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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