Here are some ways of modeling various Shapes:
Homotopy Type  As Cell Complex  

Circle  
nsphere  
Torus  
Möbius Band  
Projective Space  
kleinBottle 
For more information about constructing shapes as cell complexes see the page here.
NSphere
Here is more detail about constructing the nsphere using paths.
0sphere  1sphere  2sphere  3sphere 
and so on...  
The 0sphere consists of 2 points. There is no nontrivial structure for higher dimensions.  The circle (1sphere) consists of:
There is no nontrivial structure for 3D and above. 
The hollow sphere (2sphere) consists of:
There is no nontrivial structure for 3D and above. 

Could the circle be constructed using just one base point and one line? 
Contractibility of Singletons
see page here.