What's the area of a Möbius strip?

Syth

From http://vitanuova.loyalty.org/2002-07-03.html.

Möbius strip area

Jonathan Walther wrote to ask me an interesting question about the area of a Möbius strip.

I've been having a debate with a friend about how to calculate the area of a moebius strip, where the moebius strip is constructed by taking a 1" by 10" area, twisting it, and joining the ends.

I have been maintaining that the area remains the same; that is, 10 square inches. My friend insists it is 20 square inches.

After discussion with my friend it became apparent that our different calculations came from our having different concepts of "area". He used a strip of paper to "illustrate" the moebius strip, and I feel this gave him erroneous intuition in this case.

My observation was that if you did make the strip from paper, you would need 20 sq. in. of paint in order to paint the whole thing. If you used 10 sq. in. of paint, you would have 10 sq. in. of surface unpainted.

However, Jonathan argues that this is a misinterpretation if the Möbius strip is seen as having zero thickness, because then points on one "side" are actually identical with the corresponding points on the "other side". He suggests that, on a zero-thickness strip, you can go only 10" before you return to your starting point. (On a strip made of paper with non-zero thickness, you must go 20" before returning to your starting point.)

Does anybody have a view to clarify this? It does just seem like a question of how to define surface area, but maybe there is a particular definition of "surface area" or "Möbius strip" which is somehow preferable.

(Normally you deal with areas in a plane, and the definition is easier. Is there something handy from multivariate calculus here?)

What does anyone else think the area of a Möbius strip is? I'd think it's 10, since ideally you'd dealing with a zero dimensional strip.

ecksor

If you're dealing with a real piece of paper then I'd be inclined to follow the argument that if you joined the ends without twisting, then it would be 10 on the inside and 10 on the outside so it is 20 overall, and twisting won't make any difference to that.

If you're dealing with a zero dimensional strip (which I assume you mean to not have 'sides' as such) then it doesn't sound like something that has an area at all.

Syth

If you're dealing with a zero dimensional strip (which I assume you mean to not have 'sides' as such) then it doesn't sound like something that has an area at all.

Area is only 2 dimensional, so it's quite possible that a totally flat surface could have an area.

ecksor

Well then, tear it, unfold it and get your 10x1 rectangle or equivilent and argue over whether covering one side also covers the other side.

Rredwell

First of all, you have to clarify what u mean by "area". It's very easy to confuse the "paint" area and the "paper" area; our intuition tells us the two are the same, but are they?

2-d shapes have an area, but is a Mobius strip even 2-dimensional? (Maybe it's 2.5-d!)

I'm really stumped by this, but like all problems in mathematics, the answer stems from a very simple premise.

Capt'n Midnight

How about trying to tile it with square tiles ?
the rule being to not put a tile where there is one already.

it's like trying to argure as to whether a circle is infinte length or if it's 355/113 times the diameter. - so if you view it as a circle then it has Infinite area.

if you insist that it does not have infinite lenght then it must have finite thickness. - if you measure the circumfrence of a circle you go around the long way you don't say it's zero because the starting and end points are next to each other - similarly even if you say the thickenss of the strip tends to zero you have to do two loops to get back where you started - otherwise you'd have a torus.

Syth

Capt'n Midnight wrote:

it's like trying to argure as to whether a circle is infinte length

How can a circle have infinite length?

Capt'n Midnight

Since area x height gives volume there is another way of working out the area of a mobius strip, cut a klein bottle down the middle and work it out
http://www.kleinbottle.com/sliced_klein_bottles.htm

Certified Y2K compliant and made entirely from Baryonic materials.

SkepticOne

It boils down to whether you measure both sides of the original strip that makes up the mobius strip before it is joined or whether you just measure one side.

In a normal surface that isn't joined with a twist like the mobius strip, for example a rectangle drawn on the flat plane, you wouldn't measure both 'sides'. Your assessment of area would simply be length times breadth. You only measure one 'side' of the surface and would consider it odd if someone insisted it was 2 x length x breadth.

The situation becomes slightly confusing with the mobius strip because the thing only has one side even if the strip making it up had two sides before being joined up with the twist. Do you measure the single side of the mobius strip or the one side of the strip making it up?

Since a basic property of the mobius strip is that it is non-orientable, this would suggest that only one side of the original rectangle should be measured. If you place a 'p' and a 'q' (it's mirror image) next to each other on the mobius strip, non-orientable means that you should be able superimpose the 'p' onto the 'q' by moving one of the letters along the strip until it is back where it started from. If the original strip making up the mobius strip had two sides (e.g a bit of paper) then this could not be done as the 'p' would be on the other side to the 'q' and would not be superimposed. Therefore the original strip only has one side and so the answer is 10 square inches.

mrhappy42

A cylinder is the same, its not a true surface but a surface with a boundary Henle 1994, p. 129.

The area formulae is given here (item 10) http://mathworld.wolfram.com/MoebiusStrip.html