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 :

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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 \(√2\).

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\((3,2), (7,5), (17,12), (41, 29), (99,70), (239,169), (577,408), …\) 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 :

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There is a silver rectangle :

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Which has a silver spiral :

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

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A close-up showing the rhombi and squares :

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

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How to make your own silver rectangle.

Paper sizes are rectangles in the proportion \(1 : √2\)

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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.

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

 
© OldTrout 2016