Jin-Ho ParkSchool of Architecture University of Hawaii at Manoa 2410 Campus Road Honolulu, HI 96822 USA
Fig. 1. Reconstruction of a portion of a handwritten page from Schindler's notes for Church School lectures, 1916. Schindler's layout has been followed. Editorial additions are shown bracketed in lower case [ ]. L, length. W, width. H, height. Fig. 1 shows a portion of Schindler's Church School lecture notes. It emphasizes that architects should be "conscious" of mathematics, since "everything [comes] under its laws." In architecture, mathematics is "usually employed to find proportions." Having said that, Schindler then states, "form in general shall have no rule but its expression." He rejects rules that determine the proportion of columns -- "any size and thickness possible." And rules regulating the proportioning of rooms -- "rooms do not have to follow rules." Such formulae are "merely playing with numbers" (Fig. 2). As a former student of Otto Wagner, Schindler placed priority on materials, structure and functional expression. These factors determined their own dimensioning according to performance. Fig. 2. "Rooms do not have to follow rules." Schindler's example of the kind of rule to be rejected. Left, room plan of ratio 3:2. Right, end wall of ratio 2Ö2:1. One Ö2 rectangle on the end wall is shown tinted. Schindler also rejected the conventional wisdom of his time concerning regulating lines. The repetition of the same ratio throughout the parts of a whole is seen as producing a confusion of scale. This is illustrated in an example that Schindler devised after Robinson. In a three-part tower, each part is governed by identical regulating lines, simply called "diagonals." This results in elements different in scale, producing undesirable ambiguity. Schindler labeled this "wrong." In contrast, when the height of each element is the same, a constant scale is maintained, but the proportion changes as indicated by the non-parallel diagonals.[2] Schindler labeled this "right." He concluded: "No diagonals as comparison equals proportion" - a clear rejection of the hallowed principle of "geometric similarity" and its representation as regulating lines (see Fig. 1 above). Figure 3. Robinson's tower example with captions. Left, "Failure of the method of similar dimensions, as applied to a storied-tower." Right, "Correct method of proportioning a storied-tower."
In Fig. 4, Robinson illustrated the "system of perfect similarity," in which the rectangles share common diagonals through a center point. "The resulting rectangles are all of the same character, all elongated or all shortened …." Fig. 4. Robinson's diagrams showing "geometrical interpretations of arithmetical ratios." The diagonals that Robinson mentioned were, in his time, the
well known regulating lines of triangulation. Hendrik Petrus
Berlage's use of the "Egyptian" triangle in the elevational
treatment of his celebrated Amsterdam Stock Exchange building,
1898, remains a prime example of the contemporary application
of triangulation. Berlage's 1908 essay "The Foundations
and Development of Architecture" provides an excellent summary
of geometrical and proportional design at the end of the nineteenth
century with respect to both historical analysis and design synthesis
[Whyte and de Wit 1996: 165-257]. Otto Wagner, on the contrary,
makes no specific mention of geometry or proportion in Figure 5. Regulating lines in 1916 villa, Le Corbusier, 1916. Schindler's approach runs counter to that of Le Corbusier. Born in 1887, the same year as Schindler, Le Corbusier used regulating lines in his early works (Fig. 5), transforming the principle of geometric similitude into Le Modulor and the persistence of the golden section in his work after the second World War. Nevertheless, the two architects agreed on the human size as a "fundamental unit." Both rejected the artificiality of the metric system. According to Schindler, the divisions of the meter are "too small for conception." He saw the meter as an architecturally meaningless measurement, an "absurdity," and the foot as an unsuitable measure for architecture.[3] In his 1916 lecture notes, Schindler recommended that the architect "choose his own "unit" out of different conditions" such as "lot, expression of building, purpose, and so on. Unit to be subdivided 1/2, 1/4." This initial idea is developed later in the 1946 "Reference Frames in Space."
Fig. 6. Reconstruction of portion of handwritten page from Schindler's notes for Church school lectures, 1916. Editorial comments are shown in brackets [ ]. Writing of the stretched string, Robinson observed that "the subdivisions will give the notes of the gamut: 1/2, 8/15, 3/5, 2/3, 3/4, 4/5, 8/9."[4] These terms give the ratios of the octave in the modern, intonation scale. He continued:
It should be noted that classical Greek musical ratios did include the numbers 7 and 11, but not 13. Ratios 6:7, 7:8, 10:11, 11:12, are cited by Ptolemy. Indeed, all such ratios from 1:2 to 24:25 with the exception of 12:13, 13:14, 22:23 occur in Ptolemy's work [Barker 1989: 270-301]. At the time that Robinson was writing it was understood that simple ratios 1:2, the duple, 2:3, the hemiolic, and 3:4 the epitritic, had been used by the Greeks in their architecture. Berlage mentioned a French writer, Charles Chipiez, who attributed these ratios in 1891 to the principle dimensions of the Parthenon. Berlage also quoted the British authority, James Ferguson, writing in the 1860s:
Instead of Robinson's "series", Schindler used the
term "row", a sequence, "a following of unequal
units with definite changes." A page from Schindler's notes,
Figure 6, illustrates a particular "row", which is
equivalent to the classical sequence of subsuperparticular numbers
[March 1999: 91-100]. If a/b is a term in the "row",
the next term is (a + 1)/(b + 1). Schindler's use of the term
"row", Robinson gives the series: 1/3, 2/4, 3/5 4/6, 5/7, 6/8, 7/9, 8/10, 9/11, 10/12, 11/13. 1/4, 2/5, 3/6, 4/7, 5/8, 6/9, 7/10, 8/11, 9/12, 10/13, 11/14. 1/5, 2/6, 3/7, 4/8, 5/9, 6/10, 7/11, 8/12, 9/13, 10/14, 11/15. Schindler (see Fig. 6) abbreviated the first two of Robinson's sequences to the rows "1/2 ...11/12" and "1/3 ... 11/13 etc.," and remarked, "Each row approaching 'unity' = 1" [Robinson 1898-9: 310]. The fraction 11/13 is a most unmusical ratio according to Robinson. A fact that Schindler repeated: "No 7, 11, 13 in musical intervals." It is as if Schindler is making the point that architecture does not have to be bound by the restrictions of musical intervals. This break from musical theory is supported by Daniel Barbaro who had said as much in his commentaries on Vitruvius in the sixteenth century [March 1999: 91-100]. It is also in sympathy with that "emancipation of dissonance" which both Helmoltz, as a scientist, and Schoenberg, as composer, were advocating for modern music [Dahlhaus 1987: 120-127]. It is interesting to note that alternate elements of Robinson's and Schindler's second rows can be reduced by common factors, whereas their first rows cannot. Reduction of the fractions in the first row can be seen embedded in the second row: 1/2, 2/3, 3/4, 4/5, 5/6, ... Once reduced, all values are "root" values in the classical sense [March 1998: 58]: 1/2 is the root of the sequence 1/2, 2/4, 3/6, 4/8, 5/10, and so on. The distinction is illustrated using Robinson's and Schindler's example of 9/19, which is preceded by 8/18 and begins with the core value 1/11, the first ratio in the tenth row. If 8/18 is reduced to its equivalent root value 4/9, the core value becomes 1/6, no longer 1/11, since the sequence of the fifth row begins 1/6, 2/7, 3/8, 4/9, ... etc. A different core value now applies.[5] This shows that the row preserves scale while exhibiting distinct ratios. The row avoids the problem of scale ambiguity posed by regulating lines and illustrated by Schindler in Figures 3 and 4. Fig. 7. Robinson's diagrams showing "geometrical interpretations of musical ratios." Fig. 7 shows Robinson's diagrams which exhibit a relationship with musical ratios. He explains:
Schindler's diagrams (see Fig. 6) differ from those of Robinson in that he makes each annulus one unit wide, not half a unit. This appears to be an anomaly since the numerical rows that he abbreviated follow those of Robinson. A rearrangement of the diagrams in the manner of the ancient Greek "gnomon" produces L-shaped additions each of which is a unit wide (Fig. 8). Note the lines at the bottom, shown bold. The length of these lines may be used to name the series, or row. The line length of the first row, row-1, is 1, the second, row-2, is 2, and so on. Fig. 8. Variation of Robinson's diagrams above in which the "gnomons" are one unit wide. The core line is shown bold. Left, is a diagram illustrating the third row since the core line is 3 units long; right, the first row with a unit core. It is significant that Schindler illustrated a set of nested squares representing the zeroth row, row-0, with terms like a a/a. In Figure 6, Row-0, which Schindler illustrated in his lecture note, is represented by squares 1/1, 2/2, 3/3 … . These are square multiples of the unit, and find expression in Schindler's own designs.
Fig. 9. Photograph of the Shampay house model showing the living room and dining room fenestration patterns.
Figure 10. Photograph of the How house ceiling structure Figure 11. a. Plan of the How house ceiling beams; b. decomposition as the first terms in row-0 to row-8), or 1/1 to 1/9; c. subshapes of squares in row-0, or 1/1, 2/2, ... 9/9; d. subshapes of rectangles in row-1, or 1/2, 2/3, ... 8/9.
The regulating principle, or "geometric similarity", gives 8 different equivalences:
Fig. 12. Photograph of the Braxton-Shore house model.
Imagination is the key to Schindler's idea of "conception." In one of his Church School lecture notes he writes:
For Schindler, designing was essentially a mental process. Architectural drawings merely record the results; the physically executed work gives it expression. This is why he placed so much emphasis on his reference frames in space. This three-dimensional conceptual frame allowed the architect to define spaces and consider their relationships one with another -- first in his head, not on paper.
Robinson gives an example of dimensional change using the method of regulating line, stating 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 high and ten wide another part that is only eight wide shall be about, or exactly, five and six-tenths high" [1899: 298] (Fig. 13). This represents a linear scale change of 80%; but Robinson does not take the opportunity to point this out, but this is precisely what Schindler objects to -- a change of scale. Figure 13. Robinson's example of "arithmetical proportion of breadth to height, indicated by diagonals."
[2]
This same example is shown in the notes for the Chouinard lectures,
Los Angeles, 1931. Interestingly enough, both Schindler and Neutra
were listed for the Chouinard history in the calendar for 1931/32.
See http://calarts.edu/alumni/chouinard/.
[3]
See a recent discussion in [Tavernor 2002]. [4]
Robinson used both notations [5] Schindler's theory of the "row" is mathematically
generalized in March [2003]. [6] For the detailed analysis of the Shampay house,
see [Park and March 2002]; also, see [Park 2003]. [7] For the detailed analysis of the Braxton-Shore project,
see [Park and March 2003]; [Park 2002] also interprets the room
ratios in relation to the Fibonacci and Lucas sequence.
Dahlhaus, Carl. 1987. Gutheim, Frederick, ed. 1941. Mallgrave, Harry Francis, trans. 1988. March, Lionel. 1993a. Proportion is an Alive
and Expressive Tool …. Pp. 88-101 in ______. 1993b. Dr. How 's Magical Music Box.
Pp. 124-145 in ______. 1999. Architectonics of Proportion:
a Shape Grammatical depiction of classical theory. ______. 2003. Rudolph M. Schindler: Space
Reference Frame, Modular Coordination and the "Row".
March, Lionel and Judith Sheine. 1993. Park, Jin-Ho. 1996. Schindler, Symmetry and
the Free Public Library, 1920. ______. 2002. Rudolph M. Schindler's Braxton
House: The Fibonacci and Lucas Sequence. ______. 2003. The Stylistic Characteristics
of the Shampay House of 1919: A Formal Analysis. Submitted to
Park, Jin-Ho and Lionel March. 2002. "The
Shampay House of 1919: Authorship and Ownership," ______. 2003. Space architecture: Schindler's
1930 Braxton-Shore project. Robinson, John Beverley. 1898-9. Principles
of Architectural Composition. Sarnitz, August. 1988. Schindler, Rudolph Michael. 1946. Reference
Frames in Space. Tavernor, Robert. 2002. Measure, Metre, Irony:
Reuniting Pure Mathematics with Architecture. Pp. 47-61 in Whyte, Iain Boyd and Wim de Wit (trans). 1996.
Architectural Research Quarterly (arq),
Environment and Planning B: Planning and Design, Journal
of the Society of Architectural Historians, Journal of
Architectural and Planning Research, and the Nexus Network
Journal.
top of
pageCopyright ©2003 Kim Williams Books |
NNJ HomepageAutumn
2003 indexAbout the
AuthorSearch
the NNJOrder
Nexus books!Research
ArticlesThe
Geometer's AngleDidacticsBook
ReviewsConference and Exhibit ReportsReaders'
QueriesThe Virtual LibrarySubmission GuidelinesTop
of Page |