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

see page here.


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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

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.

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

Other Books about Curves and Surfaces

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