# Maths - Shapes

Here are some ways of modeling various Shapes:

Homotopy Type As Cell Complex
Circle n-sphere Torus  Möbius Band Projective Space kleinBottle

### N-Sphere

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

 0-sphere 1-sphere 2-sphere 3-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? ### 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, a) (b,p) = (p @ i, 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 so that i is now in context). The second component should be -- a square connecting a to p and this is exactly what the above -- square for p @ i /\ j gives us```

 metadata block see also: Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory: Correspondence about this page Book Shop - Further reading . Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.       Introduction to Topological Manifolds (Graduate Texts in Mathematics S.) Other Books about Curves and Surfaces

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