 |
| Von Koch's Snowflake Curve |
In general,
geometric objects are usually classified according to the number of their
dimensions, like this:
- Points have zero dimensions.
- Lines (straight lines or curves) have one dimension.
- Surfaces have two dimensions.
- Volumes have three dimensions.
Furthermore,
mathematicians often work with objects that have more than three dimensions,
which are very difficult for us to imagine.
At the beginning of the 20th century, the Swedish mathematician Helge von Koch discovered a strange curve (the von Koch snowflake), which has the following properties:
- It is continuous in every point, but has no
derivative at any point.
- Although it is a closed curve, its perimeter
is infinite.
- Although it is a line, its dimension is not 1.
In 1919, the
mathematician H. Hausdorff proposed a new concept of dimension, which would be applicable
to these strange curves and would distinguish them from ordinary lines and
surfaces. According to his definition, too complicated to detail here, all typical
lines, open or closed, have dimension equal to one; all typical surfaces have
dimension two. But curves similar to von Koch’s snowflake can have fractional
dimensions, between one and two. For example, von Koch’s snowflake has the
following dimension: (log 4)/(log 3) = 1.2618595071429...
Von Koch’s
curve was not the first strange curve that mathematicians discovered. At the
end of the 19th century, Giuseppe Peano devised another curve, even more
strange, which bears his name, and which is defined as the limit of a set of
successive curves, the sixth of which can be seen in the attached figure. This
curve, like all those preceding and following in the succession, is evidently a
line with a starting point and an ending point, located at two opposite
vertices of a square. A point moving along this line would not have freedom of
movement except in one direction. Therefore, it appears that this is a
one-dimensional object. But it is also evident, if we observe the succession of
the curves, that the line ends up filling the square. We have
here, therefore, a line that completely fills a surface, without ceasing,
apparently, to be a line.
 |
| Sixth approximation to Peano's Curve |
How many
dimensions does Peano’s curve have? On the one
hand, it appears to have one dimension, because it
is the limit of a succession of curves whose dimension is always 1. On the
other hand, it seems to have two dimensions, as the curve in the limit passes
through all the points of a square. And indeed, if we calculate its Hausdorff
dimension, it comes out to be 2. Peano’s curve is also strange for not having a
derivative at any point, but that is another story.
Many years
after Peano and von Koch discovered their curves, the Polish-American
mathematician Benoit Mandelbrot proposed the name fractals for these curves (and for other mathematical
objects).
Just as Peano’s
curve, despite being a curve, fills a surface and therefore has dimension 2,
there are fractal surfaces that fill volumes, and therefore have dimension 3.
And there are fractal volumes that fill all the available space, and therefore can
be said to have dimension 4.
Let's look at
the attached figure, which I've taken from the book Scale: The universal laws of life and death in organisms, cities and
companies, by Geoffrey West, from
the Santa Fe Institute. It represents the metabolic rate (in watts) of several
species of birds and mammals, depending on their body mass. The two axes of
the figure are logarithmic, which means that the straight line where the
metabolic rates of all those animals are approximately located, is actually a
power curve, whose exponent is the slope of the line, i.e. approximately ¾.
West explains the figure thus:
Elephants
are roughly 10,000 times (four orders of magnitude) heavier than rats;
consequently, they have roughly 10,000 times as many cells. The 3⁄4 power
scaling law says that, despite having 10,000 times as many cells to support,
the metabolic rate of an elephant (that is, the amount of energy needed to keep
it alive) is only 1,000 times (three orders of magnitude) larger than a rat’s.
How can this be?
According to West, the circulatory system of birds and mammals is made of blood vessels with a hierarchical structure, with the aorta and vena cava at
the top, plus various vessels of decreasing diameter, ending in capillaries of
microscopic diameter. This defines a fractal structure in three dimensions. Every
cell in the body receives food from a nearby capillary, so the blood vessels must
fill all the available space (the entire volume of the body), just like Peano’s
curve fills the surface of a square. Therefore, the Hausdorf dimension of our
circulatory system is equal to 4. This would explain the exponent ¾ in the
growth of metabolism as a function of body mass in birds and mammals.
More about this, next week.
No comments:
Post a Comment