The Floating Tower : realistic or science fiction project?

ANALEMMA TOWER is a real project of Clouds Architecture Office and consists in a floating tower.
How can a building float in the air? The tower will be hanged with heavy-duty cables to an asteroid (previously catched) orbiting the Earth.

The building orbits the Earth according the asteroid orbital way.
The orbital path will have a shape like an 8 and a period 24h long, this means that the tower won't be every time at the same place, but will travel above major cities.

In meantime NASA communicated that in the future harnessing and manipulating asteroids will be possible.
Same NASA is planning to capture and redirect an asteroid in 2021 (not so far!)

I think this is one of the most impressive project that I ever seen.
I'm really thinking about it: it would be possible or not?
Maybe this specific project won't be realized in a short time, but the fact that also NASA is working on asteroids catch , let me believe that in future will can travel using asteroids!

For more detailed information and imagines, here the link of my source:
http://www.cloudsao.com/ANALEMMA-TOWER