Landscape and Urban Planning, 2 1 ( 199 1) 8 l-90 Elsevier Science Publishers B.V.. Amsterdam

81

he utility of fractal geometry in la dscape design

Bruce T. Milne Department q f Biology, Universit_v of New Me_Gco, Albuquerque. NM 8 713 I, ClS.4

(Accepted 14 September 1990)

ABSTRACT

Milne, B.T., 199 1. The utility of fractal geometry in landscape design. Landscape Crban Plann., 2 1: 8 l-90.

Natural landscapes possess remarkable regularity in the patterning, sizes, shapes, connectedness. and density of patches. Landscape ecological studies that explore the fractal geometry of nature have found many examples of consistent variation in landscape pattern with scale. Fractal landscape patterns affect the distributions and movements of animals, and there- fore may be an important characteristic to include in designed landscapes. A computerized design system enables fractals to be created interactively and automatically overlayed on images or existing maps. The fractal designs are then used in simulations of foraging animals to determir how the new design affects the movements, energetics, and locations of species. The combination of computerized frdctal designs and ecological simulation models may enhance both the ecolog- ical relevance and aesthetic value of the designed landscape.

The planner or landscape designer seldom knows how the designed landscape affects the movements of animals, the distribution of spe- cies, the rates at which resources are depleted, or distu.rbance and community assembly. Al- though some quantities such as the number of bird species in forest islands may be estimated empirically (e.g. Van Dorp and Opdam, 1987 ), no simple way exists for determining how landscape structure (e.g. hedgerow intersec- tions, Forman and Baudry, 1984) modifies species distributions. Even if such procedures were available for all landscapes and all spe- cies, the design and construction of new land- scapes could be enhanced further by simple and rapid procedures for configuring landscape elements (e.g. trees, fields, boulder outcrops) in ways that emulate fundamental character- istics of natural landscapes. Here, the funda- mental characteristic of interest is the remark- ably consistent way in which patch density,

forage biomass, patch edge length, and patch shape vary as a function of the resolution, or scale, at which measurements are made (Man- delbrot, 1983; Milne, 1988, 199 la; Voss, 1988).

Ecological phenomena such as the spread of fire, nutrient redistribution, hydrological cy- cling, pollutant flow, predation, and nest par- asitism are affected by landscape geometry (Risser et al., 1984; Forman and Godron, 1986; Turner, 1987). Often, these ecological interactions are modified by several factors that operate simultaneously at different scales. For example, Gambel oak is common at dry, southerly latitudes, but paradoxically it reaches grecttest abundance at higher, wetter elevations Neilson and Wullstein, 1983 ). Interactions

between m oisture availability and factors af- fecting oak seedling establishment make water a poor. predictor of oak abundance. Indeed, landscape patterns are regulated by many pro- cesses that ;*ange from the day-long process of seedling germination, to successional changes

0 169-2046/9 l/$03.50 0 199 1 Elsevier Science Publishers B.V. All rights reserved.

82

lasting decades after a fire, to the variation of the solar constant stretching over millennia (Delcourt et al., 1983). Spatially, these and other processes may impact square centime- ters or the entire globe. As a rule, short-lived processes (e.g. seedling establishment ) OCCUPY small areas, whereas persistent processes leave marks over hundreds of square kilometers. Thus, landscapes are affected by processes that vary across many temporal and spatial scales.

Animals and plants interact with a given landscape at a variety of scales (Woodward, 1987 ). Animal species differ in home range area, density, metabolic rates, ingestion rates, reproductive rates, and locomotion, all of which are readily predicted from measure- ments of body mass (Peters, 1983). Conse- quently, body mass is a surrogate for the scale at which mammals, birds, grazing animals, and reptiles interact with the environment (Milne et al., 199 I ). Put another way, a given land- scape pattern, be it natural or designed, is per- ceived quite differently by various species, de- pending on the scale at which they use the landscape (Morse et al., 1985 ),

The regularity of temporal, spatial, and biotic scaling relationships makes it possible to as- sess whether a .andscape and the organisms contained withi it function in a way that is compatible with the natural scales that the or- ganisms are adapted to. For example, the Ser- engeti, with its hundreds of thousands of hec- tares and multitudes of grazing animals, is one of the last remaining landscapes in the world that supports 2 megafauna one would expect in a large natural g;assland; the Serengeti ecosys- tem persists because the region is the correct size for supporting large beasts.

In contrast, a 100 kg animal such as a sabl? antelope at a typical zoo is granted less than 0.15% of the 3 km* the animal would occupy in the wild (see I-Iarestad and Bunnell, 1979 ) . Contrasting a cattle ranch with a dairy farm re- veals similar discrepancies of scale. The west- ern ranch, with perhaps 60 ha per animal, is much closer to the scaling of a natural herbi-

R.T.MILNE

vore system than the dairy. The dairy farm is successful only because the farmer imports or grows sufficient feed to pack the animals into a fraction of their natural home range area. Ar- tificial systems can only be maintained if an external source of food energy and supplemen- tal resource cycling procedures are built in. Every departure from the natural scale exacts a cost in management effort.

Below, a general method for generating nat- urally scaled fractal landscape patterns is pre- sented. Fractal designs preserve many aspects of natural landscapes by ensuring that the spa- tial scaling of the landscape is built in. The de- signs increase the chance that temporal and biotic scaling will follow. After constructing the fractal design, a naturally scaled simulation of mammalian grazing can be used to explore the potential biological implications of the design. Together, the design and simulation provide the designer with ecologically meaningful interpretations of the new landscape. This ap- proach to design focuses on the pervasive frac- tal character of the landscape, and attempts to maintain a corlsistent change in ecological phenomena with scale, rather than attempting to design and control the smallest nuances of landscape function and structure.

TAL G LA

Fractal geometry is one of the widest reach- ing mathematical developments of the 20th century. Mandelbrot ( 1983) and others have observed striking regularities in the shapes of coastlines, the perimeters of rain clouds. the roughness of terrain, and the shapes of plants (Burrough, 198 1; ovejoy, 1982; Feder, 1988). The regularity is readily described by fractals, which are mathematical representations of na- ture. Fractals often exhibit statistically regular patterns that occur when a quantity such as patch density at one location is predictable from measu.rements of the same quantity nearby. For example, the coast of Maine is characterized by peninsulas, many of which have smaller peninsulas attached, ad infini-

UTILITY OF FRACTAL GEOMETRY IN LANDSCAPE DESIGN 83

turn. The fractal geometry of the coastline im- plies that the presence of a peninsula, 15 km long, ensures the presence of yet smaller spits jutting off from the large one (Milne, 199 1 a).

More significantly, coastlines and other pat- terns that exhibit fractal structure obey strict relationships between the scale of observation and the quantity measured (Mandelbrot, 1983; Stanley, 1986 ) . For example, fractal scaling was found in analyses of remotely sensed im- agery obtained for the Sevilleta National Wild- life Refuge in New Mexico. Regions occupied by 0- 10% grass cover exhibited regular de- creases in grass density as the spatial resolu- tion was decreased (Milne, 199 1 a). However, areas trampled heavily by cattle 15 years be- fore the study, and then left ungrazed by cattle, did not exhibit simple fractal relationships (Milne, 199 1 c). Natural grasslands without heavy disturbance exhibited archetypal fractal relationships while disturbed areas did not. This contrast suggests that natural configura- tions of plants and resources, similar to those found in undisturbed landscapes, can be cre- ated by using fractal patterns to design landscapes.

There are many alternative scaling relation- ships for a given landscape. For example, one could examine the relatifDnships between patch area and perimeter with changes in area (Krummel et al., 1987; ONeill et al., 1988), the density of convolutions on the coastline, or the density of pixels on a digital image of a landscape (Milne, 1988, 1991a). Each of the quantities (i.e. perimeter to area ratio, convo- lutions per kilometer, and density) are pre- dicted to vary as a function of the length scale used, according to the ge era1 fractal scaling relation

Q(L)=kL (lb

where Q(L) is the quantity measured using a length scale L, k is a constant, and D is the frac- tal dimension of the quantity (Stanley, 1986:, Voss, 1988 ) . This expression applies in all cases considered here. If animals and plants are en-

visioned as operating at different scales, due to differences in home range area or dispersal distance, then eqn. ( 1) describes how the spe- cies perceptions of a resource vary with scale.

For landscape analysis, a versatile fractal method has been developed that provides both an estimate of the fractal dimension describing the concentration of pixels of a given cover type and a visualization of scale-dependent pixel density on a digital representation of the land- scape (Milne, 199 1 a). The method entails constructing a digital image of a cover type or class of interest (e.g. forest ). Then a square window is centered on each pixel of the class, and the number of pixels of the class within each window is counted. The procedure yields the frequencies at which m = 1, 2, . . . L2 pixels of the class are found on the image within win- dows of length L. The counts are formed for a series of L values ranging from very small (e.g. 3 ) to large (e.g. 3 1- 10 1) . By transforming the frequencies to probabilities, the probability density function

1 Pb%L)=l (2)

is formed, which requires that the sum of the probabilities p@z,L) of finding m pixels in windows of length L equals 1 .O summed across all values of m from 1 to n(L); n(L)SL (Voss, 1988 ). Next, the quantity to be used in eqn. ( 1) to estimate the parameters k and D is the sum of each p( m,L) value multiplied by the respective m value

n(L)

Q(L)= 1 uzp(mJA (3) tn= I

Thus, Q(L) is the expected number of pixels (i.e. the first moment of the distribution ) found in a window of length L, and Q(L) in- creases as a power of L with an exponent equal to the mass fractal dimension of the mosaic of pixels (Voss, 1988 ). The mass fractal dimen- sion and the constant k describe how the num- ber of pixels varies with scale.

Spatial information about the scale depen- dence of pixel density can be represented by

84

tallying the number of neighboring pixels of a given class included in each window of a given length. By displaying the respective tallies for each pixel on a gray scale image, high numbers of neighbors are revealed for dense clusters of pixels. and few neighbors are found for sparsely clustered pixels. If this analysis is performed for each of three values of L, and the bright- ness of each gray-scale image is resealed such that the maximum total number of visits by windows of length L is displayed as brightly as the image display permits, and the three im- ages are overlayed in the red, green, and blue color planes of the image processor, then the resulting image reveals the spatial pattern of scale-dependent density at each scale (Milne, 199 1 b). Theoretically, this analysis reveals where, on a fractal distribution of food re- sources, animals with home ranges of length L are likely to go to find food that is at a high density when perceived at the animals char- acteristic scale. Interestingly, the image also shows that animais operating at different scales are likely to go to different regions to find high concentrations of resources.

GN CTA SCAPES

An interactive method is required to design fractals. A useful method should facilitate creativity while maintaining the scaling behav- ior of naturally occurring patterns. Bamsley ( 1988a, b) developed the mathematics and software for so-called iterated function sys- tems (IFS) that allow fractals of an infinite variety to be constructed easily (Fig. 1). By coupling the IFS method with existing corn-- puter-aided design software, geographic infor- mation systems, or remote sensing image anal- ysis packages, the IFSs become another tool for producing designs with aesthetic value and ecologically meaningful structure.

Iterated function systems work by rotating, stretching, and translocating points in a coor- dinate system by means of simple equations, not at all unlike the equations used to compute

, i :

( / djj

, , 1=

..i )

_, 4 ... (A)

,

B.T. MlLNE

ii, I!I :(.I

:

Fig. 1. Exampies of fractals generated using the iterated func- tion system method. The panels illustrate the continuous de- pendence of the pattern on small chdnges in the IFS. Each panel was made by slightly alteringthe functions used to cre- ate panel (A).

perspective projections. Surprisingly, the equations of the IFS .may be applied randomly to a set of starting coordinates, and after many iterations, the set of transformed points may take on a spatial configuration reminiscent of natural patterns. Each of the k= 1,2, . . . n func- tions of the IFS is of the following form

w(k) =[,:;:I=[; ;][;]+E] in which the original coordinates of a point specified by x and y are transformed by the ro- tational matrix containing the parameters a, 6, c, and d, and then translocated (i.e. moved lat- erally or verticaily ) by the vector containing parameters e andJ The design of the IFS and the pattern it creates are controlled by the pa- rameter values. Applying the rules of matrix multiplication and addition, the new values of x and y are

x = ax+by+e

and

y' =cx+dy+/-

UTILITY OF FRACTAL GEOMETRY IN LANDSCAPE DESIGN 85

Thus, after one iteration, each of the points in the original square boundary of Fig. 2 (A) take on coordinates indicated by the polygon within the original boundary. The dots on the curved lines of Fig. 2 (A) are the successive locations of the vertices of the polygon for several itera- tions of the transformation. The rotation, stretching, and translocation in Fig. 2 (A) oc- cur if the parameters are set such that a= 0.6, b=O.l, c=O.2, d=0.4, e=O.l, andf=0.3. After many iterations, the polygon converges to the so-called fixed poirt shown inside of the pol- ygon. The fixed point is the point whose value

is not affected by the function, and it may be thought of as an attractor of all points within the original boundary; points travel along paths whose shapes are determined by the lunction. The function effectively warps the space within the coordinate frame such that the shortest distance along which a transformed point moves towards the fixed pointis not necessar- ily straight, but may be curved.

An IFS with several functions leads to much more complex patterns, because the functions are applied randomly to the current coordi- nates of a point (Fig. 2 (B ) ). The curved lines

b ,

/ /

I

FPI

I

FP2

Fig. 2. Graphical explanation of how iterated function systems create fractal patterns. (A) Transformation of the original bound- ary by a pair of equations shrinks and rotates the boundary to create a new polygon (transformed boundary). Reiteration of the transformation projects all points toward the fixed point. (B) The fixed points produced by two functions, showing how trajec- tories toward fixed points vary between functions. (C) A series of points created by applying functions 1 and 2 in the order 1,2, I, 2,2, 1. Dashed lines represent the trajectory each point would make if it were subjected to repeated iterations of the respective functions. Intersections indicate where the procedure switched from one function to the next. (D) The resulting fractal pattern after 500 iterations.

86

in Fig. 2 (B) indicate that function 1, with fixed point 1, tends to attract points originating from within the coordinate frame towards the fixed point along routes that are shaped quite differently than the routes leading to fixed point 2, which is the fixed point of the second function. Rarely will the fixed points of each function be equal, and consequently the appli- cation of a randomly chosen function forces points off the fixed points, thus perpetuating the series of transformations.

A point follows a jagged trajectory if the transformations are applied in the order 1, 2, 1,2,2, 1 (Fig. 2 (C) ). The point begins at po- sition 0 at the top of the coordinate frame, and is transformed using function 1, which drives the point closer to fixed point 1. Then, in a manner that is analogous to flipping a coin, a

B.T. MILNE

random number is used to choose function 2, thereby resulting in a new trajectory towards fixed point 2, but only as far as point 2. Ran- dom selection of function 1 drives the point to- ward the first fixed point, and so on. Once the transient behavior of the trajectory disap- pears, the point oscillates back and forth in the region between the fixed points, and traces out a pattern (Fig. 2 (D) ). Thus, the parameters of the functions determine all possible routes the point takes to create the fractal pattern, and manipulation of the parameters regulates the final pattern.

Fractals created by an IFS have four funda- mental properties that are of interest to the de- signer, and a fifth characteristic of ecological significance. First, a small change in the pa- rameters of the IFS results in small, continu-

Fig. 3. Fractal created using a three-function IFS and then symbolizing each point based on the number of the last function used to create the point. The straight lines connect consecutive points to illustrate thar the image is not created one row or column at a time.

UTILITY OF FRACTAL GEOMETRY IN LANDSCAPE DESIGN

ous changes in the fractal pattern produced, thereby facilitating animation (Barnsley, 1988b). A wide range of different patterns can be produced from simple manipulations of the IFS (Fig. 1).

Second, each point on the attractor has an address. For the designer, the addresses pro- vide a means of assigning different qualities to each point. The address is a list describing the order in which the various functions w(k) for k= 1,2, 3, . . . n were chosen randomly to create the point. For example, the points can be sym- bolized based on the last function that was ap- plied to create a point. Thus, a point created by the string of functions in the order 2,2,1,3,1 could be represented by the numeral 1 to indi- cate that function w( 1) was used last. Substi- tuting different colors, symbols, landscape ele- ments (e.g. tree species, land cover types), or even other IFS for the numerals provides a tex- turing scheme that maintains a high degree of fidelity with the overall pattern (Fig. 3 ). The hierarchical structure of the fractal can be in-

3

-200 c

it 0100. 3

A

A

A

A

A

tm m 66 -Ip

G 6 2'0 Windcw length

7 15

0.0 _- - -- -

0 50 100 152 2oc) Number of piwls

Fig. 4. Statistical behavior of the fractal used to make Fig. 3. showing the distribution of the number of pixels found in 3 X 3, 7 x 7, and 15 x 15 pixel windows. The inset illustrates the scale dependence of the expected number of pixels (i.e. the first moment of the frequency distributions) in a window as a function of window width; squares represent an analysis of the pattern in Fig. I (A), triangles show the results for the fractal used to make Fig. 3.

87

corporated into the design by using the penul- timate function number to color a pixel. This produces a much more intricate texture, with each of the n possible symbols nested within the prominent clusters that would have ap- peared if the points had been mapped using the symbol for the last function, rather than the penultimate function.

Third, the fractal can be magnified, subject to the numerical accuracy of the computer, to reveal additional details at high resolution. Thus, one imagines zooming in on a fractal that represents different cover types to reveal indi- vidual trees within the cover types (Bamsley, 1988b ). A sophisticated rendering of the im- age would enable oblique views at any scale.

Fourth, the fractal created by an IFS ex- hibits the same kind of scaling found in natural landscape mosaics. For example, the pattern in Fig. 3 exhibited a strong dependence on the number of image pixels found within sampling windows of various lengths (Fig. 4). The ef- fect of scale was much greater for the image in Fig. 3 than it was for Fig. 1 (A), as indicated by the slope of the curves (Fig. 4, inset ). Eco- logically, the scale dependence suggests that organisms that gather resources with different home range areas will experience exponen- tially different concentrations of resources (represented bv the pixels) in a given fractal landscape (M&e et al., 1989). The reason for this difference relates to the low fractal dimen- sion of 1.1 for Fig. 1 (A ), indicating a very simple, linear structure of the fractal (recall that lines are one-dimensional). In contrast, the dimension of 1.4 for the fractal used to construct Fig. 3 indicates the greater tendency for the fractal to fill the plane, thereby offering much greater variation in the clustering of pix- els at all scales.

Finally, fractals generated in this manner are necessarily phenomenological. In nature, frac- tals such as snowflakes and dendritic patterns are thought to be formed by microscopic me- chanical processes (Nittman and Stanley, 1986; Meakin, 1988 ). For example. dendritic

88 . B.T. MILNE

features can be explained as the steady forma- tion of macroscopic features from microscopic asymmetries, which occur when a molecule is added to the exterior of a growing mass of mol- ecules. Any variation in the process that leaves more molecules in one growing branch than another biases the future addition of mole- cules to the larger mass. Thus, the dendritic feature grows into a shape reflecting the histor- ical events which led to its creation (Nittman and Stanley, 1985; Meakin, 1988). The IFS does not have the mechanistic qualities char- acteristic of natural fractal patterns. Rather, an IFS produces patterns that fortuitously resem- ble familiar objects, or objects of the imagination.

greater density than large mammals (Peters, 1983 ), they have smaller home ranges, and much higher metabolic rates. These, and other ecological, behavioral, and physiological char- acteristics were used to regulate animal den- sity, home range area, speed of movement, ingestion rate, metabolic rate, and the time scales at which the species foraged in a simu- lation model.

ECOLOGICAL CONSEQUENCES: ALLOMETRIC HERBIVORY

The animals were placed randomly on the landscape and allowed to eat within a fractal pattern. The movements of animals through- out the landscape were determined by a simple rule. If the animal obtained enough forage within its home range to offset its metabolic costs, then it was allowed to remain in place and feed until the next time period. However, animals finding insufficient food were allowed to move in a random direction to another home range location.

Ultimately, any design should be evaluated to determine its ecological function relative to species distributions or the flows of resources through the landscape. Each landscape design implies specific interactions with water and nutrient runoff from one area to the next, as the wooded vegetation intervening between farm fields, for example, may filter nitrogen and phosphorus, thus reducing pollutants in water courses (Correll, I 983 ) . Likewise, envi- ronmental gradients a e lobes of wooded peninsulas may alter the distribution of spe- cies, and thereby regulate the number of spe- cies (Milne and Forman, 1986). Finally, the economics of foraging may differ between spe- cies living in the same landscape, depending on the scale at which the animals gather resources (Milne, et al., 1989, 1991).

Successful animals tended to cluster to- gether on the landscape in areas with high con- centrations of food, while hungry animals moved about in search of food (Fig. 5). The fact that 3 kg animals used much smaller home ranges, lived at higher densities, and had higher metabolic costs resulted in a rapid depletion of resources on the landscape (Fig. 5 ). Large an-

A model of allometric herbivory was devel- oped to assess the consequences of fractal landscape patterns on the foraging success of small mammalian herbivores (Milne et al., 199 1). Allometric herbivory refers to the way in which the scale at which animals forage is predictable from the animals body mass. For example, small mammals are present at much

CJ 60, E ,-

: *

40 L 0

i- __~_ .__l-- _-.I 125000 25ocoo Time (win)

Fig. 5. Variation in the percent qf simulated animals foraging in a fractal landscape as modeled for 180 days.

UTILITY OF FRACTAL GEOMETRY IN LANDSCAPE DESIGN 89

imals were more mobile on average, reflecting the difficulty they experienced in locating high concentrations of food within their larger home ranges. Thus, the fractal geometry of the land- scape resulted in very different perceived pat- terns of food distribution for each species, and consequently altered the foraging success of animals that operated at different scales.

CONCLUSIONS

In nature, complex landscape patterns are formed by a multitude of processes, including the birth and death of plants, soil erosion, grazing, and disturbance by fire. The interac- tions between random events and constraints on growth processes create patterns of seem- ingly infinite complexity (e.g. Stanley, 1986 ) that exhibit remarkably consistent variation with scale, e.g. Fig. 4. While foraging or search- ing for mates and nest sites, organisms are con- fronted with heterogeneous distributions of re- sources or habitat; the fractal geometry of the distributions may alter movement patterns (Wiens and Milne, 1989) or abundance (Morse et al., I985 ). Theoretically, the mag- nitude of the effect on animals is related to the scale at which species perceive the environment.

Tools are needed both for the analysis of landscape designs and for the creation of land- scape patterns that have geometrical regularity similar to the regularity found in natural land- scapes. Ecological research is needed to deter- mine the consequences of large changes in re- source density with scale vs. small changes with scale. Until such research is complete, simula- tion models based on allometric herbivory, or other properly scaled ecological processes, are one way in which ecologically sound assump- tions about animal activity can be coupled with computer-aided design packages to evaluate whether designed landscapes have desirable ecological characteristics. The generation of fractal patterns by iterated function systems should be regarded as one of the best available

techniques for creating patterns with the same statistical properties observed in natural landscapes.

ACKNOWLEDGMENTS

I thank A.R. Johnson and Y. Marinakis for helping decipher the mathematics of iterated function systems. The comments of an anony- mous reviewer helped clarify an early manu- script. This work was supported by the Na- tional Science Foundation (grant nos. BSR- 8806435 and BSR-86 1498 1) and the Depart- ment of Energy (grant no. DE-FG04- 88ER607 13 ).

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