Finding a point in an infinite space

[warty goblin]

Warty Goblin's method does bisect the the plane every time and you can easily extend it, so it would work for continuum simply by asking not just about higher digits but the fracions as well. The only thing it does not do is to give a good approximation until it obtains the highest digit.

Unless you run it backwards, and start by asking about the leading term of the binary representation of each coordinate. Only real change is asking whether the digit you just determined was the last before the decimal or not, instead of whether it's the last digit.