when any someone who is asked to comment about: how many surface’s shapes are there? he or she could tell you this: if only closed surfaces are considered, there’re two types:
- orientable
- non orientable
Orientable are the sphere (the 2-sphere), the torus, double torus, etc… But non orientable are the projective plane: an abstract simple construction given by sewing a mobiusband with a 2-disk, 
Let’s abstract with the symbols 
With these we have an algebra of sets that looks like that
where we have abstracted: 
Even to them who know those ideas come into surprise when they are told that there’s no only one three dimensional shape: is 
Even if you think and say that 
Would you like to see a sample of the great variety of them? urgently follow this link here please!
This link gonna give you a rough idea of several types of 3-spaces looking a little more close at them,… some are now celebrated in 3d-topological technology…
http://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html
the 8 geometries
http://www.matematica.unimore.it/0ricerca/Gruppi/Gr_geometria.htm
Gruppo di ricerca