 |
Virtual
Ecosystems -- Nature's Geometry
|
|
"Philosophy
is written in this grand book - I mean universe - which
stands continuously open to our gaze, but which cannot
be understood unless one first learns to comprehend the
language in which it is written. It is written in the
language of mathematics, and its characters are
triangles, circles and other geometric figures, without
which it is humanly impossible to understand a single
word of it; without these, one is wandering about in a
dark labyrinth."
- Galileo (1623) |
A Fractal Terrain |
|
The use of virtual ecosystems as
a tool for understanding real ecosystems is really a
study of dynamic geometric patterns. We've seen that
computer models like cellular
automata can be used to generate and recognize
characteristic system patterns that we've called attractors
. These attractors can give us some clues about the
ecosystem under study. But, this kind of pattern
recognition is a qualitative process.
It would be nice if we could glean more information by
quantifying pattern characteristics. Since patterns are
geometric objects, quantification requires some sort of
geometric study to add further information. Geometry is
a mathematical language that describes, relates, and
manipulates shapes and patterns. Geometry is concerned
with making our spatial intuitions objective.
The classic geometry has been the Euclidean geometry.
Euclidean geometry portrays the world using points,
lines, circles, squares, cubes, and other primitive
objects. Euclidean geometry provides a first
approximation to the structure of physical and natural
objects. Euclidean geometry concisely describes man made
objects (circles, squares, cones, ellipses). But, it
yields cumbersome and inaccurate descriptions of natural
shapes and processes.
Euclidean geometry is the source of our common worldview
of dimension. We describe our world in terms of zero,
one, two, or three dimensions. A point is zero
dimensions, a line is one-dimensional, a flat plane is
two-dimensions, and the real world that we live in is
considered three-dimensional. This concept of dimension
represents only a rough approximation of our natural
world and an expanded definition of dimension is needed
to more precisely describe that world.
Euclidean geometry is also an insufficient descriptor of
our world because it has little to say about the
relationships between real world objects and how those
relationships are formed. The self-organization
and emergent
behavior pages emphasize that system patterns
(attractors) are the result of relationships at a local
level. The cellular automata pages show some of the
rule-based patterns. Euclidean geometry cannot help in
describing these local interactions or the resulting
patterns.
In the late 19th century, it became clear to
mathematicians that other geometries were needed to more
precisely visualize the natural world and analyze
natural patterns. While new geometries were developed,
it wasn't until 1975 that Benoit Mandelbrot introduced
fractal geometry. Assisted by major advances in computer
graphics technology, fractal geometry has become
recognized as the language of nature's irregular shapes.
In addition, fractal research has helped reconnect pure
mathematics research with the natural sciences and
computing. And, fractal geometry has become an important
conceptual tool in most of the natural sciences
including physics, chemistry, biology, geology,
meteorology, physiology, and materials science.
This web page provides an overview of fractal geometry.
Subsequent pages detail the important concepts of fractal
dimension and self
similarity . The major differences between Euclidean
geometry and fractal geometry are:
| EUCLIDEAN
GEOMETRY |
FRACTAL
GEOMETRY |
| Traditional
- over 2000 years old |
Modern - only 25+ years
old |
| Based
on characteristic size or scale. Objects are
described in terms of radius or line length. |
No specific size or
scaling. Form looks the same at all levels of
magnification (self-similarity). |
| Describes
man made objects such as circles or cubes. |
Appropriate for natural
shapes (e.g. trees) and system patterns (strange
attractors are fractal) |
| Described
by formula. For example, the formula for a
circle ( r2=x2+y2
) completely describes the circle. |
Described by recursive
algorithm or rule. A fractal image can only be
drawn using an iterative simulation. |
| Integer
dimension (0,1,2,3) only |
Dimension is a real
number (e.g. 1.56321) that describes the
complexity of the pattern. |
There are two major areas where fractal geometry is
handy for studying ecosystems.
First, by knowing the fractal dimension of a pattern,
the complexity of patterns in ecosystems can be
determined. For example, the fractal dimension of a
swimming shark's track can indicate whether or not the
shark is hunting. The fractal dimension of a coral
colony will indicate whether certain critical levels have
been reached.
A second use for fractals is the simulated construction
of natural objects such as mountains, trees, and coral
reefs. While the movie industry has used fractal
technology extensively, the scientist can discover
things about natural growth processes by modeling
"fractal forgeries" of nature.
To explore this subject further visit the fractal
construction , dimension
, fractal
dimension , and self-similarity
pages. In addition, real fractal applications are
portrayed in the applications section. See the pullout
menu at the left for a list of applications. |
|
