Classical Generation of a Fractal
Applications of fractal geometry, in which a suitably selected
generator is applied through stages of successive iteration to an
initiator, are legion. We consider one simple case here,
generated in a classical manner, and then suggest one alternative for
its generation using another procedure. As an example, we
explore the self-similar geometry of
fractals in order to envision scaling networks to different dimensions
differing levels of space-filling by streets depending on surrounding
see how self-similarity might generate street
pattern, consider the application of a V-shaped generator, applied to a
to generate successive stages of finer and finer grid pattern (Figure
1, Arlinghaus 1985, 1990). The
animations of Figure 1
illustrates the process through successive stages of generator
application and scaling. Figure 1a illustrates the general
process while Figure 1b suggests how to visualize the detail of
application of the fractal generator. The
limiting position of such space-filling occurs when this iteration of
transformation is carried out infinitely. Using
Mandelbrot’s characterization of fractional
space-filling is captured as D
where n is the number of
and K is the
number of self-similar regions. Thus, in the case of Figure 1, D =
= 2. When the process is carried out infinitely, the entire
area over which it is carried out becomes filled with network
edges. In order to suggest network scaling and pattern, one would
wish to truncate the iterative process and stop it after a finite
number of steps, perhaps determined by smallest parcel size according
to zoning category.
Figure 1a. A
style of hierarchy generated through
self-similarity applied to a square initiator. D = 2. General
Figure 1b. Visualization of
detail of generator application in Figure 1a.
the generator, but keeping the same
initiator (a square) produces a different pattern (Figure 2). The
animations of Figure 2
illustrates the process through successive stages of generator
application and scaling. Figure 2a illustrates the general
while Figure 2b suggests how to visualize the detail of application of
the fractal generator.
Here, in the infinite process, the fractional dimension
is D =
= (2/3)*log(3)/log(2) = 1.0619248. The
choice of the three-sided generator
produced a pattern shaped similarly to the choice of the two-sided
but did so in a much less compact manner. Hence,
the space-filling of the lines and associated
are less than in the previous case, suggesting orientation of street
pattern to parcel in yet another manner. Again, though, to
realize the model in relation to street networks and parcels, the
iteration would need to be truncated, presumably in relation to the
urban character of the space under consideration.
style of hierarchy generated through self-similarity applied to a
square initiator. The dotted lines show the polygon generated (or given
in the previous state). The solid lines
show the generated edges. The dashed lines
are inferred (but not generated) pattern. Thus,
the value of D = 1.0619248 applies only to the generated (solid)
pattern. General iteration sequence.
Visualization of detail of generator application in Figure 2a.
Geometric Generation of
Fractals: Eigenvalue Geometry
method of making direct calculations of
fractional dimension relies on using a generator applied at successive
to an initial shape. Successive generator application to new
shapes creates an infinite process that converges to some finite value
fractional dimension. The previous
section showed two examples. Generator
selection, to produce a desired outcome (pattern or number) is an art.
this section, we reconsider the example of Figure 2 generated above
with respect to different underlying geometric
processes. With generator scaling and
shape selection, as above, only an envelope of the associated pattern
generated. Advantages of the strategy
suggested below are:
full pattern, and not merely its
envelope, is generated.
systematic geometric process with
roots in linear algebra may offer interesting related connections.
scaling effect is evident in the process above as
the generator is successively scaled to fit polygon sides.
An alternative procedure for generating the
fractal sequence of Figure 2 employs a single shape, a “4-star graph,”
when translated across the plane creates the same pattern as in Figure
sequence of images in Figure 3 illustrates the process: begin with a
(Figure 3a), apply a shift it to the right (Figure 3b), continue
shift the image in the same direction as the original transformation
Then, using a shift orthogonal to the first one, apply it to a 4-star
shift it (Figure 3d), continue stretching to shift the image in the
direction as the original transformation (Figure 3e). Finally, use the
shift directions to fill in the rest of
space (Figure 3f).
a geometric interpretation of a linear algebra
context, the shift directions function as eigenvectors
of a transformation to create the
pattern when they are
applied to a 4-star graph. The eigenvalues of the transformation are
amounts the eigenvectors are stretched or shrunk. In this example, one
consider each basic eigenvector to have length 1, in which case other
eigenvalues would be the set of positive and negative integers. If,
this synthetic geometric form is superimposed on some underlying
system, then the eigenvalues might be different, although still
multiples of the basic one. Notice that the visualization of Figure 3
these vectors are “lurking” in Figure 2—they are the edges of the
which the generator is applied!
a. Initial 4-star.
b. Vector to shift pattern.
c. Successive application of vector
d. Vertical shift.
e. Successive application of vertical vector.
f. Use of the pair of vectors to fill in pattern.
sequence in Figure 3 generates the first layer
of pattern. To illustrate how to create a subsequent layer, and all
focus on a single generated polygon (teragon) from Figure 3f (shown in
Begin, again, by introducing a single small 4-star graph (Figure 4b).
Apply a vector
to transform it (Figure 4c). Continue the application in the same
positive or negative (Figure 4d). Apply an orthogonal vector to move
graph (Figure 4e). Continue the application of that vector in positive
negative directions (Figure 4f). What
causes the shifts here are the orthogonal vectors; in this case, the
in selecting the initial pattern, the 4-star, rather than in selecting
generator. Here, generators are directed
straight line segments.
b. Initial 4-star.
c. Vector to shift pattern.
d. Successive application of vector.
e. Orthogonal shift.
f. Successive application of orthogonal vector.
of the vectors in Figure 4 expand
pattern to fill space. Figure 5a shows this space filling together with
pattern of basic vectors in relation to each other: both orientation
length. Figure 5b shows that the process creates 8 scaled-down teragons
to the first teragon. Notice that the
full pattern, and not merely its envelope, is generated by this process.
This alternative strategy is faithful to the
initial ideas involving self-similarity.
Left (a): One set of two orthogonal eigenvectors is used to generate
each layer of pattern. Right (b): The required eight self-similar
regions are displayed, suggesting the method for continuing the
infinite process, using 4-star graphs. It is identical to Figure
2c, generated in the classical manner.
6 shows that the classical method for fractal generation using a
generator on an initiator yields the same result as using shifts
induced by vectors.
|Figure 6. Figure 5b, from the
vector approach, is overlain on Figure 2c from the classical approach.
ideas suggested in the iteration sequences in
Figures 4 and 5 might be viewed as broadly similar
to downtown planning: fill an area
with buildings as allowed by zoning, and put in as many roads as
facilitate movement; superimpose a park if desired, but as an overlay
fundamental strategy; and, use the value of D = 2 as the upper bound
which space-filling by roads is not permitted, except in situations
are multiple vertical layers of street network (beyond the scope of
might determine, in advance, the extent of
permitted space-filling by a road network as, perhaps, a new class of
Traditional zoning basically mandates such filling by buildings and
types. Then, one might use the appropriate fractal dimensions,
absolute (rather than as relative) values based on the iteration of
self-similar generators applied to an initiator. Figure 7 illustrates
might interpret these ideas in a simple landscape, much as Burgess
simple example, based on the mathematics readily available to him.
the central ring of Figure 7, the street pattern
envelope is generated by the iteration sequence of Figure 1; the
initiator is shaded a solid yellow color and is identical to the
initiator in Figure 1. The second frame in the animation sequence in
Figure 1 is displayed in Figure 7 as a blue shape. Grid pattern
is filled in with fractal iteration, as suggested in the animation of
Figure 1. Here, the sequence employed
leads to D = 2
when carried out infinitely (lines would fill everything).
It is truncated here after two steps and its edges form the suggested
road network with streets becoming successively narrower as the pattern
the outer ring of Figure 7, the street
pattern envelope is generated by the iteration sequence of Figure 2;
The initiator is shaded solid orange. The
second from of the animation yields the brown shape--follow the
generator edges with your eye. Again, narrower streets come from
the edges of smaller (later) iterates. Grid
pattern is filled in through successive generator application. Here the sequence
employed leads to D =
1.0619248 when carried out infinitely (there
gaps between lines—some space would not be filled).
two iteration sequences displayed in Figure 7
mesh quite well at the boundary separating one region from the other.
roads from the core region flow naturally into roads in the rim. Some
more local angled roads in the core region flow into the intersections
larger roads along the ring perimeter. These locales might be natural
install traffic circles (avoiding costly traffic lights). In Figure 7,
suggest locations for traffic rotaries, with the largest ones marking
where the largest roads come together, and so forth, creating a
rotaries marked by stars of various sizes. Others
of the more local angled roads do not
continue into the outer
ring. These roads become natural cul-de-sacs, and offer privacy or
needs to specific land use types in the core. In
Figure 7, circles mark the locations of
cul-de-sacs arising naturally in the shift of the underlying grid
might follow the form created by the meshing or failure to mesh of the
underlying geometry, giving parcels on cul-de-sacs special zoning that
privacy or security concerns.
Fractally-bounded parcel and road pattern development. The
iteration sequence of Figure 1 generates the road network in the
central circle. The iteration sequence of Figure 2 generates the
road network in the outer ring.
two-ring pattern displayed in Figure 7 might of
course be extended to include more rings, reflecting variety in zoning
In such extension, traffic circles/rotaries would emerge at boundaries. Cul-de-sacs would emerge in moving from one
ring to another, and, again, zoning patterns might follow their
generating sequence for the extension of pattern
might be as follows: create a fractal pattern that generates a road
then truncate this pattern generation in accordance with parcel size as
determined by land use and zoning. Then, align patterns at the edges of
generated adjacent rings. Rings are but one pattern.
Oddly -shaped regions might offer different challenges, but still
the same style of argument might be employed. Typically,
the adjacent patterns do not mesh because
different iteration sequences, reflecting variety in space filling
land use, are employed. One approach, as suggested in Figure 7, might be to align core road network boundary
points and insert traffic circles where appropriate. The left-over
become cul-de-sacs, which might be used to advantage for privacy needs
residential land uses, for industrial uses wishing a relatively low
for a future society less dependent on frequent use of a personal car
and perhaps more dependent on delivery vehicles of various sorts.
unlike the case of much contemporary urban planning, good reason exists
cul-de-sacs—as a useful protrusion, unlike an appendix, of an
The case of Figure 7 was generated using the geometric animations of
Figures 1 and 2. They might equally be generated using the
alternative eigenvalue approach. To determine which approach is
most effective in varying situations is an open question and one that
is currently under study. The answer may lie in what technology
is available to those executing the visualizations.
space-filling needs of different zoning
categories produce distinct neighborhoods of road network. Within-ring
accessibility along a grid network is encouraged, while
accessibility is reduced, perhaps fostering a safer environment. Such
approach also might offer a visual “interest” factor in an otherwise
overall grid pattern. Naturally, one might vary the urban model, the
partitioning urban areas, and the general planning context (beyond
land use). Fractally-bounded
characterizations might help to guide real-world planning process when
it is integrated into any conceptual model for urban planning.
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Employment, and the Economy (Chene Street History Study); Daniel
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Ashbel Smith Professor in the School of Economic, Political and Policy
Sciences, at the University of Texas, Dallas.