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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 x3=x+1. The sides labeled in the diagram satisfy:

b=1ac=a(1a)d=1a(1a)

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

a1=bd=1a1a(1a)

Multiplying out the equation gives:

a3a2+2a1=0

The claim is that a=1p2 because p>1 and clearly a<1, so the aspect ratio convention was switched in my diagram. Substituting this for a gives:

p6p4+2p21=0

which multiplies out to:

1p2+2p4p6=0

Now, substituting p3=p+1 gives:

1p2+2p(p+1)(p+1)2=0

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