Construction of the Möbius strip and other surfaces

Take four paper rectangles. The proportions are not important: you can take as dimensions for the sides approximately 5cm and 30cm. Choose one of the rectangles and glue together the opposite small sides: you obtain a cylinder. Do the same for the second rectangle, but twist the paper strip "once" (i.e. halftour) before glueing: you obtain what is called the Möbius strip. Do the same for the other rectangles twisting respectively twice and three times before glueing. You can see animations which suggest the constructions by clicking on the corresponding figures in the next table. The figures and animations were made with Mathematica and translated to Java using appletsfrom Live Graphics3D. If you have never used these applets, start by looking at the corresponding help.

n

Animations

Figures

Animations

n

Central
cut

Non-centered
cut

Orientability

Keep in mind in particular that:

  1. a double click of the mouse interrupts or restarts the animation;
  2. stereoscopic viewing (of the double image created with the "s" Key) does not require special glasses: only the capacity to fuse together the two images. It is easier to manage with small near images than with large ones and the size and distance of the images depend on the screen resolution being used;
  3. on the first column is the number of twists before glueing;
  4. on the column Animations, the small figures are links to animations which suggest the corresponding glueing process and the third line gives the size of the corresponding files to import (besides those of the applet "classes");
  5. on the column Figures, the images link to the final figures obtained, with the border marked;
  6. on the column Central cut, the small figures are links to animations which suggest the longitudinal cut of each strip along the central circle;
  7. on the column Non-centered cut, the small figures are links to animations which suggest the longitudinal cut of each strip starting from a point on the strip which is at a distance of 1/3 of the width of the strip from the border;
  8. on the column Orientability, the question of the non-orientability of some surfaces is raised.

Paint the surfaces obtained, to conceal the glueing area. Take two of the paper models and try to find what there is in common between them and what is different. Do that for each couple. The ideal would be to identify a property in relation to which some of these models can be considered equivalent and then try to identify what distinguishes these equivalent models.

As illustration of what has been said, links to animations, which correspond to cutting (and separating) a Moebius strip longitudinally, respectivaly along the central circumference and along a line locally dividing the Moebius strip in three parts of equal height, have been placed in the previous table. You are advised to proceed in the following order:

  1. Try to imagine what is obtained, in each case, for \(n=0\) (easy), \(1\), \(2\), \(3\), ...;
  2. Make the experiment with paper and scissors to check whether your conjecture is correct;
  3. Reconstruct with the animations what you did with the scissors.