Maths - Shapes

Here are some ways of modeling various Shapes:

  Homotopy Type As Cell Complex
Torus torus
Möbius Band   mobius band
Projective Space   projective

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


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
Correspondence about this page

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cover Introduction to Topological Manifolds (Graduate Texts in Mathematics S.)

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