About Dimension
The Chaos Hypertextbook™
© 1995-2003 by Glenn Elert
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3.2 Topological Dimension
By definition, the null set (
) and only the null set shall have the dimension -1. The dimension
on any other space will be defined as one greater that the dimension of the object
that could be used to completely separate any part of the first space from the
rest. It takes nothing to separate one part of a countable set from the rest of
the set. Since nothing () has dimension -1, any countable
set has a dimension of 0 (-1 + 1 = 0). Likewise, a line has
dimension 1 since it can be separated by a point (0 + 1 = 1),
a plane has dimension 2 since it can be separated by a line (1 + 1 = 2),
and a volume has dimension 3 since it can be separated by a plane (2 + 1 = 3).
We have to modify this dimension a little bit, however.
Sure a countable set can be separated by nothing, but it can also be separated
by another countable set or a line or a plane. Take the rational numbers, for
example. They form a countable infinite set. By embedding the set in the real
number line, we could separate one point from any other with an irrational number.
This set is has dimension 0, which would give the rational numbers a dimension
of 1 (0 + 1 = 1). By embedding the set in the coordinate plane,
we could also use any line with an x-intercept. This would give the rational numbers
a dimension of 2 (1 + 1 = 2). We could also use planes if
we embedded the set in a euclidean three-space and so on. I think it would be
all right if we used the minimum value and called it the dimension of the space.
What about our composite spaces (
) and (
)? We want the first to have dimension 1 and
the second dimension 2. The x-shaped space is no problem. The least dimensional
entity needed to separate it would be a point even at the intersection. The point
and filled square is a bit more challenging. We need to distinguish between local
dimension and global dimension. If we use
the last definition and apply it to the set as a whole, then the space () would have dimension 0. If on the other hand, we examine it region
by region we find that the point part has dimension 0 while any part of the square
region has dimension 2. This is an example of a local dimension. The global dimension
of the whole space should be two-dimensional so we need to modify our definition
slightly. The dimension of a space should be the maximum of its local dimensions
where the local dimension is defined as one more than the dimension of the lowest
dimensional object with the capacity to separate any neighborhood of the space
into two parts.
The measure defined above is called the topological dimension
of a space. A topological property of an entity is
one that remains invariant under continuous, one-to-one transformations or homeomorphisms.
A homeomorphism can best be envisioned as the smooth deformation of one space
into another without tearing, puncturing, or welding it. Throughout such processes,
the topological dimension does not change. A sphere is topologically equivalent
to a cube since one can be deformed into the other in such a manner. Similarly,
a line segment can be pinched and stretched repeatedly until it has lost all its
straightness, but it will still have a topological dimension of 1. Take the example
below.
- Start with a line segment. Divide it into thirds. Place the vertex of an
equilateral triangle in the middle third.
- Copy the whole curve and reduce it to 1/3 its original size. Place these
reduced curves in place of the sides of the previous curve.
- Return to step 2 and repeat.
The result the is the Koch coastline, which evolves
something like this.
The Koch Coastline
With each iteration the curve length increases by the factor 4/3. The infinite
repeat of this procedure sends the length off to infinity. The area under the
curve, on the other hand, is given by the series
1+ (4/9) + (4/9)2 + (4/9)3 + ...
which converges to 9/5 (assuming the area under the first curve is 1). These
results are unusual but not disturbing. Such is not the case for the next curve.
- Take a unit square. Call it a cell.
- Divide each cell into four identical miniature copies of the original cell
- Draw a line starting in one cell so that it passes through every other cell
until it returns to the starting position. (Also make sure the line does not
stray too far from the previous iteration of the curve.)
- Return to step 2 and repeat.
The result is something like the diagrams below. (Cell lines were omitted in
the third iteration for clarity. The last diagram represents the hypothetical
result of an infinite iteration.)
Peano Monster Curve (A Variation on Hilbert's Version)
This curve twists so much that it has infinite length. More remarkable is that
it will ultimately visit every point in the unit square. Thus, there exists a
continuous, one-to-one mapping from the points in the unit interval to the points
in the unit plane. In other words, an object with topological dimension one can
be transformed into an object with topological dimension two through a procedure
that should not allow for such an occurrence. Simple bending and stretching should
leave the topological dimension unchanged, however. This is a Peano monster
curve (actually, a variation on Hilbert's version of Peano's original),
so called because of its monstrous or pathological nature. Since there are no
such things as monsters, we have nothing to fear. The Koch and Peano curves raise
questions about the meaning of dimension that will be answered in the next section.
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