Unexpected shapes

The Möbius^∞ strip

Source

   Consider a strip of plain old paper like, say, a long shopping receipt. Mathematically speaking, it’s kind of boring; it’s got the simple measurements of length and width from which we can calculate the area and a right angle at each corner but not much else. However, when we crumple it up into a ball, that same piece of paper becomes something different. All the new shapes and angles that are created make it far more interesting. The study of such twisted and scrunched up objects is called topology:

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects 1

wonderopolis.org/

   A simple and fun example of this is the Möbius strip. If you take a long strip of paper and make one half-twist (180°) and join the ends together (glue is best for this, but tape works too), then you’ve made a Möbius strip! So what? Well, first of all, let me tell you that the surface of the strip has only one side. To prove this, put a pen anywhere on the strip and draw a line down the centre; you’ll need to rotate the strip as you go. You’ll notice that you eventually make your way back to where you started without ever taking your pen off the paper; this is because the surface has only one side – try doing this with a regular piece of paper and you’ll find it can’t be done.

   Next, take a pair of scissors and cut down this central line all the way around the strip. It seems like you’re going to cut the strip in half and end up with two strips, right? But what happens?!

  

   You still end up with one strip! But, there’s a catch: it’s not a Möbius strip anymore. How can you tell? Well, take a pen of a different color and draw down the central line again. You’ll find your way back to where you started, but this time you’ll only have marked one side – this new strip has two sides!

But, wait, there’s more…

whatdowedoallday.com

    Make another Möbius strip but this time cut it along the point one-third of the width of the strip; see the example on the left. Keep cutting all the way around. What’s going to happen this time?

You now have a short Möbius strip intertwined with a longer twisted loop!

If that was a little hard to follow, check out this explainer video:

Cutting a Möbius strip in half  | AnimatedTopology |

Source

For more mindblowing unexpected shapes watch professor Tadashi Tokieda make a square out of two circles and some interesting conjoined love hearts.

Unexpected Shapes (Part 1) - Numberphile

For older learners, check out Dr. Tokieda's lectures on Topology & Geometry

This nifty Möbius Music Box is worth a look and this Möbius strip story telling demonstration is awesome.

 BONUS: Check out the Möbius Wall

∞  Named after German mathematician, and elderly Bilbo Baggins look-alike,  August Ferdinand Möbius