I learned about the Möbius Strip and the Klein bottle as a young girl from my topologist father. It is a good introduction to how strange and odd topology can be to us outsiders. You can make a Möbius strip by taking a simple length of paper, let’s say 2 cm wide by 20 cm long, and twisting one end one half turn then joining the two ends together. The intriguing result is that the strip has lost one of its two surfaces. Trace your finger around the surface and you will see. It is now a 2-dimensional creature, the same as a sheet of paper, sitting very comfortably in our 3-dimensional world. Very cool. Now make another with the opposite twist and join them together along their edges to make a Klein bottle. Oops ….not so easy. In fact impossible in our world, you need to go to 4-space, the curious and enticing hinterland of topology. Another way to make a Klein bottle is to stretch the neck of any bottle and poke it through itself and make it emerge at the bottom of the bottle. Glass blowers can do this but topologists won’t let you. They don’t allow intersection of surfaces. Surfaces have to dip and out of 4-space to bypass one another.
I welcome your comments. Please feel free to leave me a message on the Information page.