Magdy M. Ibrahim and Robert J. Krawczyk
Illinois Institute of Technology
College of Architecture, Chicago, IL USA
- Abstract
- 1. Introduction
- 2. Creating a fractal
- 3. Shape of generators and initiators
- 4. Direction and proportion
- 5. The meaning of the generator
- 6. Observations
- References
This paper investigates an approach
to the use of fractals in architectural design. Two major aspects are
discussed. First, the effect of the direction of the line segments on the
generated fractal, as well as, the proportions between the generator and the
initiator. Secondly, the meanings attached to the lines segments that
constitute the fractal are based on some known architectural organization. The
goal is to develop a basis for using fractals to suggest architectural forms.
A fractal is an object or quantity that
displays self-similarity on all scales. The object need not exhibit exactly the
same structure at all scales, but the same “type” of structures must appear on
all scales (Weisstein 1999). Fractals were first
discussed by Mandelbrot (Mandelbrot
1978) but the idea was identified as early as 1925. Fractals have been
investigated for their visual qualities as art, their relationship to explain
natural processes, music, medicine, and in mathematics (Pickover
1996).
Examining fractals, we were able to
classify them into two major categories depending on the way they are created,
and the mathematical method used to calculate them. From the drawing method
point of view, the first is line or vector fractals. These are generated from
the replacement of a group of vectors, such like the Dragon Curve, as in Figure
1a. The second are fractals that are generated as a group of points in the
complex plane, such as the Mandelbrot set and the
Julia set, as in Figure 1b.
From the mathematical point of view,
we can classify fractals into three major categories. The first, IFS, iterated
function system, like Koch Snowflake, Cantor set,
Fig. 1. Types of fractals.
In this research, we are interested
in the vector-based fractals and the use of the replacement concept and the
iterated function system as a way to generate them. These fractals have
directional and geometrical properties that make them possibility suitable for
applications in architecture. Vector-based fractals can be described in terms
of vertices and the lines connecting them. This has the potential to be used
directly as architectural elements or to simply use the vertices to define the
locations of such elements.
Chris Yessios
with Peter Eisenman (Yessios
1987) was among the first to write about utilizing fractals and fractal geometry
in architecture. Yessios described a way computers
can be introduced to architectural design as an explorer and generator of
architectural forms. He used the fractal geometry, arabesque ornamentation and
DNA/RNA biological processes as fractal generators. A fractal program was
developed that enabled him to use several generators on the same base and to go
many steps forward in the iteration process, as well as, backward. The project
that was used in this investigation was a studio project to design a building
for a competition given a specific
architectural program.
More recently, S. Durmisevic and O. Ciftcioglu (Durmisevic and Ciftcioglu 1998)
discussed another approach on how fractals can be used in the architectural
design. They proposed to use a fractal tree form as an indicator of a road
infrastructure and another fractal to determine the type of architectural forms
to place along this transportation spine. The fractal geometry was used for
architectural forms and urban planning. No other similar studies could be found
where fractals were used as architectural form generators to suggest
three-dimensional forms. The results from these two suggest that the explicit
self-similarity and the repetition of the same shape may distract from
developing interesting architectural forms, so both developed means to modify
the replication.
A vector-base fractal is composed of
two parts: the initiator and the generator. For example, the Koch
Snowflake starts with an equilateral triangle as the initiator. The generator
is a line that is divided into three equal segments. The middle segment forms
an equilateral triangle as in Figure 2.
By replacing every line of the
initiator with the full generator, we get the first iteration of the snowflake.
By iterating this operation again and again, replacing every line of the new
initiator with the full generator, we end with a figure that approximates a
snowflake. The iteration process should continue to infinity to generate a real
Koch Snowflake fractal, but as we are interested in the evolving form, we only
iterate the function for some finite number of times. Figure 3 displays the
Koch Snowflake with 3 iterations. If the generator is changed, inverted, we can
develop an entirely different form, the Koch Antisnowflake
as in Figure 3.
Fig. 2.
Generator and initiator.
Fig. 3. The Koch Snowflake and the Antisnowflake.
3. Shape of generators and initiators
There is a group of fractals that
have been formally identified, including Mandelbrot
set variations as described in his book “The Fractal Geometry of Nature” (Mandelbrot 1983) that depend on the concept of replacement.
Some of the IFS fractals are: Cantor Set,
Fig. 4. Tree fractal, Cesàro fractal, Barnsley's Fern,
Dragon Curve, H-fractal, Sierpinski curve, square and
triangle.
All of these fractals are based on
simple geometric shapes. Shakiban and Berstedt (1998) discussed a new generating procedure based
on vector calculus and modular arithmetic to generate the Koch Snowflake. The
procedure was then applied to create more generalized snowflakes rather than
the triangular classical snowflakes. They also suggested the use of n-sided
polygons, such as pentagons as initiators. In all cases the replacements made
at each iteration are consistent and unvaried.
In all these fractals, the generator
and the initiator have no specific meaning. They are mostly based on simple
geometric shapes: lines, squares, or triangles that are able to produce an
interesting arrangement. In his book, Mandelbrot
mostly used squares in his fractal generation. Mandelbrot
was interested in the calculation of the fractal dimension, and as he produced
interesting patterns of fractals, he was concerned about the comparison between
the fractal dimensions of each generated fractal. Although, as described by Carl Bovill (1996), the fractal
dimension has a meaning in evaluating the visual richness or density, for
architects it has less meaning in the generation of new forms. These shapes
have no real meaning in architecture.
Yessios’s (1987) implementation of fractal
generation was highly interactive and allowed a fractal to be developed one
iteration at a time or at multiple increments. At the same time, generators
could be changed, replaced, deleted, or inserted, at any iteration. The
generation process could go forward and backward allowing the designer to
return to an earlier state. The fractals resulted in three-dimensional models
but no explicit meaning of these generators or initiators was shown. No
indication whether they were site boundaries or circulation axes, or any other
architectural organizational element. Differing from traditional fractal
generation, this approach enabled a designer to interfere with the generated
form at any time of the process. This is a major deviation from the basic
concept of the fractal generation as an uninterrupted process. As an
experimental design tool it does offer a variation not considered by others.
Another interesting method to modify
the geometric shapes produced by a fractal comes from the random selection of
the direction and displacement of the initiator. Mandelbrot
discussed this when he was describing the random Koch coastline, and Brown
fractals and the random midpoint displacement curves. Similar application in
architectural fractals was not found.
As we started to investigate the
selection of the generator and initiator, our initial focus was on the effect
of proportion and direction of the initiator on the produced fractal as related
to the generator. Normally, the length of the generator is equal to a segment
of the initiator and the direction of the line segments in the generator and
initiator are the same as seen in the Koch Snowflake in Figure 3.
One possible variation is to modify
the direction of the entire generator or initiator or individual parts of each
and the orientation of the generator as it is placed on the initiator. Figure 5
displays the Koch Snowflake with the generator in one direction and normal and
reversed direction for the initiator. When the initiator is reversed the Antisnowflake appears. Figure 6 reverses the direction of
the placement of the generator. The fractals developed starting at the second
iteration are undocumented versions of the Koch Snowflake. Additional
variations were found by modifying the direction of individual line segments
within the generator or the initiator.
In addition to direction, the
proportions of the line segments in the generator were modified in relationship
to the initiator. To demonstrate this concept, we used a square initiator and
the normal Koch Snowflake generator. Figure 7 shows fractals that are based on
generators that range from 25% to 75% of the size of the initiator.
Fig. 5.
The direction effect on the generated fractal applied on the simple Koch
Snowflake.
Fig. 6.
The effect of the direction of the placement of the generator on the generated
fractal applied on the simple Koch Snowflake.
Fig. 7. The proportion effect on the generated fractal applied on the simple square.
5. The meaning of the generator
As previously noted, the classical
fractals have no specific meaning associated to their shapes; they are simply
forms that generate interesting fractals. Selecting the generator can be based
on architectural organizational schemes used as major axes of a site planning
or a building, or using the patterns discussed by Francis D.K. Ching (1979) in organizing spaces.
Another approach to develop a
generator is to use studies such as Durand’s. In the book “Précis
des lecons d’architecture”,
Jean L. Durand (1802) describes a compendium of neo-classical design rules.
Cross axes, grids, squares and circles are taken as primitive shapes. He
demonstrates, by diagrams, how to assemble these, into symmetrical skeletons of
construction lines, Figure 8. Durand’s rules do not provide a complete,
consistent specification of a classical architectural language, but are a
straightforward method to develop them into a grammar that does.
We selected some of these diagrams
and defined them as the initiator for the generation of fractals based on the
Koch Snowflake. The goal was to investigate if the fractal generated from them
might then be related to a rational organizational pattern.
Fig. 8. Durand
neo-classical design rules.
In addition to these initiators, we
also applied the proportion and the direction studies to them. The created
fractals now are determined by two factors: the proportion of the generator to
the initiator, and the direction of the vectors of the generator. Figure 9
displays a series based on these parameters.
It was observed that some of the
shapes have changed after several iterations into some different organizations.
The square with two horizontal compartments was converted into a three
compartments shape rather than two. And other shape that had a vertical axis
was converted into a shape that have horizontal axis of symmetry. These
generators have different vector directions, starting from all vectors right to
all vectors left.
Fractals have a great visual
richness, but despite that, they cannot contribute to the simulation of
architecture forms unless they can have meaning associated to the components
used. The concept of utilizing the fractals, as a form investigating technique
can be useful if they are rationalized so that they relate to architectural
elements like axes or masses, some architecturally related organizational
scheme.
There are more than the regular
fractals that can be produced with the same iteration concept. Investigating
the directional effect of the generator can lead to some unidentified fractals
with different visual characteristics, as well as, the change in the
proportions of the generator and the initiator. These variations indicate a
possible approach for architectural form development.
Continued research will focus on
possible meanings of the initiator and methods of transforming these fractals
into three-dimensions.
Fig. 9.
Durand’s design rules with the different generators applied to them.
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