# mathr

## Plastic rectangles

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.