Plastic number

The powers of the plastic number A(n) = ρⁿ satisfy the third-order linear recurrence relation A(n) = A(n − 2) + A(n − 3) for n > 2. Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the Padovan sequence (also known as the Cordonnier numbers), the Perrin numbers and the Van der Laan numbers,[5] and bears relationships to these sequences akin to the relationships of the golden ratio to the second-order Fibonacci and Lucas numbers, akin to the relationships between the silver ratio and the Pell numbers.[6]

The plastic number satisfies the nested radical recurrence[7]

${\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}.}$

Because the plastic number has the minimal polynomial x³ − x − 1 = 0, it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x³ − x − 1, but not for any other polynomials with integer coefficients. Since the discriminant of its minimal polynomial is −23, its splitting field over rationals is ${\displaystyle \mathbb {Q} ({\sqrt {-23}},\rho ).}$ This field is also a Hilbert class field of ${\displaystyle \mathbb {Q} ({\sqrt {-23}}).}$ As such, it can be expressed[7] in terms of the Dedekind eta function ${\displaystyle \eta (\tau )}$ with argument ${\displaystyle \tau ={\tfrac {1+{\sqrt {-23}}}{2}}}$,

${\displaystyle \rho ={\frac {-1}{z^{23}}}{\frac {\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}=1.3247\dots }$

and root of unity ${\displaystyle z=e^{2\pi i/48}}$. Similarly, for the supergolden ratio with argument ${\displaystyle \beta ={\tfrac {1+{\sqrt {-31}}}{2}}}$,

${\displaystyle \psi ={\frac {-1}{z^{23}}}{\frac {\eta (\beta )}{{\sqrt {2}}\,\eta (2\beta )}}=1.4655\dots }$

Also, the plastic number is the smallest Pisot–Vijayaraghavan number. Its algebraic conjugates are

${\displaystyle \left(-{\frac {1}{2}}\pm {\frac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{\frac {9+{\sqrt {69}}}{18}}}+\left(-{\frac {1}{2}}\mp {\frac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{\frac {9-{\sqrt {69}}}{18}}}\approx -0.662359\pm 0.56228i,}$

of absolute value ≈ 0.868837 (sequence A191909 in the OEIS). This value is also ${\displaystyle {\frac {1}{\sqrt {\rho }}}}$ because the product of the three roots of the minimal polynomial is 1.

The plastic number can be written using the hyperbolic cosine (cosh) and its inverse:

${\displaystyle \rho ={\frac {2{\sqrt {3}}}{3}}\cosh \left({\frac {1}{3}}\cosh ^{-1}\left({\frac {3{\sqrt {3}}}{2}}\right)\right).}$

(See Cubic equation#Trigonometric and hyperbolic solutions.)

Three partitions of a square into similar rectangles

There are precisely three ways of partitioning a square into three similar rectangles:[8][9]

1. The trivial solution given by three congruent rectangles with aspect ratio 3:1.
2. The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
3. The solution in which the three rectangles are all of different sizes and where they have aspect ratio ρ². The ratios of the linear sizes of the three rectangles are: ρ (large:medium); ρ² (medium:small); and ρ³ (large:small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio ρ. The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio ρ⁴.

The fact that a rectangle of aspect ratio ρ² can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ² related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.[10][11]