Maths - Shapes

Here are some ways of modeling various Shapes:

  Homotopy Type As Cell Complex
Circle  
n-sphere  
Torus torus
Möbius Band   mobius band
Projective Space   projective
kleinBottle    

For more information about constructing shapes as cell complexes see the page here.

N-Sphere

Here is more detail about constructing the n-sphere using paths.

0-sphere 1-sphere 2-sphere 3-sphere
0-sphere and so on...
The 0-sphere consists of 2 points. There is no non-trivial structure for higher dimensions.

The circle (1-sphere) consists of:

  • 2 points.
  • and 2 lines between these points.

There is no non-trivial structure for 3D and above.

The hollow sphere (2-sphere) consists of:

  • 2 points.
  • 2 lines between these points.
  • and 2 planes between these lines.

There is no non-trivial structure for 3D and above.

 
       
Could the circle be constructed using just one base point and one line? circle

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

 


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see also:

Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory:

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
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