exotic sphere

In differential topology, an "exotic sphere" refers to a differentiable structure on a sphere that is not equivalent to the standard one. John Milnor discovered 28 such structures for the seven-dimensional sphere ((S^7)) in 1956, a finding that significantly revived interest in differential topology.1

A sphere, in geometry, is the set of all points equidistant from a central point. Its surface area is calculated as (4\pi r^2) and its volume as (\frac{4}{3}\pi r^3). The concept of exotic spheres extends this by exploring different ways a sphere can be "smoothly" shaped, even if they appear topologically identical to the standard sphere.21