Plastic number

In mathematics, the plastic number ρ (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, le nombre radiant) is a mathematical constant which is the unique real solution of the cubic equation It has the exact value Its decimal expansion begins with 1.32471 79572 44746 02596 09088 54.... 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, 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. The plastic number satisfies the nested radical recurrence 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 This field is also a Hilbert class field of As such, it can be expressed in terms of the Dedekind eta function with argument , and root of unity . Similarly, for the supergolden ratio with argument , Also, the plastic number is the smallest Pisot–Vijayaraghavan number. Its algebraic conjugates are of absolute value ≈ 0.868837 . This value is also 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: (See Cubic equation#Trigonometric and hyperbolic solutions.) Dividing a square into similar rectangles There are precisely three ways of partitioning a square into three similar rectangles: The trivial solution given by three congruent rectangles with aspect ratio 3:1.