# The Plastic Number

The golden ratio is well known and is associated with the Fibonacci sequence of numbers.     $$\varphi$$ is the positive root of the quadratic equation $$x^2-x-1 = 0$$.

The exact value is given by

$$\displaystyle \frac{1 + \sqrt5}{2} \approx 1.61803$$

Its continued fraction is $$[1; 1, 1, 1, …]$$.

I have written on the silver ratio and its association with the Pell sequence of numbers.  $$\delta_s$$ is the positive root of the quadratic equation $$x^2-2x-1 = 0$$.

The exact value is given by

$$1 + \sqrt2 \approx 2.41421$$

Its continued fraction is $$[2; 2, 2, 2, …]$$.

The plastic number (or ratio), denoted by $$\rho$$ (rho), is associated with the Padovan sequence of numbers.

This sequence of numbers begins

$$1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, …$$

The Padovan sequence $$P(n)$$ is defined by the initial values

$$P(0) = P(1) = P(2) = 1$$

and the recurrence relation

$$P(n) = P(n-2) + P(n-3)$$

For example, $$P(12) = P(10) + P(9)$$ which is $$16 = 9 + 7$$.

In the above image is a visual proof that the sequence also satisfies the recurrence relation

$$P(n) = P(n-1) + P(n-5)$$

Some values of the convergents are

$$\begin{array}{l|l} Ratio & Decimal\\ \hline 1/1&1\\ 1/1 &1\\ 2/1 &2\\ 2/2&1\\ 3/2&1.5\\ 4/3&1.\overline 3\\ 5/4&1.25\\ 7/5&1.4\\ 9/7&1.\overline{285714}\\ 12/9&1.\overline3\\ 16/12&1.\overline3\\ 21/16&1.3125\\ 28/21&1.\overline3\\ 37/28&1.32\overline{142857}\\ 49/37&1.\overline{324} \end{array}$$

The limit of the convergents of consecutive Padovan numbers is

$$\displaystyle \lim \limits_{n \to \infty} \frac{P(n)}{P(n-1)} = \rho$$

$$\rho$$ is the real root of the cubic equation $$x^3-x-1 = 0$$.

A cubic equation either has three real roots ( which may be repeated) or one real root and two complex roots.

$$\rho$$ has the exact value

$$\displaystyle \rho = \frac{\sqrt[3]{108 + 12\sqrt69} + \sqrt[3]{108 -12\sqrt69}}{6}$$

with an approximate value of

$$1.324 \, 717 \, 957 \,…$$

Its continued fraction is

$$[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, …]$$

The powers of $$\rho$$

$$\begin{array}{l|l} \rho^n & Approx. Value\\ \hline \rho^{-2} & 0.569840\\ \rho^{-1} & 0.754878\\ \rho^0 & 1\\ \rho^1 & 1.324717957\\ \rho^2 & 1.754878\\ \rho^3 & 2.324718\\ \rho^4 & 3.079596\\ \rho^5 & 4.079596 \end{array}$$

The following diagram shows both of the above given recurrence relations and the plastic ratio in one dimension (the division of the vertical line on the RHS  –  $$\rho^2 : \rho^3 = 1 : \rho$$) and the plastic ratio in two dimensions (the rectangle on the LHS  –  $$\rho^4 : \rho^5 = 1 : \rho$$).

© OldTrout $$2018$$

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