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Two Interpretations of Dimensionality in Geometric Figures

Date: 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/ 
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Higher-Dimensional Geometry

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