Lionel MarchSchool of the Arts and Architecture University of California Los Angeles Los Angeles, CA 90024 USA
Nexus
Network Journal [2003]. While Schindler's "space reference
frame" is becoming better known [Schindler 1946], its relationship
to the "row" is only to be found in Park's recent investigation.
The theory of the "row" counters traditional proportional
notions, many of which are derived from the principle of geometric
similitude: a principle which is mostly represented in architectural
drawings by regulating lines and triangulation. Park gives examples
of this. Here, the simple mathematics of row theory is presented.
A short background note concludes the paper.
As an architect, Schindler had to choose a suitable module for his "unit." In most of his designs, he chose 48 in. This module might be divided, Schindler argued, by 1/2, 1/3, 1/4, without losing a "feeling" for the dimensions. In fact, Schindler used 36 in., 24 in. and 12 in. as refinements in plan, and the one third divisions, 16 in. and 32 in., almost exclusively in elevation. Consider a general module
After Robinson [1899], he suggested an interpretation for this sequence, or row, as intervallic ratios in a musical scale. Schindler suggested the general rule. If p/q is a term in a row, then the successor term is given by the rule: Thus, ( Schindler then gave another, abbreviated example: 1/3, ... 11/13, while Robinson, in addition, set out the rows beginning with 1/4 and 1/5. The fractional notation, Fig. 1. Gnomonic diagram for the row ( q
- p). The illustration shows p = 6, q =
8. Note the dotted diagonal at 45° and the bold line of length
2 which defines row (2).Now consider row (2): Every second term, read as a fraction, may be reduced to its coprime form:
These second terms belong to row (1), while the residual terms belong to a new kind of row in which the successor rule is Treating To avoid such multiple interpretations in the remaining sections
of this paper, the notation
If is
a raum in a row, then the first raum of the row is 1 x
( n) = { k x
(n + k) | k = 1, 2, ... }.A designer may consider a space, or raum
In doing so the designer will move to neighboring rows:
To generalize this; consider that
ranging from row( a = 2, 1, 0, -1, -2, and of the length b
= 1, 0, -1. There are (2.2 + 1)(2.1 + 1) = 15 neighborly rectangles,
including the original rectangle.Except for row(1) the rows include rectangles in which the two dimensions are not coprime. That is, they share a common factor. Thus in row(2), the rectangle 2x4 has the same ratio as the rectangle 1x2 in row(1). It is useful to extract the coprime rectangles in which the two dimensions are prime to one another as shown in Table 1.
Table 1. Examples of coprime raum where the two dimensions are prime to one another. Consider the simple row( This may be seen in an example using a graphic representation of anthyphairesis [Fowler 1999; March 1999a, b]. Simply put, this means subtracting successively the largest squares from a given rectangle and residuals until the procedure stops. The procedure will not stop if the rectangle is incommensurable, but this is not the case here. For example, consider the partial diagram of row(6): 1x7, ... 11x17 (Fig. 3). Successive subtraction of squares in these rectangles reveals the natural unit of each. Fig. 4 illustrates both the coprime and non-coprime rectangles of row(6). It can be seen that a particular row may contain raum proportional
to raum in other rows, as in this example. The question may be
asked: when does a raum have the proportion of the root ratio
r n modulo (s-r ) = 0.If ( For example, in row(6), the raum proportional to 1 : 2 is
given by Yet there is no raum proportional to 5 : 9 in row(6) since 5.6 modulo (9 - 5) = 30 modulo . According to Schindler, the unit of dimension, or module, is the choice of the architect. "He needs a unit dimension which is large enough to give his building scale, rhythm and cohesion. And last, but most important for the 'space architect,' it must be a unit which he can carry palpably in his mind in order to be able to deal with space forms freely but accurately in his imagination" [Schindler 1946; see Park 2003]. Robinson [1899: 298] gives an example of dimensional change using the method of regulating lines. He states that the "fundamental idea of proportion ... is that all parts share the same general character - what geometricians call "similar"; that is, that if one part is seven wide and ten high another part that is only eight high shall be about, or exactly, five and six tenths wide" (Fig. 5). This represents a linear scale change of 80%; but Robinson does not take the opportunity to point this out. Fig. 6. The geometric method gives rise to scale changes such as this 80% linear reduction in the unit module. In Fig. 6, the method of anthyphairesis, successive subtraction, reveals the "natural" unit of measure, or module of a rectangle [March 1999a, b]. This so called "geometric" method leads to two problems for Schindler. First, there is the question of uncertain scale and choice of unit. Second, there is the potential for unacceptable computation "for easy grasp." Schindler sees no good reason to accept a fraction such as 3/5. If, in figure 6, the unit is presumed to be 1 ft., then the reduced rectangle would have the dimensions 8 ft. by 5 ft. 7 1/5 in. which "it is hardly possible to visualize." By using the 48 inch unit, Schindler proposed that it is possible to "feel" the larger rectangle as 1 3/4 x 2 1/2 units, and the smaller as either 1 1/2 x 2 units, or 1 1/3 x 2 units. This degree of refinement of the dimensions allows the architect "to carry in his mind" the design concept.
Schindler's paper was positively reviewed in An illustration will demonstrate the problem of employing
the principle of geometric similitude with a modular grid. Suppose
a ratio of 3 : 5 is adopted. The first modular rectangle will
be 3 This example employs a ratio, 3 : 5, that Schindler recognized as a rational approximation to the golden section, although he rejected the value of the section in architectural design. Nevertheless, the example serves to introduce Le Modulor in which this ratio appears in the Fibonacci sequence of proportions promoted by Le Corbusier (Figure 8): It comes as no surprise that Le Corbusier's proportional palette
is a subset of Schindler's universal set {row(
Schindler sought freedom from these artificial constraints and limitations. Proportion as such was not the issue. He was more concerned with the preservation of scale throughout a work, the rhythmic relationships and the play of the unit system. Where necessary the system could be broken. Consistency was no virtue. The grid did not have to be square. "It is not necessary that the designer be completely enslaved by the grid. I have found that occasionally a space-form may be improved by deviating slightly from the unit. Such sparing deviation does not invalidate the system as a whole but merely reveals the limits inherent in all mechanical schemes" [Schindler 1946].
Figure 9. Modular lattices showing, left, the factors of 120 and, right, the factors of 48.
I have written about Schindler's dimensioning system in several papers [March 1993a, b, c; 1999a]. I was unaware of Schindler's lecture notes at the time of writing. Judith Sheine first pointed out to me the sequence 1/2, 2/3. 3/4.4/5, ... and the musical scale which she had come across in Schindler's notes when researching her monograph [Sheine, 2001]. She believed it confirmed my emphasis on the musical analogy in Schindler's work. In the process of his doctoral investigation, Jin-Ho Park came across the same notebook and made the valuable connection with Robinson's articles [1898-99]. This source material has revised my views. I am no longer so convinced by the musical analogy in the form that I first promoted it. Any ratio involving small numbers will reproduce the same ratios to be found in musical theory. I was aware of this [1993a: 94-5]. Schindler's interest in Robinson's paper on proportion shown in his lecture notes suggests a more liberal appreciation of architectural proportions than adherence to musical harmony would permit. His approach is best explained arithmetically rather than geometrically, and his lecture notes confirm this. I previously suggested that Schindler's system of dimensioning was "classical," and I believe this to be true. His son, Mark Schindler, remembers that his father would often relax in the evenings by examining "classical" architectural drawings. Schindler's lecture notes show a broad interest in architectural history. Where he introduces the notion of a modular unit and a "net for planning" based on the "largest common division," he sidebars ten dimensions of the thirteenth century Elisabethkirche, Marburg, as multiples of a 17 ft. unit. On the very next page he describes the concept of a "row." Again, it was Jin-Ho Park who brought the Schindler-Heath correspondence on modular coordination to my attention. In Figure 10. The platform of the Parthenon seen as a gnomonic diagram generated from a 1x 10 rectangle, tinted. The gnomonic rectangles belong to Schindler's row(9).
March, Lionel. 1981. A class of grids. ______. 1994a. Dr. How's magical music box.
Chapter 12, pp. 124-145 in ______. 1994b. Log house, urhütte, and
temple. Chapter 10, pp. 102-113 in ______. 1994c. 'Proportion is an alive and
expressive tool...'. Chapter 9, pp. 88-101 in ______. 1998. ______. 1999a. Architectonics of proportion:
a shape grammatical depiction of classical theory. ______. 1999b. Architectonics of proportion:
historical and mathematical grounds. ______. 1999c. Music for the eyes: Schindler
in proportion. March, Lionel and Judith Sheine, eds. 1998.
March, Lionel and Philip Steadman. 1971. ______. 1975. ______. 1975. Park, Jin-Ho. 2003. Rudolph
M. Schindler: Proportion, Scale and the "Row".
Robinson, John Beverley. 1898-99. Principles
for Architectural Composition. Sarnitz, August. 1988. Schindler, Rudolph M. 1946. Reference Frames
in Space. Sheine, Judith. 2001.
Lionel
March was admitted to Magdalene College, Cambridge, to
read mathematics under Dennis Babbage. There he gained a first
class degree in mathematics and architecture while taking an
active part in Cambridge theater life. In the early sixties,
he was awarded an Harkness Fellowship of the Commonwealth Fund
at the Joint Center for Urban Studies, Harvard University and
Massachusetts Institute of Technology under the directorships
of Martin Meyerson and James Q Wilson. He returned to Cambridge
and joined Sir Leslie Martin and Sir Colin Buchanan in preparing
a plan for a national and government center for Whitehall. He
was the first Director of the Centre for Land Use and Built Form
Studies, now the Martin Centre for Architectural and Urban Studies,
Cambridge University. As founding Chairman of the Board of the
private computer-aided design company, he and his colleagues
were among the first contributors to the 'Cambridge Phenomenon'
- the dissemination of Cambridge scholarship into high-tech industries.
In 1978, he was awarded the Doctor of Science degree for mathematical
and computational studies related to contemporary architectural
and urban problems.Before coming to Los Angeles he was Rector and Vice-Provost of the Royal College of Art, London. During his Rectorship he served as a Governor of Imperial College of Science and Technology. He has held full Professorships in Systems Engineering at the University of Waterloo, Ontario; and in Design Technology at The Open University, Milton Keynes. At The Open University, as Chair, he doubled the faculty in Design and established the Centre for Configurational Studies. He came to UCLA in 1984 as a Professor in the Graduate School of Architecture and Urban Planning. He was Chair of Architecture and Urban Design from 1985-91. He is currently Professor in Design and Computation and a member of the Center for Medieval and Renaissance Studies. He was a member of UCLA's Council on Academic Personnel from 1993, and its Chair for 1995/6. He is a General Editor of Cambridge Architectural
and Urban Studies, and Founding Editor of the journal Planning
and Design. The journal is one of four sections of Environment
and Planning, which stands at "the top of the citation
indexes." Among the books he has authored and edited are:
The Geometry of Environment, Urban Space and Structures,
The Architecture of Form, and R. M. Schindler: Composition
and Construction. His most recent research publications include:
"The smallest interesting world?", "Babbage's
miraculous computation revisited," "Rulebound unruliness,"
"Renaissance mathematics and architectural proportion in
Alberti's De re aedificatoria," and "Architectonics
of proportion: a shape grammatical depiction of classical theory."
His book Architectonics of Humanism: Essays on Number in Architecture
before The First Moderns, a companion volume to Rudolf Wittkower's
Architectural Principles in the Age of Humanism was published,
together with a new edition of the Wittkower, in the Fall 1998.
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