Why don't iPad-type tablet screen edges have a ratio of 1 : 0.70710678118??

It's a fine idea, and it's also the idea behind ISO/metric paper sizes, as Markus Kuhn explores in this article: http://www.cl.cam.ac.uk/~mgk25/iso-paper.html

It's not ideal to zoom an app and all its GUI components to half its size. It would want to scale UI elements differently. So there's no strong reason to keep the proportions the same.

Also there's all the reasons to choose other screen ratios.

Agree that half-sized UI elements would be undesirable.

I like the idea. Is it posible to make a "prototype"? I.E. modify Android to place two black strips on the borders, so the free screen has this ratio.

It surprises me that nobody on the Android side is advertising they've done this.