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 x3=x+1. The sides labeled in the diagram satisfy:
b=1−ac=a(1−a)d=1−a(1−a)
The rectangles are all similar, meaning they have the same aspect ratio, so:
a1=bd=1−a1−a(1−a)
Multiplying out the equation gives:
a3−a2+2a−1=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:
p−6−p−4+2p−2−1=0
which multiplies out to:
1−p2+2p4−p6=0
Now, substituting p3=p+1 gives:
1−p2+2p(p+1)−(p+1)2=0
and multiplying this out gives 0=0 as all the terms cancel.