Compiled with assistance from Brian Klinkenberg, University of
British Columbia
NOTES
UNIT 47 - FRACTALS
Compiled with assistance from Brian Klinkenberg, University of
British Columbia
A. INTRODUCTION
Why learn about fractals?
- fractals are not so much a rigorous set of models as a
set of concepts
- these concepts express ideas which have been around in
cartography for a long time
- they provide a framework for understanding the way
cartographic objects change with generalization, or
changes in scale
- they allow questions of scale and resolution to be dealt
with in a systematic way
Length of a cartographic line
- if a line is measured at two different scales, the second
larger than the first, its length should increase by the
ratio of the two scales
- areas should change by the square of the ratio
- volumes should change by the cube of the ratio
- yet because of cartographic generalization, the length of
a geographical line will in almost all cases increase by
more than the ratio of the two scales
- new detail will be apparent at the larger scale
- "the closer you look, the more you see" is true of
almost all geographical data
- in effect the line will behave as if it had the
properties of something between a line and an area
- a fractal is defined, nontechnically, as a geometric set
- whether of points, lines, areas or volumes - whose
measure behaves in this anomalous manner
- this concept of the scale-dependent nature of
cartographic data will be discussed in more detail
later
Where did the ideas originate?
- term was introduced by Benoit Mandelbrot to the general
public in his 1977 text Fractals: Form, Chance and
Dimension
- a second edition in 1982 is titled The Fractal
Geometry of Nature
- some of Mandelbrot's earliest ideas on fractals came
from his work on the lengths of geographic lines in
the mid 1960s
- fractals may well represent one of the most profound
changes in the way scientists look at natural phenomena
- fractal-based papers represent over 50% of the
submissions for some physics journals
- many of the studies of the fractal geometry of
nature are still at the early stages (especially
those in geomorphology and cartography)
- the results presented in some fields are very
exciting (e.g., see Lovejoy's (1982) early work on
the fractal dimensions of rain and cloud areas)
B. SOME INTRODUCTORY CONCEPTS
Euclidean geometry
C. SCALE DEPENDENCE
Determining fractal dimension
Some questions
1. what is the "true" length of a line?
2. how can you compare curves whose lengths are
indeterminate?
3. of what value are indices based on length
measurements?
- the perimeter of an area object increases steadily
with scale, but the area of an area object deviates
up and down by much smaller amounts
- are analyses based on area less scale-dependent than
ones based on perimeter?
- what does this indicate about measures of shape
based on the ratio of perimeter to the square root
of area?
- there is no complete solution to these (and similar types
of) problems
- however, use of fractal geometry (especially the
fractal dimension) does allow us to make reasonably
meaningful comparisons and indices (as illustrated
in Woronow, 1981)
- these questions are of special interest to cartographers
interested in digital representations of cartographic
features (e.g. Buttenfield, 1985)
- there are implications with respect to:
1. digitizing
- determination of the appropriate sampling
interval
2. generalizing lines
- the best method for generalizing lines may be
that method which best retains the fractal
dimension of the line
3. displaying lines at a scale greater than that
at which the line was collected
- introduce additional "information", by adding
artificial detail to the line, detail which is
a function of the fractal dimension of the
original line);
4. incorporating the fractal dimension into
traditional cartometry measures
- see Woronow (1981)
D. SELF-SIMILARITY AND SCALING
Self-similarity
Scaling
- not necessarily equivalent to self-similarity, although
the two terms are often used interchangably in the
literature
- consider a landscape, as represented by a surface and a
contour map
- on the contour map (coordinates in 2 dimensions
only) the axes can be switched without fundamentally
changing the characteristics of the landscape, i.e.
the characteristics of the contour lines
- contour lines are therefore examples of simple
scaling fractals
- in the case of the surface, with coordinates in 3
dimensions, we cannot interchange the z axes with
either of the x or y axes without fundamentally
altering the characteristics of the landscape
- since the z axis has a different scaling
parameter than the x or y axes, a three-
dimensional representation of the Earth's
surface is therefor an example of a non-uniform
(or multiple) scaling representation
- shapes that are statistically invariant under
transformations that scale different coordinates by
different amounts are known as self-affine shapes
(Peitgen and Saupe, 1988)
- the Earth's surface is an example of a self-affine
fractal, but it is not an example of a self-similar
fractal
- contour lines, which represent horizontal cross-
sections of the land surface, are examples of
statistically self-similar scaling phenomenon
(because the contour has a constant z value)
- because the land surface is self-affine and not self-
similar, those techniques which determine the fractal
dimension of the land surface itself produce values which
are different than the values produced by those
techniques which determine the fractal dimension of the
contours derived from that land surface
E. ERROR IN LENGTH AND AREA MEASUREMENTS
- scale, through its relationships with generalization and
resolution, significantly influences length and area
measurements
- problems in estimating line lengths, areas, and point
characteristics can be related to the phenomenon's
fractal dimension (Goodchild, 1980)
- estimates of area are frequently based on pixel counts,
especially in raster-based systems
- the error in the area estimate is a function of the
number of pixels cut by the boundary of the object
- boundaries with a fractal dimension greater than one
will appear more complex as the pixel size decreases
(as the resolution increases)
- the more contorted the boundary, or the higher its
dimension, the less rapid the increase in error with
cell size
diagram
- error in a pixel-based area estimate will also be a
function of how the phenomenon is distributed about
the landscape: the error in area associated with a
highly compact phenomenon will be much less than the
error in area associated with a widely dispersed,
patchy phenomenon
- Goodchild and Mark (1987, p. 268) show that:
- the standard error as a percentage of the area
estimate is proportional to a(1-D/4) where a is
the area of a pixel and D is the fractal
dimension of the boundary
- standard error will thus depend on a1/2 for
highly scattered phenomenon and a3/4 for
single, circular patches with smooth boundaries
REFERENCES
Only a very small portion of the literature is presented
here. For further references you should refer to the
Goodchild and Mark (1987) paper; recent issues of Water
Resources Research and Science also contain relevant papers
Burrough, P.A., 1981. "Fractal dimensions of landscapes and
other environmental data," Nature 294:240-242.
Buttenfield, B., 1985. "Treatment of the cartographic line,"
Cartographica 22:1-26.
Goodchild, M.F., 1980. "Fractals and the accuracy of
geographical measures," Mathematical Geology 12:85-98.
Goodchild, M.F., and Grandfield, A.W., 1983. "Optimizing
raster storage: An evaluation of four alternatives,"
Auto-Carto 6(2):400-407.
Goodchild, M.F., and Mark, D.M., 1987. "The fractal nature of
geographic phenomena," Annals AAG 77(2):265-278.
Hakanson, L., 1978. "The length of closed geomorphic lines,"
Mathematical Geology 10:141-167.
Lovejoy, S., 1982. "Area-perimeter relation for rain and
cloud areas," Science 216:185-187.
Mandelbrot, B.M., 1977. Fractals: Form, Chance and Dimension,
Freeman, San Francisco.
Mandelbrot, B.M., 1982. The Fractal Geometry of Nature,
W.H. Freeman and Co., New York.
Milne, B.T., 1988. "Measuring the fractal geometry of
landscapes," Applied Mathematics and Computation 27:67-
79.
Peitgen, H.-O. and D. Saupe (Eds.) 1988. The Science of
Fractal Images, Springer-Verlag, New York.
Richardson, L.F., 1961. "The problem of contiguity," General
Systems Yearbook 6:139-187.
Unwin, D., editor, 1989. Special issue on fractals.
Computers and Geosciences 15(2).
Woronow, A., 1981. "Morphometric consistency with the
Hausdorff-Besicovich dimension," Mathematical Geology
13:201-216.
EXAM AND DISCUSSION QUESTIONS
1. Although fractal concepts are important in understanding
the error associated with pixel-based area estimates, little
has been said about the relationship between fractals and
area estimates obtained from vector-based systems. Why?
(i.e., would the area of an enclosed figure change
significantly? It is expected that the area shouldn't
change significantly, as the self-similar detail should
increase the area as much as it decreases the area.)
2. Define "fractal". Include in your description terms such
as scale dependency, self-similarity and scaling.
3. Discuss some of the ways in which fractals have changed
our way of looking at phenomena. Based on your readings,
provide examples from a variety of fields.
4. Theoretically, fractal behavior applies to a phenomenon
across all scales. Practically, of course, there are limits
to the application of self-similarity to natural phenomena.
Where do you think some of these limits occur? (i.e.,
between what scales do you think portions of coastlines, for
example, exhibit self-similar behavior.) What are the
implications with respect to the generalization of
cartogrpahic lines, if we observe definite limits to the
self-similar behavior of cartographic features?
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