Today we celebrate the birth of August Möbius (17 Nov 1790 – 26 Sep 1868), German astronomer, mathematician and author, and the man who introduced the concept of the Möbius Strip, one of the more interesting objects in mathematical topology.
Getting into the mathematics behind the Möbius strip is beyond the scope of this blog, but suffice to say that a Möbius strip is a two dimensional surface with only one side. Mathematically it is said to be a non-orientable object.
A physical model of a Möbius strip can be constructed by taking a long, rectangular strip of paper, giving it a 180 degree half-twist and joining the ends together to form a loop. Starting at any point on the surface of the resultant loop, you can draw a line along its centre, passing along both sides of the strip and eventually returning to your original starting point, without once lifting your pen.
Moreover, cutting the Möbius strip along this centre line with a pair of scissors will not result in two separate loops, but rather a single loop, double the length of the original, and with two full twists in it. Thanks to having an even number of twists, the resultant loop will not be a Möbius strip anymore – if you draw a line along its centre, you will remain on one side of the strip, and the line will not traverse both sides.
If you repeat the cutting exercise and cut this new loop along its centre, you will end up with two separate loops, wound through each other, and each having two full twists.
Getting back to our original Möbius strip, another ‘trick’ is to make your cut 1/3 from the one edge. If you keep cutting until reaching your original point, you end up with two loops – one new, thin Möbius strip (the center third of the original), which will be the same length as the original, and one thin loop with two twists and twice the length of the original.
In the same way that a Möbius strip (with 1 half-twist) when bisected, results in a new loop with 2 full twists, loops with more half-twists will lead to different end results. Generally speaking, a loop with N half-twists (where N is an odd number), when cut along its centre, becomes a loop with N+1 full twists.
These crazy, twisty loops are not only amusing things to cut up and play with – they have lots of useful practical applications. When designed in the shape of a Möbius strip, conveyer belts can be made to last longer, as wear and tear will be shared between the two sides. Similarly, typewriter ribbons can be more effective when shaped with a half-twist. Möbius strips, with their unique geometry, also occur in physics and chemistry, and even have applications in electric circuitry.
Amusing things indeed, and surely well deserving of their own special day!