How to turn a donut inside-out.
Image 1: A diagram depicting the poloidal (θ) direction, represented by the red arrow, and the toroidal (ζ or φ) direction, represented by the blue arrow.
Image 2: Turn a donut inside-out (in the hyper-sphere) while interchanging the meridianal and longitudinal curves represented by the cuts. Credit for this one is due to Dr. Chaim Goodman-Strauss. [Source]. See more at here.
Image 3: Turning a punctured torus inside-out. [Source]
Torus ( Shape of Donuts):
- In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit.
- Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus. In fact, S3 is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle).
Search more at http://en.wikipedia.org/wiki/Torus