June 11, 2025

Monique Nguyen

11th Grade

Fountain Valley High School 

Introduction

Everyone’s seen the recycling symbol: three arrows that follow each other in some sort of triangular loop as the simplest description. But, the recycling symbol actually models a popular object with mathematical implications called a Möbius loop (or sometimes Möbius strip or band), which is essentially the recycling symbol connected, just with half-twists in the place of arrows. Now, some of the geometric properties behind these Möbius loops, such as how they only have one edge and side and are non-orientable, allow them to play a vital role in modelling the shape of another mathematical object called the Klein Bottle. This bottle has a special name because it is non-orientable like the Möbius loop, but even more interestingly, the shape also has no edges.

Möbius Loops

August Möbius, the mathematician most credited for the shape, officially discovered the Möbius loop in 1858 as he was studying another geometrical shape, a polyhedron, which is a three-dimensional shape with multiple flat faces, edges, and vertices. Möbius loops are surprisingly easy to make. They’re made by twisting a simple strip of paper an odd number of times, then taping these twists together in a loop; this gives them only one edge. To provide a visual, this would mean that by coloring a Möbius loop in, both sides of the paper would be colored in without ever crossing an edge, and this is because of those half-twists in the shape. Moreover, by cutting through the middle of the Möbius loop, this does not result in two loops of paper; it actually results in only one paper loop. Because of its properties, this curious shape connects to various fields, such as topology and mathematical orientation. 

Orientation essentially refers to an object’s angular position in space in reference to something else or a set standard. For the Möbius loop, running an object along the loop allows an object to change or flip its orientation when it returns to its starting point, while it is still in the same mathematical space as before. Neil deGrasse Tyson explains it as having a right-handed glove, and as this glove is run along the surface of the Möbius loop, it will change its orientation and become a left-handed glove when it returns to its starting position, essentially becoming a mirror image of itself. On the other hand, an object with two sides instead of one, unlike the Möbius loop, the right-handed glove will remain the same when it returns to its starting position, regardless of the path taken. This idea of orientability also allows topologists to distinguish between the two types of shapes. Topology is the field that studies the properties of objects after specific transformations, such as stretching, contracting, and twisting, rather than any cutting and gluing. Regardless of the size or shape, the mathematical properties of the shape stay the same; the properties are independent of size, so they are virtually the same object to a topologist. A donut, for example, clearly only has one hole, and by stretching the donut, it maintains the number of holes it has, so it is the same object despite a difference in size. The Möbius loop heavily influenced the expansion of this field, allowing for different discoveries in studying how the world works as well as comparing nonorientable surfaces to orientable ones. There was even a puzzle that was confirmed by Richard Evan Schwartz on whether the length from a previously proposed model, the Halpern-Weaver conjecture, is truly the smallest length for the Möbius loop to avoid something called an embedded band. In simple terms, it has no self-intersections or overlaps with itself in a three-dimensional sense.

Klein Bottle 

The mathematician Felix Klein officially discovered the Klein bottle in 1882, but it was popularized even more by an astrophysicist, author, and teacher named Clifford Stoll, who created a business selling them. For context, some may have heard of Stoll by his legendary effort in finding and stopping one of the first cases of cyberhacking or his book, The Cuckoo’s Egg, too. But, back to the Klein bottle itself, what is its connection to a Möbius loop, both mathematically and physically? Well, it’s possible to model the shape of the Klein bottle using two Möbius loops. Essentially, by combining the single edge of one Möbius loop with the single edge of another, this creates a four-dimensional shape with no sharp edges; with glass-blowing techniques, this shape is the Klein bottle, similar to a cylinder stretched and then pushed inside itself with the end of the cylinder meeting its own base. Fun fact: Some mathematicians like to put one Klein bottle inside another, sort of like Russian dolls. 

Anyway, the Klein bottle only has one surface or one side, meaning that its inside surface is the same as its outside surface. To provide a better image, by simply drawing along the Klein bottle, this will eventually color both the inside and the outside of the bottle without ever crossing an edge, quite similarly to a Möbius loop in that they are both non-orientable. Another one of its properties, which is inverting the bottle or trying to pull it inside of itself, shows that it would return to the exact same orientation at its original starting point. Additionally, back to the idea of self-intersections, there’s a clear space where the Klein bottle has to overlap itself, but in a four-dimensional space, it would be completely embedded without self-intersections; this is its true form. In layman's terms, the area right where the bottle overlaps would be as if there was no overlap; there would be no blockage when drawing a line horizontally across the bulb of the bottle or vertically along the loop of the bottle. But Klein bottles demonstrate the mathematical property where such objects must self-intersect in a three-dimensional space, differing from a Möbius loop. Topologically, the same concepts from the Möbius loop apply to the Klein bottle, where it maintains the same mathematical properties, such as number of sides, holes, and loops, even if the Klein bottle is made smaller or larger. 

Conclusion

Both the Möbius loop and the Klein bottle have made waves in the topological and geometric scenes; the Klein bottle itself is quite similar to the Möbius loop but is a more mathematically complex object in comparison, developing upon the mathematically foundational ideas that were initiated by the original loop. In demonstrating mathematically properties from orientation, self-intersections, and three-dimensions, as well as exhibiting various curious traits, clearly the loop and the bottle are essential to understanding how the physical world works and what other spaces or universes could look like. Plus, as an added bonus, Klein bottle hats and Möbius loop scarves are also for sale by Stoll in addition to his Klein bottle business, which demonstrates the properties just described in person!