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1. An outline of the important properties of the symmetries
of regular polygons;
Let 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 D_{n} for some
integer n; and the cyclic groups denoted by C_{n}.
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 D_{n} has 2n elements or operations,
while C_{n} has n elements, and so is of order
n. For example, the symmetry group of the square is the
dihedral group D_{4} 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.
C_{4} 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 c_{4} is included in D_{4}, mathematicians
say C_{4} is a subgroup of D_{4}. And in fact
C_{n} is a subgroup of D_{n} 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 D_{4} 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 C_{4} 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, D_{3} is the group
of symmetries of an equilateral triangle, which has order 6 with
its D_{1}, C_{3} and C_{1} subsymmetries.
Furthermore, we can generalize the lattice diagram of the regular
polygon, which shows its hierarchical order of subsymmetries
(Figure 3).
Gottfried
Semper: In Search of Architecture (Cambridge, MA: The MIT
Press, 1989), p.219. return
to text[4] F. L.
Wright, 'In the Cause of Architecture: Composition as Method
in Creation', unpublished essay, 1928. return to text[5] See
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
to text[6] R. 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[7] 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.
return
to text [8] In 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[9] 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[10] Park
, "Subsymmetry analysis of architectural design: some examples"
(see note 5) systematically analyzed the two buildings in terms
of their symmetrical hierarchies. return to text[11] Park (1999) provides a detailed study on symmetric operations in the Schindler Shelter (1933-1943). return to text[12] 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[13] 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[14] 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[15] 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
to text[16] Schindler's 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[17] L. March, "A Class of Grids", Environment and Planning B: Planning and Design 8 325-382,
1981. return
to text[18] See P. Klee, Pedagogical Sketchbook, Faber and Faber Limited,
London, 1953. return
to text
- 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 Sketchbook, London: Faber and Faber, 1953).
Architectural Research Quarterly)
and "Subsymmetry analysis of architectural designs: some
examples" (Environment and Planning B: Planning and Design).
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