Our first möbius strips are ready!
We experiment with theese...
A long line is drawn in the middle,
and we cut the strip through that line!
Why do we get different results?
This is puzzling!
It remains one peice!
Let's repeat the experiment!
We now create tetratetra flexagons.
We cut a "window" in the middle
Ready...
...to look through it!
We keep folding.
Add some tape in the end.
Surpise! The card is magic!
The whole group is really bussy!
We number the sides of the card
They are 4, not 2 as it seamed at first!
To unfold it is really tricky!
Some more decoration.
There! Some harts!
Topology is “rubber” geometry. A figure may change its shape continuously, but as long as it keeps some other characteristics as holes and knots, is regarded unchanged. As a branch of Mathematics, Topology is a mixture of Geometry and Analysis and notions like “continuous mapping” or “compactness” demand a good level of Analysis for their approach.
Even so, there are a few things to do in a very basic level suitable for High School work. We gave a try to those, in October 26th 2015, when we had 24 visitors in our school from Vossius Gymnasium of Amsterdam, Holland (19 students and 5 teachers), members of the Erasmus+ program Geo Future Excellence Programme (GFEP). Five of those students and their teacher Mrs. Ingrid Kemerink joined the B2 Class in Geometry to learn about Möbius Strip and TetraTetra flexagon.
The paradoxes of Möbius Strip were firstly introduced by the art of M.C. Escher, an artist from Holland, famous for his math prints. Then all students constructed quite many Möbius strips and experimented with those. They learned that they may exist surfaces with just one side (no distinction between inside and outside), that when cut in half they remain one piece and other topological paradoxes. Finally, tetratetra flexagon was introduced as a “magic card”. Students really enjoyed the task to reveal the hidden sides of the card and to watch the “magic pictures” reversing.
Below, there is a small presentation that guided their work. I hope it will guide you as well to wonderful creations! Just push the button "next" to move from one slide to the next.
Model Experimental General Leceum of Heraklion
Math Laboratory October 26th, 2015
to our Guests from Holland

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!

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

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?

Cut the strip along the line you drew in the first experiment. What happened? Did you expect that?
WATCH

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

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

 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).

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

2. Fold the left column to the right twice, as shown below.

 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”.

 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”.


...and the worksheet