Analysis and Synthesis in Architectural Designs:
A Study in Symmetry 
School of Architecture
University of Hawaii at Manoa
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Honolulu, HI 96822 USA
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Ordered designs are frequently encountered
in art and architecture. The underlying structure of their spatial
logic may be discussed with regard to the use of symmetry principles
in mathematics (Weyl, 1952). In architectural designs, the use
of symmetry may sometimes be apparent immediately by just looking
at designs, although the final design is seemingly asymmetrical;
or various symmetries are manifested in the parts of the designs,
yet not immediately recognizable despite an almost obsessive
concern for symmetry. At this point, it is crucial to develop
a formal methodology, which may clearly elucidate different hierarchical
levels of the use of symmetry in architectural designs.
an effort to do this, before proceeding to analytic and synthetic
applications, we discuss a methodology founded on the algebraic
structure of the symmetry group of a regular polygon in mathematics.
The approach shows how various types of symmetry are superimposed
in individual designs, and illustrates how symmetry may be employed
strategically in the design process. Analytically, by viewing
architectural designs in this way, symmetry, which is superimposed
in several layers in a design, becomes transparent. Synthetically,
architects can benefit from being conscious of using group operations
and spatial transformations associated with symmetry in compositional
and thematic development. The advantage of operating with symmetry
concepts in this way is to provide architects with an explicit
method not only for the understanding of symmetrical structures
of sophisticated designs, but also to give architects insights
for the construction of new designs by using symmetry operations.
The objective of this paper resides in searching
out the fundamental principles of architectural
composition. A study of the fundamental principles of spatial
forms in architecture is an essential prerequisite to
the wider understanding of complex designs as well as the creation
of new architectural forms. There are many
ideas about what the fundamental principles of form are. I stand
by Goethe's theory of metamorphosis in The Metamorphosis of
Plants. His theory centered
on the notion that there may be an ideal form or urform,
key to understanding the subsequent development
of forms. Based on the urform, a variety of new designs
can be developed. Later on, Gottfried Semper used the term "tectonics"
to describe the theoretical viewpoint of the universal principles
of forms in nature and in art, writing, "Tectonics is an
art that takes nature as a model - not nature's concrete phenomena
but the uniformity [Gesetzlichkeit] and the rules by which
she exists and creates". He was convinced
that the understanding of fundamental principles would aid the
generation of new creative work through modification, transformation
and development. In his article "In
the Cause of Architecture: Composition as Method in Creation",
Frank Lloyd Wright laid emphasis on the study of the principles
of composition, claiming that "geometry [is] at the center
of every Nature-form we see".
By looking into nature and grasping the fundamental principles
at work, architectural forms that are not imitative but creative
can be developed.
In this paper, I substantially
recount the methodology employed in my previous papers  and then
create a new design in order to demonstrate the application of
the method in formal composition. This paper is composed of the
following three parts:
1. An outline of the important properties of the symmetries
of regular polygons;
2. An analysis of an existing architectural design;
3. The synthesis of an abstract design with respect to the symmetries
associated with the square.
THE METHODOLOGY OF SYMMETRY
Symmetry operations are concerned with
spatial displacements or motions which take a shape and move
it in such a way that all the elements of the shape precisely
overlay one another, so that, despite the displacement, the shape
appears not to have moved from its original position. For example,
if we rotate a square 90 degrees around its center, it appears
unchanged, whereas if we rotate an equilateral triangle 90 degrees
around its center, it would appear to have changed its position.
The motion of rotation through 90 degrees is called a symmetry
operation of the square; it is not a symmetry of the triangle.
Mathematicians call the collection of all the symmetry operations
or motions that leave a particular geometric object fixed its
symmetry group. The symmetry group of a two-dimensional design
can be finite or infinite. Infinite groups are the symmetries
of infinite patterns such as tilings or infinite wallpaper patterns.
Infinite symmetry groups include the motion of translation which
is a shifting of the entire pattern one unit. (think of moving
an infinite tiling of squares one unit . Each square would lie
on its neighbor; the appearance of the tiling would be unchanged.)
Point groups are finite symmetry groups and correspond to a finite
design, such as a single square. Only eight operations can be
performed on the square and leave it occupying the same position
in space: four rotations through the point at its center and
reflections in four lines through this point.
us provide an elementary account of the mathematical structure
of finite symmetry groups in particular, the point groups
in two dimensions. There are two types of finite point groups:
the dihedral groups denoted by Dn for some
integer n; and the cyclic groups denoted by Cn.
The spatial transformations of a dihedral group comprise rotation
and mirror reflection; the cyclic group contains rotation only.
The point groups have no translation. The number of symmetry
operations in a finite group is called its order. The
symmetry group Dn has 2n elements or operations,
while Cn has n elements, and so is of order
n. For example, the symmetry group of the square is the
dihedral group D4 of order 8. The eight distinguishable
spatial transformations, which comprise this group, are four
quarter-turns and four reflections, one each about the horizontal
and vertical axes and the leading and trailing diagonal axes.
C4 has four spatial transformations: the four quarter-turns.
It is the symmetry group of an object like a pinwheel, which
has rotational but not mirror symmetry. Since every symmetry
operation in c4 is included in D4, mathematicians
say C4 is a subgroup of D4. And in fact
Cn is a subgroup of Dn for any n.
We can diagram the relationship between the different subgroups
of symmetries of a square (Figure 1). In the diagram, we color the square
in ways that destroy some of its symmetries. Figure 1 illustrates
all possible subgroups of symmetries; the dark square has the
full symmetry group D4 with 8 symmetry operations.
The three squares in the second row each have four elements in
their symmetry group. For example, the first square has reflection
in vertical and horizontal mirrors, and rotation through 180°
and 360°. The second square in this row, with the pinwheel
design, has symmetry group C4 of rotations through
90°, 180°, 270° and 360° degrees. The symmetry
groups of the square in the third row each have two elements,
while the square in the bottom row has only the identity motion
in its symmetry group, that is, the no rotation less than the
full turn through 360°. The structure of the diagram can
be accounted for in two ways: from top to bottom, symmetries
are "subtracted" from the full symmetry of the square;
conversely, from the bottom to the top, subsymmetries are "added"
to achieve higher orders of symmetry. These two opposing but
complementary ways of using the diagram support our approach
to the subsymmetry analysis and synthesis of architectural design.
The lattices of subsymmetries of other polygons such
as an equilateral triangle, pentagon, etc. can be considered
as well in this hierarchical order as shown in Figure 2. A polygon with n
edges has at most dihedral symmetry of order 2n, where
the order of a finite group is the number of elements, here the
number of symmetry operations. The subgroups of the symmetry
group of a regular n-gon are ordered in the lattice diagram
as shown in Figure 3. For instance, D3 is the group
of symmetries of an equilateral triangle, which has order 6 with
its D1, C3 and C1 subsymmetries.
Furthermore, we can generalize the lattice diagram of the regular
polygon, which shows its hierarchical order of subsymmetries
COMPOSITION IN THE ARCHITECTURE OF RUDOLF
used proportion and symmetry to govern the spatial organization
of his architecture. His proportional method was explicitly described
in his article, "Reference Frames in Space".  Schindler chose a 48-inch (4-foot) increment
as his unit module. Based on this simple computation of the given
module together with its multiples and fractions (1/2, 1/3, and
1/4), it was possible for him to derive all necessary architectural
dimensions not only for major rooms but also for minor details,
including doors and furniture. Using such a proportional system,
architects can think, calculate and draw architectural space
without mechanical tools.
Schindler's debt to symmetry, particularly the hybrid
use of various subsymmetries in a single project, is astonishing.
Although he never wrote any article on symmetry, symmetries are
deeply imbedded in his designs throughout his career. Whereas
Schindler's early designs rely mainly on rigorous orthogonal
planning, his later works seem to deny any consistent principles
of symmetry. However, deeper investigation reveals that this
is not so; for example, in his later works, he increasingly exploited
the diagonal axis. The application typifies the "butterfly
patterns: a design, which split into two wings with certain angles
along the main axis".
The interweaving of ideas of proportion and symmetry
is one of the major compositional tools throughout Schindler's
career.  The Free Public
Library Project (1920) and the How House (1925) stand out from
Schindler's other work during this period in their conscious
play around the diagonal axis and a 48-inch unit system. Whereas
the Library project is composed through the superimposition of
a variety of subsymmetries without using the pure rotational
subsymmetries such as C4 and C2, the How
House is mainly determined by reflective symmetry along a diagonal
axis 6  (Figure 4). Later on, in the Schindler Shelter, he utilizes
symmetry operations, in particular rotation and reflection, to
generate plan prototypes as well as their variations.
In the analysis that follows,
we isolate individual elements of the design that rely on a symmetrical
order to identify the local and overall symmetry of the project.
Whereas in most cases the whole design is seemingly asymmetrical,
despite an almost obsessive concern for symmetry in its individual
parts, the analysis demonstrates how various symmetry operations
are involved in each of the parts of the design, revealing the
underlying structure of its spatial order. Thus, the architect's
use of symmetry principles in the design becomes more evident.
THE ANALYTIC EXAMPLE
In this section, we take an architectural project
that appears to use symmetry and scrutinize the plan to discover
its underlying symmetry structure. Schindler's Popenoe House,
designed in 1922 but later demolished, is the subject of our
analysis. To begin the analysis, I examined the archival materials
on the house, including sketches, drawings, details and photographs,
housed at the Architectural Drawing Collection, University of
California, Santa Barbara. I then enhanced them through the reconstruction
of new drawings. This research process is essential to understand
the formal and spatial aspects of the work, to clarify Schindler's
design ideas, and to evince the application of his methodology
in formal composition.
The single story desert cabin was planned for Paul
Popenoe at Coachella, California. It is one of three cabins (the
cabin for Popenoe, the cabin for Philip Lovell, and the Carton
Park House) that Schindler developed during the period. The basic
character of the designs derives from that of the Kings Road
House (1921-2) where "a marriage between the solid permanent
cave and the open lightweight tent" is a key feature. In fact, the three cabins have in common their
designs and the use of materials (in Gebhard's words, "the
cave-tent shelter of concrete, wood and canvas"), which
relate the projects to the climate, region and the surroundings.
The Popenoe cabin is a single
room type (Figure 5). The living room, located in the center of
the house, is surrounded by various activity areas, including
kitchen, bedrooms and utility room. The rooms are divided by
thin internal walls and sliding doors, providing spatial flow
and flexibility within a minimum area and optimizing the use
of interior space. Four porches, each with its own entrance door,
are situated in a spiral manner around the main body of the house,
providing intermediate spaces for an interaction between interior
The design on a rectangular site
is extremely simple, both structurally and spatially. However,
in its transparent interplay of forms, it is one of the most
striking examples of Schindler's lifelong use of proportions
and symmetry to govern spatial organization. The fundamental
48-inch unit system and symmetry operations determine all the
major decisions of the spatial organization as well as the architectural
The primary layout of the house lies
within a 22-foot by 22-foot square, over which is laid a 48-inch
module grid system. The sides of the square are subdivided into
6-foot, 10-foot and 6-foot segments, concentrically, which produces
an A-B-A rhythm of the parts that relate to each other as 3:5:3.
The concentric spatial schema forms an absolute four-fold symmetry
with the square, invoking a classical design vocabulary. The parti, or underlying geometrical
scheme, is a tool that governs spaces with regularity and shows
the extreme clarity of its geometrical origin (Figure
on the parti, the major spaces of the house are juxtaposed.
The seemingly random spatial arrangement of the final design
is in contrast with the complete geometric regularity of the
parti, and its overall spatial configuration is asymmetric.
Four additional screened porches, indicated on the
plan as Living, Sleeping, Kitchen, and Dining, connected by corridors,
are disposed in the pinwheel type of C4 symmetry around
the central square plan. These wrap around the primary square.
The four porches and corridors are designed so that they can
be closed sometimes. The length of the porch wings increase in
increments of 3 feet, 4 feet, 6 feet, and 10 feet as one moves
clock-wise around the house. The entire floor plan forms a spiral
shape, which reinforces the impression of rotation. Whereas the
basic composition of the four porches seems to derive from the
strict pinwheel type of C4 symmetry, the final design
is not absolute (Figure
7). The architect's original idea for
the composition is clearly shown in Figure 7b, where four porches,
including the corridors, entrance doors and window openings,
are determined by C4 cyclic symmetry. Thus we see
how the asymmetric final design derives from a disciplined understanding
of the principle of rotational symmetry and is not merely arbitrary.
The fireplace is set along
the D1 diagonal axis (Figure
diagram in Figure 8a illustrates that the diagonal of the square
defines the axis of the living space. The fireplace could have
been located on any side in the living room, yet Schindler has
chosen to place it in the corner of the living area. The diagonal
setting provides the dynamic sense of spatial depth and spatial
experience of openness. In addition, the details of the fireplace
reiterate the architect's use of the diagonal symmetry as shown
in Figure 8b.
The symmetric possibilities of the
design are further evidenced by the arrangement of the ceiling
structure. It reinforces the cyclical character of the design.
Furthermore, above the ceiling structure, a rectangular volume
in part of the living room, kitchen and clothes closet is raised
3 feet, which is highlighted as indicated in Figure 9. It is
set along an orthogonal axis of D1. Functionally, this accommodates
clerestory windows to bring light into the center of the house.
The whole floor plan exhibits
only C1 symmetry. It can be said that in the final
design, there is an abundance of symmetries within the parts
while the strict symmetry of the whole is negated. This approach
is outside the classical design lexicon, in which the strict
symmetrical disposition of the building parts is emphasized and
where the combination of local and global symmetry is the driving
force behind the organization of the design. As demonstrated
in this analysis, although the plan does not exploit all the
possibilities of the subsymmetries of the square, various subsymmetries
with rotation and reflection are stratified into a single story
design, extremely rare in architectural design. It appears that
Schindler initially set up a symmetrical frame for the project,
only to break it, arriving at a final, asymmetrical design that
meets all the functional requirements.
AFTER ANALYSIS, SYNTHESIS
The underlying reason
for analysis is to be able to do synthesis. In this section,
we build up a single building design, making use of the symmetries
and subsymmetries of a square motif. Then, all the subsymmetries
with their hierarchical structure are made evident in individual
floors of a new project, including the full symmetry group D4
of the square and the identity.
First of all,
we need to choose a minimum building element. Essentially we
want to provide the clearest possible picture of the design as
a whole. The minimum element is composed of dots representing
columns and rectangles representing floors. And then, as was
done in the analytic example, we take the subsymmetries of the
square with a grid system. The subsymmetries of the square are
used as compositional tools. A grid system is used as an underlying
parti to juxtapose building elements. Any other regular grids
or tessellation can be used but a 'dimensional grid' is implemented
in our exercise. In his paper "A Class of Grids", Lionel
March defines and catalogues a series of grids where he regulates
"three linear elements A, B, C. which had distinct integer
dimensions with a < b < c. The grids were designed to accept
permutations of these elements".
Among many distinctive possible composites, the permutation of
the 3, 4, and 5 linear elements which produce a polyrhythmic
grid, 3 + 1 + 1 + 2 + 1 + 1 + 3, is chosen (Figure
this exercise it is assembled in a 5 x 5 array in this exercise
(Figure 11a). The square of the grid has D4
Placing the element on top of the grid,
we arrange columns and floors differently at each level to form
symmetric designs illustrating different subgroups or subsymmetries
of D4. Each floor level is designed in terms of different
subgroup or subsymmetry of the square. Then, we stack the levels
up, floor by floor, in a standard height, creating a 3-D schematic
We start at the lowest level
of the symmetrical order. The first floor consists of the full
symmetry of the square, which is D4 symmetry (Figure 11b). Columns and floors are set along the full
symmetry of the square with four rotations and four reflections.
The second floor illustrates the pinwheel type rotational design
characteristic of C4 symmetry (Figure
positions are the same as the previous D4 configuration,
but the floors are arranged based on the four quarter-turns of
C4 symmetry. Compared to level one, floor planes are
shifted to maintain rotational symmetry, while destroying the
mirror symmetry of the first level.
level represents the half-turn C4 symmetrical design
(Figure 11d). Although columns are set as for D2
symmetry, the two floors are set along the half-turn rotation,
breaking the mirror symmetry. The fourth floor incorporates the
two mirror reflections, which form the subgroup D2
of D4 (Figure 11e). Columns and
floors are set along two mirror axes, which are perpendicular
to each other. The fifth floor is composed using a single mirror
axis (Figure 11f). The design looks similar to the level below,
but upon a closer inspection, we see that columns are set along
the D2 orthogonal axis, and floors are shifted from
the central axis destroying the vertical mirror axis. The sixth
floor illustrates a second D1 symmetry group (Figure 11g). Here the mirror axis is placed at a diagonal
to the grid. The top-level floor (seventh floor) has only 360°
rotation as a symmetry, which is denoted as C1 (Figure 11h). Mathematicians would say this level has only
the trivial symmetry group.
The final design achieved
by the synthetic method appears in a three-dimensional representation
in Figure 12.
have described and demonstrated methods of analysis and synthesis
in architectural composition, based on the algebraic structure
of the symmetry groups of a regular polygon. Using this method,
the Popenoe House has been analyzed to show a unique application
of symmetry operations with regard to the spatial organization.
In the design, Schindler explores a variety of subsymmetries
superimposed one upon the other around a single central point
in an architectural design.
A similar geometric
order and symmetric technique is applied to the generation of
a new design. However, the synthetic process differs a little
from the analytic one. Rather than a simple rhythmic grid of
the same unit , we have used
a more complex polyrhythmic grid derived from the permutation
of three distinctive linear elements. Also, whereas the analytic
example shows the superimposition of various subsymmetries in
a single floor plan, the synthetic design has a different type
of subsymmetry in each floor plan. The synthetic example is an
experimental design using the method of symmetry groups to generate
one among many other possible designs.
see that symmetry is an effective method for reading spatial
order of complex designs as well as for composing new designs
 A previous version of this paper
was published as "Subsymmetry Analysis and Synthesis of
Architectural Designs" in Bridges 2000: Mathematical
Connections in Art, Music, and Science Conference Proceedings
2000, Reza Sarhangi, ed. (Winfield, KS: Southwestern College,
2000), pp. 79-86. return
 J. W.
von Goethe, Versuch die Meamophose der Pflanzen zu Erklaren (Gotha)
translated by A. Arber (1946) Chronica Botanica 10 (2) 63-126,
 Quoted in W. Hermann, Gottfried
Semper: In Search of Architecture (Cambridge, MA: The MIT
Press, 1989), p.219. return
 F. L.
Wright, 'In the Cause of Architecture: Composition as Method
in Creation', unpublished essay, 1928. return to text
J. Park, "A Formal Analysis of R. M. Schindler's Free Public
Library Project", MA Thesis, University of California Los
Angeles, 1995; J. Park, "Subsymmetry analysis of architectural
design: some examples", Environment and Planning B: Planning
and Design, 27, 2000. return
M. Schindler, "Reference Frames in Space", Architect
and Engineer 165, 1946. This article was written in 1944
but published in 1946. Schindler noted that Frederic Heath "spurred
him to organize his idea of the unit plan." Much earlier
in his 1916 lecture note, Schindler already indicated that he
was concerned with a simple unit with its subdivision: "architect
to choose his own 'Unit', unit to be subdivided 1/2 and 1/4."
See R. M. Schindler, Lecture Note (manuscript), the Architectural
Drawing Collection, University of California, Santa Barbara,
1916. In this period he was also interested in the square unit
as well as the rectangle, triangle, and circle. He believes that
its application depends on "different expression of building".
But in his lifelong practice, unlike Frank Lloyd Wright, he never
used any other geometric forms of unit other than square. return to text
 In "Reference Frames in Space",
Schindler argues that the architect who wishes to deal with the
phenomenon of space has to have not only an innate talent, but
also a method which helps to visualize the space forms in his
mind, and to improve 'his mental image' of the space. The rationale
for Schindler's proportional system is that with the system,
the forms of space are freely conceived and precisely measured
in the architect's mind through the process of visualization.
his "Lecture Note" (see note 6), Schindler writes about
symmetry as the "simplest means of expressing an organism:
adjusting two units with one axis", and "relations
of two units". To him, symmetry is understood as the harmony
that results from the relationship of two units. This simple
definition might exert a deep influence upon his development
of the butterfly patterns of spatial organization. return to text
 See J. Park, "The Architecture of Rudolf Michael
Schindler-the Formal Analysis of Unbuilt Work", Ph.D. dissertation,
University of California Los Angeles, 1999. return to text
, "Subsymmetry analysis of architectural design: some examples"
(see note 5) systematically analyzed the two buildings in terms
of their symmetrical hierarchies. return to text
 Park (1999) provides a detailed
study on symmetric operations in the Schindler Shelter (1933-1943).
return to text
No explanation of the
circumstances surrounding the architect's intentions for the
project is provided by the archives, nor is there any description
of the project. For short explanations of the projects, see D.
Gebhard, Schindler, Peregrine Smith, Inc. Santa Barbara
and Salt Lake City, 1980; J. Sheine, R. M. Schindler,
Editorial Gustavo Gill, Barcelona, 1998. return to text
 Gebhard, Schindler, p. 52.
Gebhard writes some descriptions on the cabins, saying "Schindler's
unwillingness to use the machine to modify the climate and environment
worked far better in these vacation houses than his own Hollywood
house. All three were located in the dry inner valley between
the coast and the central desert, and they were not meant to
be used when the weather was either too cold or too hot."
return to text
 The idea of locality is a significant
issue in Schindler's space architecture in his lifelong practice.
In the discussion of the "location" , Schindler emphasizes
the relationship between "the parts and the whole"
where "nothing is alone, everything is connected relationships".
See Schindler, "Lecture Note" (see note 6). Subsequently
in his 1934 article "Space Architecture", Schindler
also emphasizes the significance of the building as "the
product of a direct impregnation by the nature of the locale",
criticizing Wright's sculptural approach for the Imperial Hotel.
See Schindler, "Space Architecture", Dune Forum,
44-46, 1934. Later on, in answer to a questionnaire from the
School of Architecture at the University of Southern California
(1949), Schindler describes the relation of house and lot: "The
conventional house is conceived as a solid mass growing out of
the ground. The space house as a space form becomes a part of
room formed by the lot, the surroundings, contours, and firmament."
Schindler's emphasis on the site is not self-contained manifestations
of physical environment. Then, in an unpublished article written
in 1952, he emphasizes again that "Since a composition in
space deals with the out-doors as its raw material, it is obvious
that the building should melt down into its surrounding that
these define the character of the interior as well"; see
Schindler, "Visual Technique", unpublished, the Architectural
Drawing Collection, University of California, Santa Barbara.
return to text
 A similar parti is found
in his early Free Public Library project (1920). The balcony
floor plan of the Library project is equally subdivided into
the nine squares where overall spaces are distributed concentrically
for functional necessity; see J. Park, "Schindler, Symmetry
and the Free Public Library, 1920", Architectural Research
Quarterly 2 72-83, 1996. return
use of pinwheel symmetry in an individual building is earlier
than that of Wright. Lionel March states that "Frank Lloyd
Wright does not use this symmetry for a parti until St.
Mark's Tower of 1929"; see L. March and P. Steadman, The
Geometry of Environment, RIBA Publications Limited, London,
1971. Yet, Wright used the symmetry much earlier in a housing
design, Quadruple Block Plan (1901). return to text
 L. March, "A Class of Grids",
Environment and Planning B: Planning and Design 8 325-382,
P. Klee, Pedagogical Sketchbook, Faber and Faber Limited,
London, 1953. return
FOR FURTHER READING. The following works cited in this
article can be ordered from Amazon.com by clicking on the title
- W. Hermann, Gottfried
Semper: In Search of Architecture (Cambridge, MA: The MIT Press, 1989).
- David Gebhard, The
Architectural Drawings of R. M. Schindler : The Architectural
Drawing Collection, University Art Museum, University of California,
Santa Barbara (Garland Publications, 1993).
- David Gebhard, Schindler (Santa Barbara and Salt Lake City:
Peregrine Smith, 1980).
- Paul Klee, Pedagogical
Faber and Faber, 1953).
SITES ON THE WWW
Mathematics and the Liberal Arts
Mathematics and the Liberal Arts
Symmetry Groups (Wallpaper groups)
and Tesselations for Children (links to 150 sites)
and the Shape of Space
tutorial on symmetry to download from Birkbeck College
Crystals and Polyhedra
of Polygons from Viewpoint of "Symmetry"
The Architecture of Rudolf Schindler
Buildings Online: Schindler
an assistant professor in the School of Architecture at the University
of Hawaii. He currently teaches architectural design studios
as well as a series of courses on Design and Computation. He
earned his BS in architecture from Inha University, Korea, and
his MA and Ph.D. in architecture from UCLA. He is the first recipient
of the R.M. Schindler Fellowship of the Beata Inaya Trust Fund,
1996/7, and twice recipient of the Chancellor's Dissertation
Fellowship, 1998/99. The focuses of his academic research are
on "The Architecture of R. M. Schindler - Unbuilt Works",
"Design and Computation", including Fundamentals of
Architectonics: Proportion, Symmetry, and Compartition, Shape
Grammars, Virtual Reality, and Digital Media. He is author of
numerous articles, including "Schindler, Symmetry and the
Free Public Library, 1920" (Architectural Research Quarterly)
and "Subsymmetry analysis of architectural designs: some
examples" (Environment and Planning B: Planning and Design).
The correct citation for
this article is:
Park, "Analysis and Synthesis in Architectural Designs:
A Study in Symmetry", Nexus Network Journal,
vol. 3, no. 1 (Winter 2001), http://www.nexusjournal.com/Park.html
Copyright ©2001 Kim Williams
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