A central visualization problem in pure geometry is to create pictures of manifolds situated in space, as well as images showing how they might look from the inside.
Informally speaking, surfaces are 2-manifolds and volumes are 3-manifolds. A few examples of 2-manifolds constructed by gluing and twisting strips of paper may help clarify the definition in the Glossary. When we glue the ends of a strip of paper together after a twist, we get a (one-sided) surface called a Möbius band. If we glue opposite pairs of edges of a square together, we get the 2-torus, a 2-manifold that looks like the surface of a donut. A manifold is embedded when it is situated in a space without self-intersections or singularities. When we give one of the edges of a square a twist before gluing, we get a Klein bottle, which, being a closed, one-sided surface, cannot be embedded in 3-space. We can embed it in 4D space, but its projection to 3D must be self-intersecting. When we glue volumes instead of surfaces, we construct 3-manifolds. Gluing together the walls of a cube in opposite pairs yields a 3-torus. If we twist one volume before gluing, we obtain a 3-dimensional Klein bottle.