A point has dimension 0, a line has dimension 1,
and a plane has dimension 2. But did you know that
some objects can be regarded to have "fractional"
dimension?
You can think of dimension of an object X
as the amount of information necessary to
specify the position of a point in X. For instance,
a block of wood is 3dimensional because you need
three coordinates to specify any point inside.
The standard Cantor set has fractional
dimension! Why? Well it is at most 1dimensional,
because one coordinate would certainly specify where a
point is. However, you can get away with "less", because
the object is selfsimilar. At each stage, you only
need to specify which 2 out of 3 segments a point is in.
Mathematicians have developed a notion of "dimension"
which for the standard Cantor set works out to be:
ln(2)/ln(3) = 0.6309...
Most other "fractals" have fractional dimension;
for instance a curve whose boundary is very, very
intricate can be expected to have dimension between
1 and 2 but closer to 2. This concept has been applied
in other sciences to describe structures that appear to
have some selfsimilarity, such as the coast of England
or gaseous nebulae in interstellar space.
Presentation Suggestions:
It makes sense to do this Fun Fact
after doing the one on
the standard Cantor set.
The Math Behind the Fact:
Actually, the notion of "dimension" can be extended to
crazy sets in many different ways. One notion is
box dimension, and
another is Hausdorff dimension.
These notions agree for the standard Cantor set
and many other sets.
How to Cite this Page:
Su, Francis E., et al. "Fractional Dimensions."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
