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.
To begin, let's look at the definition (which sometimes veers to a description) of a Möbius strip (or variant title), which may be thought to be a simple, straightforward task, almost not worth the trouble, as obvious it may be. After all, it's nothing more than a strip of paper joined at the ends with a half twist, is it not? However, defining is not as straightforward as may otherwise be thought. Although strictly this may not be thought to be necessary, as anyone with any interest in it can define it as above, there are various concerns and quibbles as to sides, twists, folds, surfaces, edges, opaqueness, dimensions and material. On all these opinions differ. In particular, Wayne Kollinger, a little-known authority, who deserves to be at the forefront, has thought this through more than most. Therefore, I now examine some of the definitions, from a variety of reputable sources.
A. by Wikipedia, Wolfram MathWorld, Britannica, Collins English Dictionary, David Mitchell, David Darling, and Michael A. Henle
A. Wikipedia
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist…The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip.
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
Wayne Kollinger rebuts this (on Quora).The problem is that a model and an example are different things. A model is representation of something; it is not the real thing. A plastic model of a fighter jet is not a fighter jet. An example is an instance of something; it is the real thing. An actual F-22 Raptor is an example of a fighter jet; it is one of many different kinds.
A. Wolfram MathWorld
The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322–323)... Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).
https://mathworld.wolfram.com/MoebiusStrip.html
A. Britannica
Möbius strip, a one-sided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist.
A. Collins English Dictionary
a one-sided continuous surface, formed by twisting a long narrow rectangular strip of material through 180° and joining the ends.
Wayne Kollinger rebuts this (on Quora). The problem is that a Mobius strip is a two dimensional surface; it has length and width but no thickness. All materials are three dimensional; there is no such thing as a material that doesn’t have some thickness.
For a moment forget the paper model of a Mobius strip. It’s leading you astray.
A Mobius strip is two-dimensional; it has length and width but no thickness. Because paper has thickness a paper model of a Mobius strip is three dimensional. There is a better starting point than the traditional paper model from which to reason…
Wayne Kollinger
A. by David Mitchell (origami/paper folder expert)
When made of paper a Mobius Band can be considered to be a folded paper object even though it does not contain any creases.
Although not exactly a definition, an interesting point (by accident?) is made that the paper is in ‘continuous deformation’ and so is not ‘smooth’. There is apparently some condition about ‘smoothness’.
A. by David Darling (Astronomer, science writer, and musician)
The Möbius band is a simple, mathematically important, and wonderfully entertaining two-dimensional object, also known as the Möbius strip, that has only one surface and one edge and is therefore of great interest in topology.
A. by Michael A. Henle (Professor of Mathematics and Computer Science at Oberlin College, US)
The Möbius strip, like the cylinder, is not a true surface but a surface with boundary.
Observations
All, more or less, give the same definition as is commonly accepted, a strip of paper joined at the ends with a half-twist. Paper is usually given as the material for reasons of convenience (availability and cheapness). However, there are other factors to consider; what about strips with multiple twists? Are these also to be considered Möbius strips? And if so, should they, with increasing twists, be of dwindling interest than the basic (½ twist) model? First, I classify the strip in terms of the number of half-twists, i.e. 0, ½, 1, 1½, etc. 0 is a cylinder easily seen, with increments involving the half twist i.e. ½, 1½ etc. are all Möbius strips 1, 2 etc, although having the superficial appearance of a Möbius strip, with twists, are not, and can be described as a topological cylinder (easily tested).
A. by Wayne Kollinger, MightyMeepleMaster, Wikipedia, and MacTutor
A. 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).
A. by Wayne Kollinger, Alan Bustany
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. by 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 the 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 number 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.
A. by Wayne Kollinger
It’s easy to make a Mobius strip. However, most people who think they know how don’t really know.
Paper model of a Mobius strip #1
Give a strip of paper a half twist (180 degrees) and join the ends.
This is the paper model of a Mobius strip that everyone first learns to make. The strip can be twisted either clockwise or counter-clockwise to model Mobius strips that are the mirror image of each other. The paper strip can be twisted any odd number (1,3,5,7,etc.) of half-twists to model additional Mobius strips.
Paper model of a Mobius strip #2
The recycle symbol is a good example of a Mobius strip that can be modeled by folding paper rather than twisting paper.
Gently fold a strip of paper over itself without creasing it. The two arms of the strip should be skewed at about a 60 degree angle to each other. Next gently fold each of the arms at a 60 degree angle so that the ends of the paper strip meet and can be joined.
The result is a paper model of a Mobius strip. Any odd number of skewed folds can be used to make a Mobius strip model. the angle of skew will vary with the number of folds.
Using the folding method you can make all the same Mobius strip models that you can make using the twisting method and more.
Paper model of a Mobius strip #3
Start by making an annulus with a large center hole. An annulus is a disk with a hole at its center. It is a flat ring. A washer is an example of an annulus. So make a large paper washer.
Remove a section of the ring equal to about a third of the annulus.
Take a strip of paper and join it to one end of the cut ring. Give the paper strip a gentle fold and join it to the other end.
The result is a paper Mobius strip model with a single fold.
Paper model of a Mobius strip #4
Take a strip of paper and tie a flat knot in it. Join the ends.
You will find you have created a flat loop with five folds. Five is an odd number so you have a model of a Mobius strip. You might want to check it out by drawing a line along the surface of the model.
Glass model of a Mobius strip #1
You will need a glass model of a Klein bottle to start with so unless you are an expert glass blower you will want to go online and purchase one.
When it finally arrives cut the Klein bottle model to create two halves that are the mirror image of each other.
Check and you will discover that each half is a model of a Mobius strip.
Glass model of a Mobius strip #2
You will need to go online again and purchase another glass model of a Klein bottle.
A Klein bottle is a two dimensional surface that is non-orientable, has only one side and like a sphere has no boundary (edge). In three dimensions a Klein bottle is a surface that passes through itself like a ghost through a wall. Glass is a three dimensional material and cannot pass through itself like a ghost through a wall. Therefore a model of a Klein bottle has an opening in its glass wall so it can pass through itself.
This time you need to enlarge the opening so that the glass does not touch as it goes through the wall. The opening now has an obvious edge.
A Mobius strip is a two dimensional surface that is non-orientable, has only one side and has a boundary (edge).
What you have is a glass model of a Mobius strip masquerading as a model of a Klein bottle.
Actual Mobius strip #1
Start with a paper model of a Mobius strip - give a strip of paper a half twist (180 degrees) and join the ends.
Take a second piece of paper that is twice as long as the first. Run it over the surface of the paper model. Like a line drawn on the surface it will travel all the way around the model and come back to meet itself. In the process it will completely hide the model from view. Join the ends of this second strip. Because it is wrapped around the model we will call it a wrap strip.
Remove the paper model from between the wrap strip without disturbing the wrap strip. (You will have to cut the model crosswise in order to remove it)The wrap strip is now in contact with itself. Where the model was there is now nothing. Or rather, there is a two dimensional space, a space that has length and width but no thickness. It is the same shape as the paper model. This space is an actual Mobius strip. It is a Mobius strip made of nothing.
You can never see or touch this Mobius strip. You can’t draw on it or paint it. Ants can’t walk on it. It exists; it is real. But if you open the wrap strip, it is gone.
Actual Mobius strip #2
Start with a wrap strip that surrounds a Mobius strip. Bond the wrap strip to itself. The two dimensional space is gone and the wrap strip becomes a model of a Mobius strip.
Where the Mobius strip used to be is the boundary that marks the center of the model. This boundary is a two dimensional surface in the shape of a Mobius strip. It is a Mobius strip.
Every model of a Mobius strip contains a Mobius strip in the form of a boundary at its center.
Just as a box contains chocolates but is not the chocolates a model of a Mobius strip contains a Mobius strip but is not a Mobius strip.
A model of a Mobius strip can be made into a wrap strip by cutting it along its center boundary. When that is done it surrounds a true two dimensional Mobius strip.
Actual Mobius strip #3
Start with a paper model of a Mobius strip - give a strip of paper a half twist (180 degrees) and join the ends.
It is traditional to cut a paper model of a Mobius strip lengthwise down the center and discover that instead of becoming two rings it remains a single ring even though it is no longer a model of a Mobius strip.
Cut a model of a Mobius strip down the center but do not let it fall open; keep it in its original position so that it still models a Mobius strip.
The cut down the center of the model is a two dimensional space; it has length and width but no thickness. It is surrounded by only one thing - the model. The cut is surrounded everywhere by the continuous surface of the model. The cut is a two dimensional surface that has only one side; it is an actual Mobius strip.
Actual Mobius strip #4
Imagine a line segment attached at its center to a circle. As it travels around the circle it slowly rotates 180 degrees so that when it returns to its starting point it is upside down. The path traced by the line segment is a Mobius strip.
[Picture]
Models of this kind of Mobius strip are often made by carving them from wood or stone or by casting them in metal.
Actual Mobius strip #5
Imagine a line segment attached at its center to a circle. As it travels around the circle it slowly rotates 90 degrees so that when it returns to its starting point it is at right angles to how it started. When it travels around the circle a second time it is now upside down and the path it traces is a Mobius strip that passes through itself.
This kind of Mobius strip can be modeled by placing a continuous series of holes in the surface of the model so that on the second time around the surface interlinks with itself.
[Picture]
A. by Wayne Kollinger
There is a lot more to be learned from the traditional experiment of cutting a paper model of a Mobius strip lengthwise at one-third its width than you might first think.
The one-third is an arbitrary number; in point of fact it could be anything less than one-half. What you are really doing is making the model smaller by cutting off its edge. Everything on one side of the cut is part of the model. Everything on the other side of the cut is associated with the edge.
Before you cut, draw a line where you are going to cut. You will notice something interesting. The line seems to be on only one side of the model. If you travel across the surface from edge to edge, only one edge has a line. If you travel from the line straight through the paper, where you come out there is no line.
This is because a paper model of a Mobius strip is not actually a Mobius strip. A Mobius trip is a two dimensional surface; it has length and width but no thickness. The paper model is a three dimensional object; it has length and width and thickness. A Mobius strip has one side and one edge. The paper model has two sides and two edges.
The word edge, like many words, has different meanings in different contexts. The edge of a Mobius strip is a one dimensional line. The edge of a model of a Mobius strip is a two dimensional surface; it is actually a second side that gets called an edge. Look at the iconic image by Escher of ants crawling on a model of a Mobius strip; you will see that the edge is indeed a second surface that even smaller ants could crawl along.
The two surfaces of the model meet at two edges (one dimensional lines). The line you drew marks one of the edges (one dimensional line). You could draw a second line to mark the other edge.
The good news is that there is a Mobius strip buried inside the model. Consider a sheet of paper. It has a top surface and a bottom surface. The half-way point between the two is marked by a two dimensional boundary. When you make a paper model of a Mobius strip this boundary takes the shape of a Mobius strip. It is an actual two dimensional Mobius strip. This means that what happens when you cut the model accurately reflects what happens to the Mobius strip.
After you’ve drawn a line on the model marking where you are going to cut, color in the area between the line and the edge (two dimensional surface). Draw a matching line on the other “side” and color between it and the edge as well. When you cut, you will be cutting between the colored edge and the uncolored Mobius strip model. It should be clear from this that when you cut you will get two separate objects.
The cut leaves the Mobius strip model with a single edge (two dimensional surface). However the edge strip has its original edge (two dimensional surface) plus the edge resulting from the cut. Because the edge strip has two edges and two sides it is not a model of Mobius strip and does not contain a Mobius strip.
Once you’ve made the complete cut you will no doubt notice that the edge strip and the Mobius strip model are linked loops. How did this happen?
The center line of a Mobius strip is equivalent to a circle. You can think of the center of the Mobius strip model as being a circle as well.
Trace the edge of a Mobius strip model starting on the outside. At some point it will cross over the center and run along the inside. When it returns to that point it will cross under the center and travel along the outside again until it meets itself.
You can mimic this with bits of string. Join a bit of string to itself to form a circle. This represents the center of a Mobius strip model. A second bit of string that is at least twice as long as the first will represent the edge of a Mobius strip model. Run the second string along the outside of the first and touching it. About half way around cross over to the inside. Travel all around the inside still touching and then cross back under to the outside. Continue along the outside until the string meets itself and can be joined. Separate the two string loops and you will see they are linked.
You can simplify this. Make a circle with a bit of string. Take a second bit if string; pass it over and then under the the first and then join its ends to make a second loop. It is obvious that in doing this you have linked two loops.
The edge of a Mobius strip model is linked around the center in just this way. But because the two are joined along an edge, it only becomes apparent when you separate the two by cutting along that edge.
A. by Alon Amit, Tom McFarlane, Wayne Kollinger
A. by Alon Amit, PhD in Mathematics (2016)
No. A torus is a closed surface, while a Möbius strip is not.
There are various models of a Möbius strip, but they either have a boundary or they are non-compact. Think of a standard model of a Möbius strip you make from paper: it has an edge. A torus has none. Therefore, they cannot be topologically the same.
A. by Tom McFarlane, M.S. in Mathematics, University of Washington, 1994
The torus is orientable, while the Möbius strip is not. Since orientability is a topological invariant, they can not be topologically equivalent.
In addition, the Möbius strip has a boundary, while the torus does not. Since the number of boundaries is a topological invariant, they can not be topologically equivalent.
A. Wayne Kollinger (2017)
Yes and no. But mostly no.
It depends on whether you’re thinking of a true Mobius strip or a paper model of a Mobius strip which almost everyone mistakes for a Mobius strip.
A true Mobius strip is a two dimensional surface. It has length and width but no thickness. It is a loop with a half twist which results in its having only one side. Side in this context means adjacent area. There is only a single area adjacent to the Mobius strip.
A torus is also a two dimensional surface. It has length and width but no thickness. It is hollow hoop. As a result it has two sides (adjacent areas) and inside and an outside.
Because a true Mobius strip has one adjacent area and a torus has two they are not topologically equivalent.
A paper Mobius strip (or any model of a Mobius strip) is a three dimensional object. It has length and width and thickness. The paper model has a rectangular cross section. If you consider only the surface of a paper Mobius strip, it is a hollow hoop, albeit one with a rectangular cross section. Topologically the edges resulting from the corners of the rectangle are insignificant and the model of the Mobius strip is equivalent to a torus.
This means that anything you can do to or with a paper model of a Mobius strip you can do the equivalent with a model of a torus.
A. by Andy Baker, Roger Pickering
A. Andy Baker. Studied at Glasgow University. (2023)
No, but there is a homotopy equivalence between them. This means that all the standard invariants of Algebraic Topology give isomorphic answers when applied to them
A. by Roger Pickering. Studied Topology at University of Warwick, graduated 1970 (2023)
No. A circle is a one dimensional object that can be embedded (drawn) in a 2-dimensional space, the plane.
A Möbius strip is a two dimensional object that cannot be embedded in a plane (unlike a disk, for example).
Created 24 July 2024 (from existing text of 2 May 2024).