^{3}+1)/(3x^{2}-1),}

which is the update step of Newton's method applied to ⁠

x

3

−

x

−

1

=

0.