Value of Plastic Constant

Theorem
The plastic constant $P$ is evaluated as:

Proof
By definition, the plastic constant $P$ is the real root of the cubic:
 * $x^3 - x - 1 = 0$

Recall Cardano's Formula:

Here we have:

Hence:

and so:

Then:

The number can then be calculated.

Since $S \ne T$, the other two roots $x_2, x_3$ has non-zero imaginary parts $\pm \dfrac {i \sqrt 3} 2 \paren {S - T}$.

Hence the root above is the only real root.