Two Interpretations of Dimensionality in Geometric FiguresDate: 03/16/2004 at 11:52:04 From: Marianne Subject: Teacher needs examples of 1 dimensional and 2 dimensional I am a high school math teacher who could possibly be teaching geometry in the fall. I would like to teach the students about dimension. I understand that the x-axis alone is 1 dimensional. Throw in the y-axis and there are 2 dimensions. Add a z-axis and there are 3 dimensions. Overall I have a good understanding of dimensions. A line is 1 dimensional, a square or rectangle is 2 dimensional, and a cube is 3 dimensional. My question is what if you throw in parabolas or circles or the absolute value function, etc.? A parabola can only be drawn in 2-space, but lines are also drawn in 2-space unless the slope is 0 or does not exist. A circle is kind of like a parabola, but it is very much like a square, so I am thinking it is 2-dimensional. My conclusion is that the only 1 dimensional object is a straight line, and a point is 0 dimensional, but I am not confident that I am correct. Can you please clear this up for me? Date: 03/16/2004 at 12:57:26 From: Doctor Peterson Subject: Re: Teacher needs examples of 1 dimensional and 2 dimensional Hi, Marianne. I think you have a very good understanding of this, and you are correct that a point is 0 dimensional. But there's lots more to dimensionality as well. The fact is that dimensionality can be taken in two ways: what we may call intrinsic dimensionality, which is essentially the number of dimensions it takes to identify the location of a point WITHIN the object, and extrinsic dimensionality, which is the number of dimensions of the (minimal)space it is "embedded" in. In the latter sense, a parabola is two-dimensional, since it can't be embedded in a Euclidean space of less than two dimensions; that is, it doesn't fit in a line. In the former sense, it is one-dimensional, since one number is enough to identify where you are along the parabola. Note that the parabola can be thought of as a bent line; bending affects the extrinsic dimensionality, as I am calling it, but not the intrinsic dimensionality. The latter is therefore a topological property, one that doesn't depend on notions of straightness and length; it is what is described here: http://mathworld.wolfram.com/Dimension.html To see how lower and higher dimensions relate to each other, take any geometric object (like a point, line, circle, plane, etc.), and "drag" it in an opposing direction (drag a point to trace out a line, a line to trace out a box, a circle to trace out a cylinder, a disk to a solid cylinder, etc.). The result is an object which is qualitatively "larger" than the previous object, "qualitative" in the sense that, regardless of how you drag the original object, you always trace out an object of the same "qualitative size." The point could be made into a straight line, a circle, a helix, or some other curve, but all of these objects are qualitatively of the same dimension. The notion of dimension was invented for the purpose of measuring this "qualitative" topological property. Here is a page on which I discussed some of these ideas, and also included a link to another page which includes another link. All of those are worth reading: Mobius Strips, Spheres, and Dimensionality http://mathforum.org/library/drmath/view/62980.html What I have called extrinsic dimension is a geometrical property, in which lengths and angles count. We call a triangle, for example, two- dimensional, because although the line segments that form it are one- dimensional, in geometry we are interested in the properties it has IN THE PLANE; we are interested not just in the object itself, but how it embeds. So it is an object we study as part of a plane, in two-dimensions. This page discusses the geometrical concept of a "two- or three- dimensional figure": Defining Geometric Figures http://mathforum.org/dr.math/faq/formulas/faq.figuredef.html A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are one-dimensional, but they lie in the plane, which is two-dimensional. Note that here the whole figure made up of segments and curves is considered two-dimensional because it lies in a plane, though the edges, even when curved, are acknowledged as one-dimensional. The trouble is that, although mathematicians tend to think about the topological idea of dimensions, anything before college tends to use only the geometrical perspective. From that point of view, your parabola is a two-dimensional figure, since you are drawing it on paper, and it doesn't lie along a line. A helix, really a one- dimensional object, would have to be called a 3-dimensional figure in this sense, since you can't draw it on paper except as an image of something in three-dimensional space. An additional complication arises when we talk about a circle, which we have defined as just the curve and not its interior, but then ask about its area, which is really the area of the disk formed by the circle and its interior. The names of most two-dimensional figures that are really one-dimensional objects are often used sloppily to refer to the corresponding two-dimensional object, like the disk, just because we are focusing not on the object itself at this level, but on the whole figure; and similarly for three-dimensional figures like a cylinder. Now, I made up the terms "intrinsic and extrinsic dimension"; they aren't standard terms, perhaps because we don't commonly bother thinking about both ideas at the same time! But here is one place where I found the same idea discussed: What is Topology? http://www.pepperdine.edu/seaver/natsci/FACULTY/KIGA/topology.htm These objects are examples of curves in the plane. In some sense they are two dimensional since we draw them on a plane. In another sense, however, they are one dimensional since a creature living inside them would be only aware of one direction of motion. We might say that such shapes have extrinsic dimension 2 but intrinsic dimension 1. So what's my recommendation? If you have a text, follow its terminology; perhaps make some mention of this issue, which may stimulate the better thinkers in your class; but try not to let it confuse everyone. That may be hard! 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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