Four Weird Mathematical Objects

Here I discuss four interesting mathematical problems (mostly involving famous unsolved conjectures) of considerable interest, and that even high school kids can understand. The field itself has been a source of constant innovation -- especially to develop distributed architectures, as well as HPC (high performance computing) and quantum computing to try to solve (to non avail so far) these very difficult yet basic problems. And for those interested in mathematical logic and measure theory, here is an interesting paradox, which somehow allows you to duplicate a ball made out of gold, into two balls, each having the same size as the original ball (though it does not double the mass.) The Banach–Tarski paradox is a theorem which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.