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Discussions of the Möbius Strip as a concept/artefact in its own right

Of interest is the Möbius strip as a concept/artefact in its own right.  As I have found, there is more to the Möbius strip than meets the eye. Much more! Although it may be easily and commonly described, as a strip of paper joined at the ends with a half twist, there are various matters to address here. Although strictly this discussion may not be thought necessary, as anyone with any interest in it can define it simply as above, there are various concerns and quibbles as to defining sides, twists, folds, surfaces, edges, opaqueness, dimensions and material. And are we talking about a mathematical ideal or a model in three three-dimensional space? There is much more to this than it first seems. On many of these aspects, even among respected mathematicians, opinions differ. Perhaps somewhat surprisingly, I have found the best discussions not in books or articles but rather on Quora, an online general question-and-answer site (and not especially of a mathematical focus as may be thought, given the quality of the answers).  In particular, Wayne Kollinger, a little-known authority, who deserves to be at the forefront of the field, has thought this through more than most, and of which I lean heavily on in the discussion, along with the thoughts of (mathematicians) Alan Bustany and David Joyce, who give a series of insightful discussions on fundamental aspects. Kollinger's exposition is outstanding for clarity. All these people have much better mathematical credentials than I have and so are in a much better position to address fundamental matters.  Pleasingly, most of the discussions are at a popular level. Therefore, I now examine some of the defining questions. 

In some places, the answers have been edited for the sake of clarity but only in a minor capacity, such as typos or other. The text remains essentially as intended by the respective authors. 

Q. 1. Is a Möbius strip a two- or three-dimensional object? (Four answers)

A. by Wayne Kollinger
A Möbius strip is a two-dimensional surface. I can think of two reasons it can be mistaken for being three-dimensional.

(i) A Möbius strip exists in three-dimensional space.
Consider a straight line. It is one-dimensional. It has length, but no width or thickness. A curved line is also one-dimensional. However, if you want to draw a curve you need a two-dimensional surface in which to draw it. A curve exists in two dimensions. Mathematicians say it is embedded in two dimensions.

A Möbius strip has length and width but no thickness; it is two-dimensional. However, because it is made with a twist it cannot exist in a surface; it needs space to exist. Mathematicians say it is embedded in three dimensions.

(ii) It seems that when they first encounter a Möbius strip almost everyone is told that it is possible to make one from a strip of paper. Paper is three-dimensional; it has length, width and thickness. It is easy therefore to imagine a Mobius strip as being three-dimensional.

A paper Möbius strip is a model of a Möbius strip. It is a Möbius strip the way a plastic model is a fighter jet. It is a representation of the real thing but it is not the real thing. The model is three-dimensional but the thing it represents, a Mobius strip, is two-dimensional.

When you accept both the mistaken idea that a paper model is an actual Mobius strip and the correct idea that a Möbius strip is two-dimensional what happens is that it becomes easy to confuse the properties of the two.

For example, draw a line on a paper model of a Möbius strip and it travels around what seems to be two sides and comes back to meet itself. Many people conclude this shows that the model has only one side. It doesn't. If you were the size of an ant it would be obvious that there is room on the "edge" of the paper model for a second line. This second line shows that the paper model has two sides.

Another example, make a model of a Möbius strip from a strip of paper that is 6″ long. It represents a Mobius strip 6″ in length. However, if you draw a line on the model and measure it, it is 12″ long. This creates a problem because it seems to show that the Möbius strip is both 6″ and 12″ long. A Möbius strip seems paradoxically to be twice as long as it is.

One more example, too many people think of a Möbius strip as an object because a model of a Möbius strip is an object. A Möbius strip is a two-dimensional surface. It exists as a concept and not as a physical object.

These and other similar problems could be avoided by studying an actual two-dimensional Mobius strip instead of a three-dimensional model of one. Fortunately, it is easy to make an actual Mobius strip.

Start by making a paper model of a Möbius strip. Give a strip of paper a half twist (180 degrees) and join the ends together.

Take a second strip of paper that is twice as long as the first. Run it over the surface of the model. Like a line drawn on the model it will travel all the way around the model and come back to meet itself. Join together the ends of the second strip. We can call this second strip a wrap strip. The wrap strip completely covers the model and hides it from sight.

Without disturbing the wrap strip remove the paper model from between it. (The model will have to be cut to be removed.) Where the model was there is now nothing and the wrap strip is now in contact with itself. Or rather there is a two-dimensional space where the model was. It is a space that has length and width but no thickness and is the same shape as the model. It is an actual two-dimensional Möbius strip embedded in three dimensions of space.

This, not the paper model, is the Möbius strip that has one side, one edge and is non-orientable.

A. MightyMeepleMaster (Reddit)

The point here is to distinguish between the mathematical object "Möbius strip" and its physical counterpart.
The physical object is, of course, three-dimensional since it's simply crafted from paper. But this is only an illustration, nothing more. A real Möbius strip is an abstract, two-dimensional topological object which you can describe mathematically.

A. Wikipedia

He [Möbius ] is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space.

A. MacTutor
A Möbius strip is a two-dimensional surface with only one side. It can be constructed in three dimensions as follows. Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180-degree twist. 

My thoughts. The question is fundamentally whether we are considering a mathematical ideal concept (as two dimensions) or as a physical model (as three dimensions).

Q. 2. How many sides does a Möbius strip truly have?

A. by Wayne Kollinger

A plastic model of a fighter jet is not a fighter jet. And a paper model of a Möbius strip is not a Möbius strip, no matter how many well-intentioned amateur mathematicians say it is.

A Mobius strip, a real Mobius strip, is a two-dimensional surface; it has length and width but no thickness. It has only one side. But to understand this you have to first understand that the word side, like so many words, has more than one meaning and then you have to understand the meaning that applies to a Möbius strip.

You are right when you say that a regular sheet of paper has six sides. (It also has twelve edges.) These sides are physical surfaces. You can draw on them, paint on them, and ants can walk on them. These are not the kind of sides that a Möbius strip has. A Möbius strip is an abstract surface, it doesn't have a surface.

If I tell you there is a tree growing on the west side of my house, you don't think I mean it is literally growing on the surface of my house; you know that it is growing in the area adjacent to my house. Side often means adjacent area. A sphere has two adjacent areas, an in side and an out side. A Möbius strip has only one adjacent area and so it has only one side.

It is easier to understand a Möbius strip and the nature of its side if you make one.
1 Make a model of a Möbius strip. Give a strip of paper a half twist and join the ends.
2 Take a second strip of paper that is twice as long as the first and slide it over the surface of the paper model. Like a line drawn on the paper model it will travel all the way around and come back to meet itself. Join the ends of this second strip. We can call this a wrap strip because it wraps around the Möbius strip and covers it completely.
3 Remove the paper model from between the inner surface of the wrap strip. There is now nothing where the paper model was and the wrap strip is in contact with itself. Or rather, there is a two-dimensional space with the same shape as the model where the paper model was. This space has length and width but no thickness. It is completely surrounded by one object, the wrap strip, so it has only one adjacent area. It is an actual Möbius strip.

It should be obvious that this Möbius strip does not have a physical surface. There is nothing to draw on, or paint on, there is nothing for an ant to walk on. It is an abstract mathematical surface.

A sphere is also an abstract mathematical surface. It has two sides, an in side and an out side. Imagine cutting it in half to make a semi-sphere. The semi-sphere has two sides. Imagine flattening the semi-sphere; it still has two sides. Imagine stretching the flattened semi-sphere, giving it a half twist and joining it to itself. The out side and the in side match up and join together to form a single side. The result is a Möbius strip.

The Mobius strip has a single side, a single adjacent area but it is not part of the Möbius strip.

A paper model of a Möbius strip has two sides, a large surface that is often called a side and a smaller surface that often called an edge.

A true Möbius strip has only one side, an adjacent area that entirely surrounds it.

A. Alan Bustany (MA in Pure Mathematics & Theoretical Physics, Trinity College, Cambridge)

That truly depends on your definition of "side". Something that is not as trivial or simple as you might think.

A 2-dimensional manifold is something that locally looks like a plane, a small piece of regular 2-dimensional Euclidean space. Globally it can be shaped in many different ways. For example, it can be:
Infinite and unbounded like the Euclidean plane
Finite and unbounded like a sphere or the surface of Earth
Bounded like a disk
Have holes like an annulus
Have holes like a torus, and so on.

What we normally think of as the side of an object would be a 2-D manifold or surface of some description. Manifolds have a very important property called orientability. A surface is orientable if you can choose a normal vector consistently and continuously across the entire surface. Equivalently you can define a consistent notion of "clockwise" on the surface. An orientable surface has a consistent direction that distinguishes one "side" of the surface from the other.

A Möbius strip however is a non-orientable manifold. Starting with a normal vector at some point on the strip, you can "slide" it continuously around the strip and back to the same point where it is pointing in the opposite direction. Hence you cannot consistently assign such vectors to points on the strip.

You might say the strip has only one side. But you might also insist that "sides" are orientable surfaces, in which case a Möbius strip does not have a side.

Topologically we can classify 2-D manifolds by the number of "holes" in them. It turns out that all non-orientable manifolds can be created from orientable manifolds by filling a "hole" with a Möbius strip (identifying the single edge of the strip with the edge of the hole). Non-orientable surfaces then appear in this classification as intermediate between orientable surfaces with a Natural numbers of holes.

In some sense a Möbius strip is half-a-hole, so it is not surprising that we have difficulty saying how many sides it has.

Created 2 May 2024. Last Updated  2 May 2024