## 1. Plastic section and plastic number.

In a way analogue to that done in the golden section of a segment
we can make the **plastic section** of a segment AB by finding its point S such that the cube with edge AS
is equivalent to the rectangle parallelepiped with sides AB, SB and AB + AS.

This definition is due to the Benedictine architect-monk Hans Van der Laan(1904-1991)
which he applied to some of his projects.

If you indicate with λ the measure of the segment AB and with σ the measure of the segment AS you have

thet is equivalent to

Let *x* be the ratio between λ and σ. You have

If you now represent in the same Cartesian plane the curves with equations
*y=x*^{3} and *y=x+1* you observe that they intersect in a single point with abscissa between 1 and 2: therefore the equation admits a unique real solution.

The real solution of the equation is called plastic number or
plastic constant and, here, this number is denoted by *P*.
The equation is called "plastic equation".

To calculate *P* you can solve the plastic equation with the
method of Cardano by proceeding as follows

You rewrite the equation using the variable *w*

You obtain

So the real roots of the equation in *w* are

For simplicity, complex roots are omitted. By replacing one any of the two values *w*_{1} and *w*_{2}
in the expression of *x*, with a calculator, you obtain an approximation to the solution *P* of the equation:

*P*≈1.3247179572447460...