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:
- 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, , by its similar border. The Kleinbottle is the surface got from sewing two mobiusbands along with their boundaries.
Let’s abstract with the symbols to the orientable surface of -genus and by the non orientable surface of -genus.
With these we have an algebra of sets that looks like that
where we have abstracted: , for a punctured surface , a closed surface with a open disk removed. And for sewing in the border
Even to them who know those ideas come into surprise when they are told that there’s no only one three dimensional shape: is the only three dimensional manifolds in math?
Even if you think and say that is another together with others little more easy to imagine.
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…