Imaging maths - Inside the Klein bottle

September 2003


  1. The following two images show different closed tubes. Which of them is orientable, and therefore a torus, and which is a non-orientable Klein bottle? The applet versions allow you to rotate the shapes and to switch on normal vectors to make it easier to investigate.

    Two copies of the Klein bottle. (Applet) The degree of symmetry may be increased. Here three bottles are joined. Which numbers of bottles give non-orientable surfaces? (Applet)
  2. There is no unique geometric realization of a Klein bottle. The following interactive applet leaves all the decisions up to you! Experiment with different parameter settings.

    The interactive Klein bottle. (Applet)

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