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Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1.

$\sum\limits_{a=1}^\infty \frac{1}{a (a+1)} =1$

I believe it's still an unsolved question whether the infinite rectangles can fit in the unit square. If it's been solved, I'd love to see the paper.

There is a method by Paulhus for packing infinite fractional squares into a rectangle.

We can start with a rectangle with an area slightly greater than all the fractional squares. $$\sum\limits_{a=2}^\infty \frac{1}{a^2} = \frac{\pi^2}{6}-1 \approx 0.644934067 \approx 0.644934069 \approx \frac{545}{467} \times \frac{21}{38}$$

His coding method seems to work very well, as seen below.

However, I haven't been able to figure out quite how to modify the code to intelligently do oblongs instead of squares. Can anyone modify the code and get the first 2000 or so rectangles into the unit square? Or has that already been done?