Mobius Strips, Spheres, and Dimensionality
Date: 05/28/2003 at 10:16:58 From: Ruth aged 5 Subject: Mobius Strips Is a Mobius Strip 2-D or 3-D? Or is it 1-D? I find it confusing because we start off with a 2-D strip of paper with a clear height/width. When we twist and join it, we still have a clear width, but we are unsure about length, though you could take this as one circuit of the loop. Depth is very confusing, though, because at one point you have the twist, which seems to have no depth at all. And the rest of it does make a shape which seems 3-D, but the 'depth' of the object is the same bit we measured for the width. Please can you help me? Thanks.
Date: 05/28/2003 at 22:57:41 From: Doctor Peterson Subject: Re: Mobius Strips Hi, Ruth. Actually, the same questions can be raised about a sphere (meaning the surface, not the solid object we call a ball). It is only a surface, making it 2-dimensional (that is, any point on the surface can be located by only 2 dimensions, such as latitude and longitude). But it is embedded in 3-dimensional space, so that we can also assign each point on the sphere to its (x,y,z) coordinates. The sphere has no thickness (ideally, that is - just as your paper really has some thickness, but we ignore it for the sake of the mathematical concept), so that any small part of it looks like a piece of a plane. Its two- dimensionality depends on that "local" property rather than on the "global" idea of size. Note that the concept of dimensions doesn't really have anything to do with whether the object as a whole has a length, width, and thickness or depth, or whether there is a boundary when you go around it. The sphere has no edges at all, so that within the sphere itself you could only measure how far you go before you loop back on yourself. In the three-dimensional space in which it resides, you can describe it by only one dimension, its radius. But dimensionality relates to how you can specify a location in it, not to the overall size of the object. Here is another discussion of related ideas: Dimensions http://mathforum.org/library/drmath/view/55345.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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