Value of Plastic Constant

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Theorem

The plastic constant $P$ is evaluated as:

\(\ds P\) \(=\) \(\ds \sqrt [3] {\frac {9 + \sqrt {69} } {18} } + \sqrt [3] {\frac {9 - \sqrt {69} } {18} }\)
\(\ds \) \(=\) \(\ds 1 \cdotp 32471 \, 79572 \, 44746 \, 02596 \, 09088 \, 54 \ldots\)


Proof

By definition, the plastic constant $P$ is the real root of the cubic:

$x^3 - x - 1 = 0$

Recall Cardano's Formula:

Let $P$ be the cubic equation:

$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$

Then $P$ has solutions:

\(\ds x_1\) \(=\) \(\ds S + T - \dfrac b {3 a}\)
\(\ds x_2\) \(=\) \(\ds -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}\)
\(\ds x_3\) \(=\) \(\ds -\dfrac {S + T} 2 - \dfrac b {3 a} - \dfrac {i \sqrt 3} 2 \paren {S - T}\)


where:

\(\ds S\) \(=\) \(\ds \sqrt [3] {R + \sqrt {Q^3 + R^2} }\)
\(\ds T\) \(=\) \(\ds \sqrt [3] {R - \sqrt {Q^3 + R^2} }\)


where:

\(\ds Q\) \(=\) \(\ds \dfrac {3 a c - b^2} {9 a^2}\)
\(\ds R\) \(=\) \(\ds \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}\)


Here we have:

\(\ds a\) \(=\) \(\ds 1\)
\(\ds b\) \(=\) \(\ds 0\)
\(\ds c\) \(=\) \(\ds -1\)
\(\ds d\) \(=\) \(\ds -1\)


Hence:

\(\ds Q\) \(=\) \(\ds \dfrac {3 \times 1 \times \paren {-1} - 0^2} {9 \times 1^2}\)
\(\ds \) \(=\) \(\ds \dfrac {-3} 9\)
\(\ds \) \(=\) \(\ds -\dfrac 1 3\)
\(\ds R\) \(=\) \(\ds \dfrac {9 \times 1 \times 0 \times \paren {-1} - 27 \times 1^2 \times \paren {-1} - 2 \times 0^3} {54 \times 1^3}\)
\(\ds \) \(=\) \(\ds \dfrac {27} {54}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2\)


and so:

\(\ds \sqrt {Q^3 + R^2}\) \(=\) \(\ds \sqrt {\paren {-\dfrac 1 3}^3 + \paren {\dfrac 1 2}^2}\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac 1 4 - \dfrac 1 {27} }\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac {27 - 4} {108} }\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac {3 \times 23} {3 \times 2^2 \times 3^3} }\)
\(\ds \) \(=\) \(\ds \sqrt {\dfrac {69} {18^2} }\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt {69} } {18}\)
\(\ds \leadsto \ \ \) \(\ds S = \sqrt [3] {R + \sqrt {Q^3 + R^2} }\) \(=\) \(\ds \sqrt [3] {\dfrac 1 2 + \dfrac {\sqrt {69} } {18} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\dfrac {9 + \sqrt {69} } {18} }\)
\(\ds T = \sqrt [3] {R - \sqrt {Q^3 + R^2} }\) \(=\) \(\ds \sqrt [3] {\dfrac 1 2 - \dfrac {\sqrt {69} } {18} }\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\dfrac {9 - \sqrt {69} } {18} }\)


Then:

\(\ds S + T - \dfrac b {3 a}\) \(=\) \(\ds \sqrt [3] {\dfrac {9 + \sqrt {69} } {18} } + \sqrt [3] {\dfrac {9 - \sqrt {69} } {18} } - \dfrac 0 {3 \times 1}\)
\(\ds \) \(=\) \(\ds \sqrt [3] {\dfrac {9 + \sqrt {69} } {18} } + \sqrt [3] {\dfrac {9 - \sqrt {69} } {18} }\)


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.

$\blacksquare$


Sources

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