Three interesting and mind bending mathematical experiments on spheres.

The Banach–Tarski paradox is a theorem in set-theoretic geometry which states: Given a solid ball in three-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. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not “solids” in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. How it works is as fascinating as the paradox itself

The paradox of creating 2 identical spheres from 1 was also used as the key plot element in Futurama.

Turning a sphere inside out without tearing it

In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means “turning inside out”). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere’s surface) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false

Visualizing being inside a spherical mirror.

You’ve definitely seen infinity mirrors where one mirror is placed in front and one behind you to create the illusion of a long hallway of repeating images. But what if you were inside a round room that was coated like a mirror?