3 ms·

Is it common to refer to the Möbius strip as a 2D surface? You can't uniquely identify each point on its surface using only two dimensions, so I wouldn't call

Is it common to refer to the Möbius strip as a 2D surface? You can't uniquely identify each point on its surface using only two dimensions, so I wouldn't call it a 2D surface at all. It lives in 3D, therefore it is a 3D surface, no?

pndmnm 15y ago
Any small portion of the Möbius strip "looks like" a chunk of 2D object, but the entire object cannot be placed directly into 2-space. The surface of a sphere is a similar example.
4ad 15y ago
Yes you can refer to each point by only two dimensions if you use a particular non-euclidian metric.
bmm6o 15y ago
It's commonly called a non-orientable 2 dimensional manifold. Locally, it looks like a smooth deformation of the plane. The non-orientable bit is what distinguishes it (globally) from something more like a cylinder.
dfranke 15y ago
You can't uniquely identify each point on its surface using only two dimensions Yes you can. Run a sharpie along the bottom edge of the paper so that it bleeds onto both sides of the paper. Now tape the ends of the strip together with a half twist to make a Möbius strip. Place a point anywhere on the strip. Draw a line through that point such that the line meets both edges of the strip at a right angle. Measure the distance along that line between your point and the darkened edge; call this x. Now, hold the Möbius strip in your left hand so that you're pinching it by the tape. With your right hand, run your finger along the strip, starting from the tape and moving to the right. Measure how far you have to move your finger in order to reach the line you drew; this could require up to two loops around the strip. Call this distance y. The tuple (x,y) uniquely identifies your point.
- grannyg00se 15y ago
  Wow. It took me a while to figure out what you were saying, but it was well said and I eventually got it. Thanks.
  - kd0amg 15y ago
    For the record, we use a similar trick on sphere-like surfaces, as latitude and longitude.
- dfranke 15y ago
  Oh dear, I just realized I dropped a sentence while I was composing this comment and now it's too late to edit. The beginning of it should read, "Yes you can. Cut out a long rectangular strip of paper. Run a sharpie..."
- bmm6o 15y ago
  Alternately, just realize that the rectangular strip of paper has a natural set of coordinates, and nothing is lost by twisting and taping. You get a discontinuity at the tape, but that's not a problem.
wisty 15y ago
If you disagree with the terminology, that's your opinion. A lot of terminology is kind of weird. "Klien Bottle" is called that because it was called "Klien Surface" in German, but "Surface" in German kind of sounds like "Flask", and they mistranslated. The name stuck because it was kind of appropriate. I say it's a 2D surface, because while it lives in 3D, it's "thin" in one dimension. It's like saying a piece of paper is a 2D object. And when I say "it can't live in 2D", I'm kind of lying - it can live in a weird kinky 2D space, but NOT a boring 2D plane. Well, you could smash it into a 2D plane, but it would self-intersect (get all smushed together, losing its shape). And by "shape", I mean topological properties ... but I hate jargon. But a Mobius strip is a 2D surface which can't live in 2D (a boring flat 2D plane); in the same way a Klien Flask is a 2D surface which can't live in 3D; and a knot is a 1D surface which can't live in 1 or 2D. Press a knot into 2D, and it crosses itself (self-intersect). Force a knot into 2D, and it will break, or join up with itself (depending on the material). Real mathematicians (i.e. not me) will say "manifold", not surface, but it's more or less the same thing.

pndmnm 15y ago

Any small portion of the Möbius strip "looks like" a chunk of 2D object, but the entire object cannot be placed directly into 2-space. The surface of a sphere is a similar example.

4ad 15y ago

Yes you can refer to each point by only two dimensions if you use a particular non-euclidian metric.

bmm6o 15y ago

It's commonly called a non-orientable 2 dimensional manifold. Locally, it looks like a smooth deformation of the plane. The non-orientable bit is what distinguishes it (globally) from something more like a cylinder.

dfranke 15y ago

You can't uniquely identify each point on its surface using only two dimensions Yes you can. Run a sharpie along the bottom edge of the paper so that it bleeds onto both sides of the paper. Now tape the ends of the strip together with a half twist to make a Möbius strip. Place a point anywhere on the strip. Draw a line through that point such that the line meets both edges of the strip at a right angle. Measure the distance along that line between your point and the darkened edge; call this x. Now, hold the Möbius strip in your left hand so that you're pinching it by the tape. With your right hand, run your finger along the strip, starting from the tape and moving to the right. Measure how far you have to move your finger in order to reach the line you drew; this could require up to two loops around the strip. Call this distance y. The tuple (x,y) uniquely identifies your point.

grannyg00se 15y ago
Wow. It took me a while to figure out what you were saying, but it was well said and I eventually got it. Thanks.
- kd0amg 15y ago
  For the record, we use a similar trick on sphere-like surfaces, as latitude and longitude.
dfranke 15y ago
Oh dear, I just realized I dropped a sentence while I was composing this comment and now it's too late to edit. The beginning of it should read, "Yes you can. Cut out a long rectangular strip of paper. Run a sharpie..."
bmm6o 15y ago
Alternately, just realize that the rectangular strip of paper has a natural set of coordinates, and nothing is lost by twisting and taping. You get a discontinuity at the tape, but that's not a problem.

wisty 15y ago

If you disagree with the terminology, that's your opinion. A lot of terminology is kind of weird. "Klien Bottle" is called that because it was called "Klien Surface" in German, but "Surface" in German kind of sounds like "Flask", and they mistranslated. The name stuck because it was kind of appropriate. I say it's a 2D surface, because while it lives in 3D, it's "thin" in one dimension. It's like saying a piece of paper is a 2D object. And when I say "it can't live in 2D", I'm kind of lying - it can live in a weird kinky 2D space, but NOT a boring 2D plane. Well, you could smash it into a 2D plane, but it would self-intersect (get all smushed together, losing its shape). And by "shape", I mean topological properties ... but I hate jargon. But a Mobius strip is a 2D surface which can't live in 2D (a boring flat 2D plane); in the same way a Klien Flask is a 2D surface which can't live in 3D; and a knot is a 1D surface which can't live in 1 or 2D. Press a knot into 2D, and it crosses itself (self-intersect). Force a knot into 2D, and it will break, or join up with itself (depending on the material). Real mathematicians (i.e. not me) will say "manifold", not surface, but it's more or less the same thing.