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
This code represents a singleton, it is saying: for a type 'A' and an element of that type 'a' there exists a unique path to any point 'x'.   "contractibility of singletons": singl (A : U) (a : A) : U = (x : A) * Path A a x CubicalCC code from here. 
This proves it is contractible. For a type 'A' and any two elements 'a' and 'b' and a path between those points 'p'. 
contrSingl (A : U) (a b : A) (p : Path A a b) : Path (singl A a) (a,<i> a) (b,p) = <i> (p @ i,<j> p @ i /\ j) CubicalCC code from here.  The first component of the above pair has to be a path from a to b,  this is exactly what p @ i gives us (note that we are to the right  of <i> so that i is now in context). The second component should be  a square connecting <i> a to p and this is exactly what the above  square for p @ i /\ j gives us 