Möbius strip: Mathematical origami
Interesting tricks using the Möbius strip
Are you ready to explore the amazing world of topology?
Cut out a strip of paper. Flip one end and glue it to the other. The result is a Möbius strip. If you rotate one of the ends twice and glue the two ends together, then cut the tape lengthwise, you’ll get two Möbius strips linked together. If one of the ends is rotated three times, you’ll obtain a thrice-twisted Möbius strip – the international recycling symbol. Glue two Möbius strips at a right angle and cut each lengthwise – you’ll get two bonded hearts.
The Möbius strip is named after German mathematician August Ferdinand Möbius. This is an example of an object that studies a separate domain of geometry – topology, the study of the properties of an object that remain constant through various deformations. To understand the properties of a Möbius strip, first consider a paper ring. It has two sides, an inner side and an outer side. If we mark a point on the outside with a marker, we can draw a line to any other point on the outside without lifting the marker and without going over the edge. There is a continuous path between any two points on the outside. This works for the inside as well. But it is impossible to get from the outside to the inside using a continuous line that does not cross any edges.
However, if we cut this ring, flip one of the ends, and reglue it, we will get a Möbius strip. It may seem that not much has changed, but now our object has only one side! You can connect any two points with a continuous line – the new surface unites the outer and inner sides of the original strip. This can be demonstrated by drawing a line along the middle of the strip. The Möbius strip has found wide application in science and technology, such as in conveyors and printers, and it also serves as a source of inspiration for many artists, directors, and writers.