UNIT 47 - FRACTALS
UNIT 47 - FRACTALS
Compiled with assistance from Brian Klinkenberg, University of British Columbia
? A. INTRODUCTION
Why learn about fractals?
? Length of a cartographic line ? Where did the ideas originate?
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? B. SOME INTRODUCTORY CONCEPTS
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Euclidean geometry
? C. SCALE DEPENDENCE
Determining fractal dimension ? Some questions
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? D. SELF-SIMILARITY AND SCALING Self-similarity ? Scaling
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? E. ERROR IN LENGTH AND AREA MEASUREMENTS ? REFERENCES
? EXAM AND DISCUSSION QUESTIONS ? 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
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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
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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
o areas should change by the square of the ratio o 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
o new detail will be apparent at the larger scale o \closer you look, the more you see\is true of almost all
geographical data
o 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
o this concept of the scale-dependent nature of cartographic
data will be discussed in more detail later
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Where did the ideas originate?
term was introduced by Benoit Mandelbrot to the general public in his 1977 text Fractals: Form, Chance and Dimension
o a second edition in 1982 is titled The Fractal Geometry of
Nature o 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
o fractal-based papers represent over 50% of the submissions
for some physics journals
o many of the studies of the fractal geometry of nature are
still at the early stages (especially those in geomorphology and cartography) o 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)
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B. SOME INTRODUCTORY CONCEPTS Euclidean geometry
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in traditional Euclidean geometry we work with points, lines, areas and volumes
o Euclidean dimensions (E) are all positive whole numbers the Euclidean dimension represents the number of coordinates necessary to define a point
o to specify any point on a profile requires two coordinates,
thus a profile has a Euclidean dimension of two
o to define a point on a surface requires three dimensions,
therefore a surface has a Euclidean dimension of three
closely allied with Euclidean dimensions are the topological dimensions (DT) of phenomena
o on a flat piece of paper (which has a Euclidean dimension of
2) you can draw a two-dimensional figure (DT= 2), a
one-dimensional line (DT= 1), and a zero-dimensional point (DT= 0) (compare 0-cell, 1- cell and 2-cell notation)
in fractal geometry we work with points, lines, areas and volumes, but instead of restricting ourselves to integer dimensions, we allow the fractal dimension (D) to be any real number
o the limits on this real number are that it must be at least
equal to the topological dimension of the phenomenon, and at most equal to the Euclidean dimension (i.e., 0<=DT<=D<=E) o a line drawn on a piece of paper can have a fractal dimension
anywhere from one to two
the term fractals is derived from the same Latin root [fractus] as fractions; therefore: fractional dimensions
the fractal dimension summarizes the degree of complexity of the phenomenon, the degree of its 'space-filling capability' overhead - Lines of different fractal dimensions
o o o o o
straight line will have equivalent topological and fractal dimensions of 1
slightly curved line will still have a topological dimension of 1, but a fractal dimension slightly greater than 1 highly curved line (DT= 1) will have a much higher fractal dimension
line which completely 'fills in' the page will have a fractal dimension of 2
many natural cartographic lines have fractal dimensions between 1.15 and 1.30
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