In nature fractal dimension has been used to describe the complexity of plant shapes (Morse et al.
1985) and the pore/grain ratio of rocks (Wong and Howard, 1986). Studies of plants have found a positive
correlation between fractal dimension and the abundance of animals on those plants. Studies of rocks have
discovered that certain minerals have a particular fractal dimension and fractal dimension has been a useful
addition to the assessment of grain size and angularity in describing the grains of sedimentary rocks,
These findings suggest that fractal dimension could also be used to describe rocks on a biologically
relevant scale. Rocks comprise the substratum on many shores, providing the setting forthe very diverse and
productive intertidal ecosystem. A useful description of their surface topography could provide important
clues about the structure of the intertidal zone. For example, due to the pelagic nature of many larval forms
of marine life, settlement on a substratum is a crucial element of community structure and dynamics. If the
texture of the rock affects settlement, then fractals may help to provide a quantitative way of understanding
how larvae select substrata.
A second aspect of fractal dimension of rocks that could have large significance in community
ecology is due to the characteristic of exponentially increasing length with decreasing scale in fractal forms.
On a fractal surface an animal I mm long has much more surface area and volume for grazing, foraging and
escaping predators than does an animal 5 cm long; these changes in surace area and volume increase with
increasing fractal dimension. Hence a rock with a higher fractal dimension may well exhibit differences in
patterns of feeding habits and predator-prey interactions.
These issues led to three questions about the rocky intertidal zone and the biomass found there.
1)ls there a difference in the fractal dimension of different substrates (in this case, rocks)?
2)ls there a difference in the biomass of different substrates?
3)ls there a correlation between the fractal dimension of a substrate and the biomass of that same
substrate?
Source
Mean-Square
F-Ratio
P-Value
Rock
011
.019
.890
43.90
Type of Biomass
79.220
000
Rock"Type
0.309
557
.460
Erroi
.554
Rock: Granite, Basalt
Type of Biomass: Animal, Plant
Table 5. Analysis of variance between means of biomass of granite and basalt in the exposed site.
The P-value in the "Rock" row indicates the probability that the difference in biomass between
rocks is due to statistical error. The p-value in the "Type of Biomass" row indicates the
probability that the difference in biomass between the types of biomass depends on statistical
error. The p-value =O.460 indicates that there was not a siginificant interaction effect.
FIGURE LEGEND
Figure! The ranges, means and error of the measurement of fractal dimension of basalt, granite and
sandstone on the 10 cm scale and the 1 m scale. The lines extending past the error bars indicate
ranges of each set of measurements.
Figure 2
The biomass on sandstone and granite in the protected site separated according to type. Note
the much higher plant biomass on the granite.
ABSTRACT
The utility of fractal geometry as a descriptive tool in the natural sciences has increased as more and
more naturally occurring fractal patterns have emerged. Fractals have been used to describe the shape of
plants, the length of coastlines and
the porous nature of rocks. This study focuses on determining if three rock types--granite, basalt and
sandstone-in the Monterey Bay region have a fractal nature and if so, how that nature may affect the
standing biomass.
Basalt was found to have a higher fractal dimension than either granite or sandstone on the scale of
10 cm. On the scale of 1 m, both granite and sandstone have a higher fractal dimension than that measured
on the 10 cm scale. In the protected site, plant biomass on granite was significantly greater than that on
sandstone. There was no correlation between the fractal dimension of substratum and the animal, plant or
total biomass. Fractal dimension did not, in this case, prove to be a useful technique for describing substrata
in the rocky intertidal zone.
INTRODUCTION
The use of fractal geometry to describe the physical environment in ecosystems has become
increasingly useful as more and more fractal patterns are observed in nature. (Sugihara and May. 1990).
Fractal geometry has been used to model the growth of species of algae, the measurement of coastlines
(Sugihara and May, 1990) and the pore/grain ratio of sedimentary and carbonate rocks (Krohn, 1988).
Fractal geometry has several important aspects but the two that are most pertinent to ecological
studies are Ithe characteristic of self-similarity and nested patterns and 2)the characteristic of scale
dependent length. Self similarity is the concept of texture within texture. A profile, for example, may have
à certain pattern to it; if, when that profile is viewed on a smaller scale the same pattern is present, that
outline has a fractal nature. Self similarity also suggests that no matter what scale is used to view the outline
à similar pattern exists. In mathematics, fractal patterns can be exactly self-similar on every scale; but in
nature outlines and surface areas that have a fractal dimension are only statistically self similar because the
pattern is not precisely the same at each scale (Sugihara and May, 1990).
Self similarity contributes to the second important characteristic of fractal dimension: scale
dependent length. As the scale at which a fractal outline is viewed decreases the detail increases. Each
decrease in scale produces an exponential increase in the measured length of the outline (Morse et al.
1985). Total length, L, is dependent on scale length
L(r)= r exp(1-D),
(1)
where r=scale length and D= fractal dimension (Morse et al, 1985). The methodology of computing fractal
dimensions devised by Mandelbrot (1983) makes it possible to calculate dimension relatively easily: take a
profile of a surface and enclose it in a square box with a certain size, S, measured as the length of a side. To
obtain the fractal dimension at a particular scale L divide the box into (S/L)exp2 boxes each with a side
length of L. Count the number of boxes in which the line appears. Plot the natural log of the number of
boxes that contain at least one point as a function of the natural log of the scale length l. The fractal
dimension is (-1) multiplied by the slope of this line.
MATERIALS AND METHODS
This study was divided into three parts: Imeasurement of the texture among and within three rock
types Z)calculation of the sessile biomass on these three rock types 3) testing for a correlation between the
fractal dimension of and the amount of sessile biomass,
lo measure the texture of different substrata, shape was analyzed by determining the fractal
dimensions of sandstone, granite and basalt. To measure the variation of dimension within one rock type,
three sites with an visually distinct texture were selected for that rock type. A 1O cm latex mold was taken
at each of these sites. I m of copper wire was molded to the surface at the same spot. An ink print from a
cross section of each latex mold was taken, and the resulting profile was digitized and analyzed with a
computer program designed to determine the fractal dimension. Tracings of the wire mold were digitized
and analyzed with the same program.
The fractal program determines dimension in the manner described above. It draws a box around
the outline and checks to see if there is data point in that box. The program then divides the box in half on
each side and again, counts how many boxes have a data point inside. The process is repeated; each time
another division is made the number of boxes increases to a power of 2. The dimension is determined by
plotting the natural logarithm of the number of boxes entered by the outine as a function of the natural
logarithm of the scale (length of each side) of the boxes. The fractal dimension is (-1) multiplied by the slope
of the resulting regression line. The smallest scale size was 3 mm for the 10 cm measurements(now referred
to as the 10 cm scale) and 5 cm for the 1 m measurements (now referred to as the 1 m scale).
ANOVA analysis was carried out to test for a significant difference of the means of fractal dimension
among the rock types on the 10 cm and the 1 m measurements. Another ANOVA was conducted to test
for differences in the means of fractal dimension between the 10 cm scale and the 1 m scale.
Measurements of biomass were carried out at a site protected from wave exposure and at a site with
more exposure, both near Monterey, CA (39 37'N, 121 50'W). Sandstone and granite test areas selected
in the protected site and basalt and granite test areas were chosen at the exposed site. Ten 100 cm transects
of visually different textures were selected within a range of 1 to 1.5 feet below the Endocladia-barnacle
region to control for effective tidal height. The rock surface at each test area had a vertical inclination. The
test areas in the exposed site faced NNW and the test areas in the protected site faced NNE. All of sessile
species were collected in each test area and this biomass was separated into animals and plants; shaken, then
blotted six times and weighed. An acrylic mold of each test area was taken after the removal of the species,
The fractal dimension of each mold was determined through the same process described above.
ANOVA analysis was carried out to test for difference in the biomass between rock types at each
exposure. A regression analysis compared the fractal dimension of each transect to the corresponding
biomass of animals, plants and combined sources.
RESULTS
Measurements of the three rock types showed a significant difference in the fractal dimension at a
scale of 10 cm (Fig. 1). The difference between sandstone and basalt was significant according to a Tukey's
HSD test, but the other two comparisons (granite versus sandstone and granite versus basalt) were not. On
the scale of 1 m, the means of fractal dimension were not statistically different among the rock types (Fig.
1). There was however a significant difference between the two scales (10 cm and 1 m) within the rock
types. A Tukey's HSD test demonstrated that fractal dimensions of granite and sandstone at the I m scale
was greater than that at the 10 cm scale (Fig. 1). The two measurements of basalt were not statistically
different.
Biomass calculations revealed that in the protected site there was a strong difference in the pooled
biomass between the granite and sandstone. There was also a significant difference between the two types of
biomass (animal or plant) depending on which rock was analyzed. A significant interaction factor indicated
that the differences between rock types depended on type of biomass (Fig. 3). While animal biomass was
similar between rock types, plant biomass was much greater on the granite.
At the protected site, granite was dominated by Mastocarpus, Cthamalus and various limpets.
Sandstone's biomass was primarily Endocladia and Cthamalus.
There was no significant difference in biomass between the rock types in the exposed site. Howeyer.
again there was a significant difference between the mean plant and animal biomasses. Plants were more
abundant then animals
At the exposed site, basalt was dominated by Mastocarpus and Cthamalus with a substantial amount
of Pelvetia and gooseneck barnacles. The biomass of granite was mostly Mastocarpus, Cthamalus and various
limpets.
No correlation was found between the fractal dimension of the substrate and the weight of the
animal, plant or total biomass present.
DISCUSSION
The significant difference between basalt and sandstone on the 10 cm scale and the ranges of fractal
dimension for each rock on the 10 cm scale (see fig. 1) and on the 1 m scale (see fig. 1) show trends
(granite has the highest range of fractal dimension, basalt falls in the middle and sandstone has the lowest
range) that suggest an overall range of fractal dimension for a rock type. How useful this tool may be is not
certain because none of the trends in this study were significant
lwo aspects of this study may be affecting the lack of a statistically significant patteren in the data
regarding fractal dimension: Ismall sample size and 2)only one rock formation for each rock type. The
trends found at the three outcroppings in this experiment-sandstone at Stillwater Cove in Pebble Beach,
granite at Hopkins Marine Reserve and at Carmel Point and basalt at Carmel Point--may or may not be
reflective of the total range of fractal dimension for each rock type. It would be enlightening to make more
measurements at these sites as well as at other formations to establish if there is a definitive range of fractal
dimension for either of these three rock types. Further study should also include more rock types; ones that
are commonly found as substrata in the rocky intertidal and other ecosystems and even ones that do not
serve as substrate at all.
The other result that suggests that fractal dimension could possibly be useful for describing an
ecosystem's physical environment is that there is a higher fractal dimension at the larger scale measured.
The lower fractal dimension at the smaller scale suggests that fractal dimension may approach one as the
scale length decreases. The next step would be to take more measurements at both larger and smaller scales
to develop à complete picture how fractal dimension changes with scale for one rock type.
The extremely high amount of plant biomass on the granite in the protected area as compared to
that on the sandstone is unexplained by fractal dimension as there was no correlation between biomass and
fractal dimension. Further study should focus on why there is a greater abundance of Mastocarpus on granite
as it contributed greatly to the overall weight of plant biomass on granite and was not present on sandstone.
There may also be a correlation between fractal dimension and some other aspect of plant type such as
holdfast size or size of the total plant.
It is probable that other aspects of the substrate could be the cause of differences in biomass,
Hardness of rock, rate of erosion and porosity may all have an effect. For instance, it may be more difficult
for a plant to establish itself on sandstone because of the rock's softness and fast rate of erosion. The higher
porosity of sandstone may allow better water retention and thus have effect on the water available to the
plants and on the temperature of the microhabitat.
REFERENCES
Greene, Gary H. 1977. Geologic Map of the Monterey Bay Region, CA. U. S. Geological Survey.
Krohn, Christine E. 1988. Fractal Measurements of Sandstones, Shales and Carbonates. Journal of
Geophysical Research. 93: 3297-3305.
Krohn, Christine, E. 1988. Sandstone Fractal and Euclidean Pore Volume Distributions. Journal of
Geophysical Research. 93: 3286-3296.
Morse, D. R. et al. 1985. Fractal dimension of vegetation and the distribution of arthropod body lengths.
Nature. 314: 731-733.
Mandelbrot, B. B. 1983. The Fractal Geometry of Nature. Freeman: New York.
Orford, J.D. and Whalley, W. B. 1983. The use of the fractal dimension to quantify the morphology of
irregular-shaped particles. Sedimentology. 30: 655-668.
Sugihara, George and May, Robert M. 1990. Applications of Fractals in Ecology. Trends in Evolution and
Ecology. 5: 79-85.
Wong, Po-zen and Howard, James. 1986. Surface roughening and the Fractal Nature of Rocks. Physical
Review Letters. 57: 637-640.
Source
Mean-Square
P-Value
F-Ratio
Rock
4.341
.003
.019
Error
001
Rock: Basalt, Granite, Sandstone
Tukey HSD
Basalt
Granite
Sandstone
M. D.
M.D. P M.D. P
Basalt
000
Granite
-.016
.236
.000 1
Sandstone
-.032
.014
-016 .223 .000 1
Table 1. Analysis of variance of the means of fractal dimensions of basalt, granite and sandstone on the 10
cm scale. The p-value =O.019 in the upper table indicates the probability that the difference in
fractal dimesnion among the rocks was due to statistical error. A p-value «O.O5 indicates a
significant rock effect.
Source
Mean-Square
F-Ratio
P-Value
Rock
001
.840
477
Error
001
Rock: Basalt, Granite, Sandstone
lable 2. Analysis of variance of the means of fractal dimensions of basalt, granite and sandstone on the 1
m scale. The p-value indicates the probability that the difference in fractal dimension among the
rocks was due to statistical error. A p-value «O.O5 indicates a significant rock effect.
Source
Mean-Square
P-Value
F-Ratio
Rock
2.244
.002
.116
Scale
030
38.572
000
Rock“ Scale
001
1.238
298
Error
.001
Rock: Basalt, Granite, Sandstone
Scale: m, mm
Tukey HSD
Basalt m
Basalt mm Granite m Granite mm Sandstone m Sandstone mm
M. D.
M.D. P M.D. P M.D. P M.D. P M.D. P
Basalt m
000
-.043
Basalt mm
.172 .000
Granite m
.023
.908
.066 .006 .000 1
Granite mm
-059
-016 .594 -082 .000 .000
.015
Sandstone m
-010
998 .033 444 -033 .689 .049 .067
000
Sandstone mm -075 .002 -032 .061 -.098 000 -016 .574 -.065 .009 .000
Table 3. Analysis of variance of the means of fractal dimensions of basalt, granite and sandstone between the
10 cm scale and the 1 m scale. The p-value =0.000 in the upper table indicates that there was no
probability that the difference of fractal dimension between scales was due to statistical error. A
p-value «O.O5 indicates a significant scale effect.
Source
Mean-Square
F-Ratio
P-Value
Rock
26.11
27.656
000
Type of Biomass
27.836
29.483
000
40.496
Rock"Type
42.892
000
Erroi
.944
Rock: Sandstone, Granite
Type of Biomass: Animal, Plant
Table 4. Analysis of variance of the means of biomass of granite and sandstone in the protected site. The
p-value in the "Rock" row indicates the probability that the difference in biomass between rocks
was due to statistical error. The p-value in the "Type of Biomass" row indicates the probability
that the difference in biomass between the types of biomass was due to statistical error. The p-
value =0.000 in the "Rock"Type" row indicates that the difference in biomass between rocks
depended on which type of biomass was analyzed.
1.20
1.15
1.10
1.05
1.00
10 cm
Basalt Granite
Rock Type
D 1
Sandstone
40
30
20
10

Sandstone
Animal

Granite
Rock Type
Plant