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Erasmus+ students apply topological tricks (Möbius Strip and Tetra-Tetra-Flexagon)
by Irini Perissinaki

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 tetra-tetra flexagons.
We cut a "window" in the middle
Ready... 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!
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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 Tetra-Tetra 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, tetra-tetra 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.

mobius strip
Möbius Strip and… tetra-tetra flexagon

Model Experimental General Leceum of Heraklion

Math Laboratory
October 26th, 2015

to our Guests from Holland

...and the worksheet



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