# The Silver Ratio

Just as there is a golden ratio,  $$φ = (1 + √5) / 2$$, based on the Fibonacci numbers, there is a silver ratio, $$δ_s = 1 + √2$$, based on the Pell numbers.   John Pell, $$1611 – 1685$$.

The continued fraction for $$δ_s$$ is :

The Pell numbers begin :

$$0, 1, 2, 5, 12, 29, 70, 169, 408, …$$

To calculate the Pell numbers the initial conditions are $$0$$ and $$1$$, then each Pell number is the sum of twice the previous number plus the number before that.  For example,

$$29 = 2 \times 12 + 5$$.

$$P₀ = 0,\, P₁ = 1$$,

$$P_n = 2P_{n – 1} + P_ {n – 2}$$.

Successive ratios of the Pell numbers converge on $$δ_s = 1 + √2 = 2.41421356 …$$

\begin{align} 2/1 &= 2\\ 5/2 &= 2.5\\ 12/5 &= 2.4\\ 29/12 &= 2.416666 …\\ 70/29 &= 2.413793 …\\ 169/70 &= 2.414286 …\\ 408/169 &= 2.414201 … \end{align}

A Pell equation is a Diophantine equation of the form

$$x² – Ny² = 1$$

When $$N = 2$$, the integer solutions expressed as ratios $$x/y$$ provide ever closer approximations to $$√2$$.

$$(3,2), (7,5), (17,12), (41, 29), (99,70), (239,169), (577,408), …$$ are integer solutions.

Note that the $$y$$-coordinates are the Pell numbers and the $$x$$-coordinates are the current $$y$$-coordinate plus the previous one,  e.g.  $$99 = 70 + 29$$.

The Ancient Greeks and Indians knew of these ratios.  $$577/408$$ was known to the Indians in the $$3$$rd/$$4$$th century B.C.

The golden ratio is connected to a regular pentagon; the silver ratio is connected to a regular octagon :

There is a silver rectangle :

Which has a silver spiral :

In the same way that the golden ratio is connected to Penrose tilings, the silver ratio is connected to Ammann-Beenker tilings :

A close-up showing the rhombi and squares :

The rhombi have angles $$45° (π/4)$$ and $$135º (3π/4)$$ as does a regular octagon :

How to make your own silver rectangle.

Paper sizes are rectangles in the proportion $$1 : √2$$

A sheet of A4 paper measures $$210$$mm by $$297$$mm (dividing both numbers by $$3$$ gives $$70 : 99$$).

If you cut a $$210$$mm by $$210$$mm square from a sheet, then you are left with a rectangle measuring $$87$$mm by $$210$$mm. This rectangle is a silver rectangle.

\begin{align} 87\, &:\,210\\ 29\, &:\,70\\ 1\, &:\,2.414\,(3\, d.p.) \end{align}