Möbius strip: Mathematical origami

Interesting tricks using the Möbius strip

Are you ready to ex­plore the amaz­ing world of topol­o­gy?

None.

• pa­per;
• scis­sors;
• glue.

Step-by-step in­struc­tions

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.

Process de­scrip­tion

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.