Möbius Strip and… tetra-tetra flexagon Model Experimental General Leceum of Heraklion Math Laboratory October 26th, 2015 to our Guest from Holland

What is Topology? It is another way to view geometric objects. It focuses not on shape but in continuous transformations. So… for Topology, a coffee mug is identical to a donut. But a donut is not identical to a sphere, because of the hole inside the donut!

Make a Möbius Strip Möbius strip is a very strange topological object. Take a long paper strip. Twist it once about the long axis. Join the narrow ends.

Möbius Strip by M.C. Escher Maurits Cornelius Escher (1898-1972)

Some experiments: 1st Draw a line along the strip, half way from its edges. Keep the move continuous, stop when you are back where you started. Does it tell you anything about how many sides does Möbius Strip have? watch

Some experiments: 2nd Cut the strip along the line you drew in the first experiment. What happened? Did you expect that? watch

Investigation What would happen if we started to cut at a distance of 1/3 of the Möbius strip width?

Tetra-Tetra-Flexagon: a Möbius Strip Application This is how we make a magic card with magic pictures! Magic, because it has hidden sides that reveal from the center of the card. With magic pictures, because they change (reverse) as you unfold the card. Here is an example:

1st step (folding) Begin with an A4 paper sheet in landscape orientation.(1st pic.) Fold it in half vertically twice (1st and 2nd pic.) Now make 2 horizontal folds, so as to divide your sheet into 43 cells. (3rd pic.)

2nd step (cut a “window”) As your sheet is fold in half, cut halfway the two horizontal folds as shown below (1st picture). Open the sheet and cut once more to create a “window” (2nd picture below).

3rd step (“rap” the paper) For the next two moves, keep your paper oriented as shown in the first picture. “Open” the window, and fold it to the right. Then, fold the square that extends to the back of the sheet.

3rd step (“rap” the paper) Fold the left column to the right twice, as shown below.

4th step (put a tape) Turn over your card carefully. One of the squares jumps out of the card. Put some tape on this square and the one next to it to join them (just these two middle ones). Mark the face with the tape as “4”. Turn the card over and mark that face as “3”. This is where you put the tape.

5th step (more sides) Fold your card in half with face 3 outside and face 4 inside. Open along that fold to reveal a new face. Name it “2”. Repeat to reveal the last face, face “1”. “Magic pictures” that reverse, can be drawn in faces “2” and “3”.

Thank you!

Resources R-U-B-B-E-R Geometry (Topology) Examples of topologically equivalent surfaces/figures World of Escher Posters Möbius Strip by David Pleacher Jim Milner Sculpture TETRA-TETRA-FLEXAGON Flexifier

Animations A short looping animation by Vlad Holst of the endless cycle of reincarnation Mobius Strip II Animated Movie by Mike Wilson A gif animation of Mobius Strip II Cutting a Möbius strip in half