What is a figure-eight shape curve whose polar equation coordinates is ρ2=a2 cos 2θ or ρ2=a2 sin 2θ?

amazing that the brain can conceptualize infinity...or can we...without smashing into paradoxes...

Part 2 Explanation Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, assume that k guests seek a room. We can apply the same procedure and move every guest from room n to room n + k. In a similar manner, if k guests wished to leave the hotel, every guest moves from room n to room n − k.

Just saying... There is such thing as Infinite Hotel Paradox Explanation : Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.