Plastic rectangles

Plastic Rectangles diagram

I was intrigued by something I spotted on Wikipedia about the plastic number:

There are two ways of partitioning a square into three similar rectangles: the trivial solution given by three equal rectangles with aspect ratio 1:3, and another solution in which the three rectangles all have different sizes, but the same shape, with the square of the plastic number as their aspect ratio.

Wikipedia lacked a diagram so I made one (above), and here follows a proof that the aspect ratio is as claimed. The plastic number \(p\) is the unique real root of \(x^3=x+1\). The sides labeled in the diagram satisfy:

\[ \begin{aligned} b &= 1 - a \\ c &= a (1 - a) \\ d &= 1 - a (1 - a) \end{aligned} \]

The rectangles are all similar, meaning they have the same aspect ratio, so:

\[ \frac{a}{1} = \frac{b}{d} = \frac{1 - a}{1 - a (1 - a)} \]

Multiplying out the equation gives:

\[ a^3 - a^2 + 2a - 1 = 0 \]

The claim is that \(a = \frac{1}{p^2}\) because \(p > 1\) and clearly \(a < 1\), so the aspect ratio convention was switched in my diagram. Substituting this for \(a\) gives:

\[ p^{-6} - p^{-4} + 2p^{-2} - 1 = 0 \]

which multiplies out to:

\[ 1 - p^2 + 2p^4 - p^6 = 0 \]

Now, substituting \(p^3 = p + 1\) gives:

\[ 1 - p^2 + 2p(p+1) - (p+1)^2 = 0 \]

and multiplying this out gives \(0 = 0\) as all the terms cancel.