Mobius Objects for Studying Topology

These objects only have one surface and one edge.
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updated October 21, 2024

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What, you don’t believe that they only have one surface and one edge!!!!

OK, start with the simplest shape that has the triangular cross section and get a felt tipped pen so you can draw on it.  Now put an “X” on the surface anywhere and start drawing a line from that point. Don’t cheat and cross over an edge. Go around the corner staying on the surface you selected.  Keep going. Draw the line right down the middle. Around the back side.  Stay in the center.  Don’t lift up the pen.  Make sure the line is continuous.  If you have to lift the pen to flip the object continue the line where you left off. Go around some more corners.  Wow, you came back to the starting “X”.

So if you drew a continuous line you only marked one side of the object.  Now check to see if there is any part of the object that doesn’t have a pen mark.  Your pen started at one point covered the complete object, and returned to the initial point. Unless, you found an unmarked region, the object only has one side.

The same exercise will show that it only has one edge.

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