A Möbius Strip is a one-sided surface that can be constructed by taking a rectangular strip of paper, twisting it once and joining the ends of the strip. The result is a wonky shaped ring. This ring, discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, has a number of interesting properties. Most notably, a Möbius Strip is a non-orientable surface. This means that there is no way to differentiate between clockwise or anticlockwise turns. Try making one, and you’ll see!
Another interesting characteristic of a Möbius Strip is that it is a 2-dimensional surface. This seems counterintuitive because usually we can draw 2D shapes easily. When we try to draw a Möbius Strip, we use 3D drawing techniques. So, why is it a 2D shape? A 2D shape is a shape that has only two dimensions: height and width. We also cannot usually pick up a 2D shape, as it only has one side, i.e. no depth. A Möbius Strip does only have one side, pick a starting point and travel along the Strip, you will see that you end up back at your starting point, implying that there is only one side. We also clearly have a height and a width, so it seems to fit the definition of a 2D space. The difficult part is deciding if the Möbius Strip has any depth. Technically, it does have depth, as we can pick it up. However, since it only has one side, that must mean it also doesn’t have any depth. This strange paradox is an example of a 2D shape that is embedded in a 3D space. Simply put, a Möbius Strip is a 2D shape that has 3D characteristics.
Interesting things happen when we cut a Möbius Strip. If you cut a Möbius Strip down the middle, the result is not two thinner Möbius Strip, but rather one larger loop with an extra turn. Why? Because the cut down the middle does not split up the shape. The cut follows along the strip and travels from the inside to the outside in a spiral pattern. To verify this, try tracing a line down the middle of a Möbius Strip and you will notice your pencil never leaves the shape and returns to your starting point, essentially drawing a loop. When you cut an ordinary 2D shape down the middle, your pencil will reach another edge of the shape and is unable to return to the starting point. The reason for this is the half turn. Any loop with a 180-degree turn will have this property. This is also true for any loop with an odd number of 180-degree turns, such as 540 degrees or 900 degrees. These are all types of Möbius Strips.
If a loop has an even number of 180 degree turns, then the story is different. When we first cut our Möbius Strip in half, we produced one larger loop with an extra turn. So, we have two 180-degree twists, behaving like two intertwined paths. If we cut this loop in half again, we end up with two loops that are interlocked. This is because cutting down the middle splits these intertwined paths, but since these paths are twisted around each other, when we cut down the middle, they are interlocked. We cannot separate these loops without breaking them. So, when we cut a loop with an odd number of 180-degree turns (i.e. a Möbius Strip) we end up with one larger loop. If we cut a loop with an even number of 180-degree turns, we end up with two smaller interlocking loops.
What happens if we try a different cutting pattern? For example, try cutting a Möbius Strip into three pieces instead of two. Begin by marking the beginning point which will be a third of the width. Then continue cutting at this width and at some point, you will have passed your starting point, and now continue cutting keeping the width of the strip you are cutting equal to a third of the original width. Once you finish cutting you should see that instead of one larger Möbius Strip we have two loops: one larger and one smaller. Also, these strips are interlocked with each other. Due to the more complex cutting pattern (thirds instead of halves) we have a similar but more complex structure. By cutting down the third, we are cutting out strips that have more twists than before. As the cuts are uncentred, we have one larger loop with four twists and a smaller loop with only a half twist (i.e. one 180-degree turn), which is a smaller Möbius Strip. The reason these loops are interlocked is for the same reason before; due to the intertwined paths of the twists, cutting them does not separate these paths, and they still interact with each other.
Möbius Strips are an interesting way to bend your mind and deepen your understanding of shapes and space. I would certainly encourage everyone to make their own Möbius Strips and experiment with the different shapes you can create.