The Underlying Geometry in Rudolph M. Schindler's Packard House

Jin-Ho Park*1 and Joung-Lan Park2

1 Professor, Department of Architecture, Inha University, Korea 2 Fulltime Lecturer, Department of Architecture, Dongyang Mirae University, Korea

Abstract For Rudolph M. Schindler, the role of geometry was a driving force for the consistent and systematic quality underlying most of his works. It provides an order for the formal expressions that encompasses both composition and construction. Among others, the Packard House stands out the most in terms of its unique employment of triangular geometry. By fabricating a scaled model and reworking drawings, this article reveals the underlying geometry of spatial composition of the house. It also discusses that Schindler's treatment of the grid is unique, in comparison to his contemporaries, for instance Frank Lloyd Wright.

Keywords: Schindler; geometry; proportion; grid; Packard House

1. Introduction that is reinforced by the façade. However, beyond Few Architects combined ingenuity of construction these distinctions Schindler's use of geometry remains and creative space design as well as R. M. Schindler. at best incompletely understood. Sadly, the house was The Packard House excels all others of its type. demolished in 2002,4 preventing further analysis of the (Whitney R. Smith)1 actual structure (Fig.1.). Rudolph. M. Schindler's buildings are recognized as icons of twentieth-century design. Less well known is Schindler's synergetic design method, which combines a specific compositional sensibility with the practicalities of physical construction. Schindler's work demonstrates an authoritative and creative commitment to theory and practice2 - his philosophy that "form conceived by the human mind is always founded on geometrical figures," implies that architecture begins with thinking about and reasoning regarding a building, shaping and feeling it in the mind as a Fig.1. Photo Taken in 2001, a Year before it was Demolished structure based on geometric laws. This is the basis of the spatial complexity and compositional consistency Nonetheless, the Packard House has been the subject in his architecture.3 of countless examinations by Schindler scholars. Esther The Packard House of 1924 is one of his most McCoy5 characterizes the house plan as "Y-shaped." impressive designs. Based upon a triangular, three- Sarnitz points6 out that "the pitched roof is angled dimensional geometry, it is also one of the most unique. rather more than 45 degrees back from the point…." This is because of the impractical configurations Barbara Giella7 notes that there were two formal generated by unusable angular spaces: triangles devices in forming the house: first, from the shape of are an uncommon aesthetic device in architectural the lot, "the 60 degree angle used for the plan"; and design. Schindler overcomes this through the strategic second, the "horizontal regulating lines at a door height implementation of a three-sided parti in the ground of six feet eight inches." Giella also believes Schindler plan, establishing an immediate trigonometric harmony "borrowed" the design from Wright's work. Judith Sheine8 describes the layout as "three wings around a triangular central kitchen on a roughly triangular lot." 9 *Contact Author: Jin-Ho Park, Professor, David Gebhard remarked that the design was "quite Department of Architecture, Inha University, arbitrary in its exterior geometry." Generally, the #253 Yonghyun-dong, Nam-gu, Incheon 402-751 Korea commentary has been preoccupied with the exterior. Tel: +82-32-860-7597 Fax: +82-32-866-4624 Despite the uniqueness of triangular shapes inherent E-mail: [email protected] within the overall design, as yet there has scarcely been ( Received April 16, 2012 ; accepted February 13, 2013 ) an in-depth analysis.

Journal of Asian Architecture and Building Engineering/May 2013/15 9 This paper investigates the underlying geometry chairs, breakfast table, and lighting fixtures.12 applied to the design of the Packard House. In doing so, it clarifies speculation and corrects common misunderstandings of the architect's design strategy.

2. Description of the Packard House Designed in 1924, the Packard House in Pasadena, California10 was a celebrated residential work of R. M. Schindler (Fig.2.). The original owner, a teacher and attorney, commissioned the young Schindler, and he responded with an unconventional triangular composition. For some, the unique, angular interplay of the design was deemed too unusual, avant-garde, and perhaps laissez-faire, for even the socially progressive to accept. However, Rose Marie Packard described it as 'California Living'. She boasted: "We have the maximum of privacy yet the entire house can be thrown open to large group entertaining, adaptability is the word that describes it."11 Fig.3. The Packard House, 1924, the Ground Floor Plan [E. Entrance, B. Bedroom, BA. Bathroom, P. Porch, L. Living Room, K. Kitchen, PA. Playroom, D. Dining Room]

The roof of the house is pitched steeply, withstanding high velocity winds, and providing coolness in the summer and warmth in the winter.13 Although Schindler originally intended to cover the roof with thin sheets of copper, for reasons of economy Fig.2. The Packard House, ¼ Scale Basswood Model he used roll roofing in the final design.14 Constructed by Hyung-wook Ahn The basic layout of the house as an equilateral triangle was derived from the shape of the site, which A logical organization is revealed in the layout of the is roughly triangular. The tri-axis is drawn on the house, immediately discernable upon analysis of the site plan as shown in Fig.4.15 The orientation and floor plan and façade. The plan is divided into tri-axial placement of all Schindler houses are determined by zones, with a kitchen located at its hub. Three wings their respective lots; according to Schindler, "The extend radially and embrace the outside yard. From the conventional house is conceived as a solid mass central point, a spatial distribution of the floor plan is growing out of the ground. The space house as a space anchored and defined (Fig.3.). form becomes a part of a room formed by the lot, the The three wings form the playroom, parent's surroundings, contours, and firmament."16 room, and living room, each with porches. Each porch opens directly onto the patios. Windows are abundant and patio doors have minimal wall planes. Schindler explained, "The three-winged plan treats the exterior walls not as facades of a convex mass, but as inside walls of outdoor rooms (patios)." The design of the house makes maximum use of available light and prevailing breezes so necessary to capture the California climate. The children's playroom is oriented to the south to encourage the penetration of maximum sunlight. A beautifully structured A-frame ceiling enhances the spaciousness of the living room. The parent's room and the living room are located 30° west and east of north, respectively. Schindler positioned the entry, garage, and service court on the Fig.4. The Packard House, 1924, Site Plan north. A service porch on the north between the living room and parents' room leads out from the kitchen to The Packard House is characterized by the co- the service court. The basic geometry is also repeated existence of two geometric figures of spatial ordering: throughout the components of the house, including an equilateral triangle and a square. The equilateral

10 JAABE vol.12 no.1 May 2013 Jin-Ho Park triangle forms the basic parti of the house, starting thought that while it was "possible to work with a from the floor plan, and then rising to the façade. One grid, which is not rectangular… [t]o depart from this half of the equilateral triangle, with sides of 1 and √3 simple frame of reference of space, however, would and a hypotenuse of 2, governs the placement of the be justified only by the most compelling reasons for it elements of the house. Set along the north-south axis, would result in tricky patterns and complicate three- the points of the triangle serve as foci for laying out dimensional thinking."19 Schindler's rejection of other Schindler's space reference frame. Schindler calls the grid types was a rational decision, because although points "sillcocks". As a matter of fact, sillcocks are the other grids provide many opportunities for variations small water faucets typically located on the outside and emphasis,20 they create other problems.21 The of the house. These sillcocks provide water for many Packard House is a prime example of Schindler's uses such as watering flowers or washing cars. Perhaps preference for simplicity. Here, he does not overlap Schindler started designing the house with basic unit grids even when planning angles. Instead, he building systems in mind from the beginning stage of separates each wing with its own unit grid lines. the design. Schindler rationalized the juxtaposition of the rooms The square units of the space reference frame are even in an angular arrangement. He understood clearly set from the sides of the triangle to form rooms. These that angled or hybrid forms could create a number of grids are seen on his drawings (Fig.5.). This notion is different patterns resulting in more intriguing spatial summarized in his 1946 article, "Reference Frames forms. Schindler's insistence on the square grid in Space,"17 in which Schindler proposes that the distinguished his approach from that of Wright's in his space architect use 48 in. x 48 in. x 48 in. "units" of application of a unit system.22 Unlike Wright who used space to mentally define spaces and to consider their various geometric unit modules including triangular relationships with one another. modules, Schindler consistently used the 48 in. square The dimensions of rooms are proportioned according unit module.23 to this space reference frame. Schindler marks his unit system on working drawings in the form of a 48 in. square grid. Each wing is numbered and lettered on the working drawings to identify locations and sizes of the parts with respect to the whole. Although in his unpublished lecture notes for the Church School of Art written in 1916 he recommended not only squares for the unit shape but also the rectangle, triangle, circle, etc., he himself preferred to use the square grid.

Fig.6. Left, Schindler's Discontinuous Grids in the Design of the Packard House; Right, Wright's Non-Orthogonal Isometric Grids In designing the Packard House, Schindler did not record the dimensions of every individual space on his working drawings. Commonly, he marked only the width and length of major rooms with simple whole numbers. Although they are not always rectangular, Schindler approximated their dimensions and, at times, overlooked dimensions of minor or insignificant spaces, including the bath and storage rooms, which were not marked on the drawings. Other spaces were interlocked, overlapped, or zigzagged; Schindler converted these room sizes into rational numbers. Dimensions of irregularly shaped rooms were approximated by adopting whole numbers and labeling dimensions "a x b" on the drawings. It is possible Fig.5. The Packard House, 1924, the Ground Floor Plan with the that he assumed that simple whole numbers would 48 in. Square Unit Grid. Redrawn by the Author from Garland, Drawing no. 2513 [E. Entrance, B. Bedroom, BA. Bathroom, make it easier to visualize space-form in the context P. Porch, L. Living Room, K. Kitchen, PA. Playroom, of his space reference frame. Room measurements are D. Dining Room] integers derived from the unit module with multiples and fractions (1/2, 1/3, and 1/4). It is certain that Schindler's preference for the square grid Schindler used approximate measures of room sizes in is significant. Unlike F. L. Wright18, who used a rational manner, using the space reference frame to rectangular, circular, and hexagonal grids, Schindler determine the sizes of each room and, probably, their

JAABE vol.12 no.1 May 2013 Jin-Ho Park 11 proportional relationships. In later projects, Schindler increasingly integrated simple geometric patterns into his unit system. The Beach House24 for Olga Zacsek is prominent among these geometric designs (Fig.7.). Although it seems to form a butterfly pattern, the house, which is set on a rectangular site, manifests the equilateral triangle. While rooms are in various shapes, Schindler provided the dimensions of the rooms with whole numbers, such Above, Living Room Elevation as 20 ft. x 16 ft. for the living room, 9 ft. x 8 ft. for the kitchen, 8 ft. x 7 ft. for one bedroom, and 12 ft. x 10 ft. for another bedroom. Schindler's standard unit system of numbers and letters was not presented on the drawings. However, this by no means implies that the method remained in his mind without being used. His instructions for using the method in the design process are made lucid in his own words: "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 Below, Side Elevation able to deal with space forms freely but accurately in Fig.9. The Packard House his imagination". 3. Façade and Roof In the Packard House, the equilateral triangle is apparent in both the façade and the cross section of the living room, and serves to juxtapose the façade symmetrically. Although the gabled roof is a result of local restrictions27 and client requirements, the façade represents a bold expression of the architect's three- dimensional triangular schema of the house in total.

Fig.7. The Beach House for Olga Zacsek built in 1936/38. Equilateral triangles are embedded in the floor plan, and are clearly apparent in the ¼ scale model.

The residence of A. van Dekker built in 1947 is another example of Schindler's experimental application of this geometric paradigm. Situated on a rectangular site, the house is regulated by angles of 30°, 60°, 90°, and 120°. A close examination of preliminary sketches shows that the scheme developed gradually, using progressively larger angles: first 30°, then 60°, and finally, 90°. Possibly, the method of employing the various angles is a result of the drafting equipment he used, such as the T-square and 30°-60° Fig.10. The Packard House: Living Room Façade Analysis with and 45° triangles25 (Fig.8.). Combinations of these the Equilateral Triangle two standard draftsmen's triangles on the horizontal T-square produce multiples of 15°, the dominant angles Referring to Fig.10., when the high-pitched roofline in Schindler's designs.26 Here again, Schindler did not is extended to meet the baseline of the house (DC), a overlap his 48 in. grid; each wing preserves disparate perfect equilateral triangle is formed. The base points grid matrices. align with the sidewalls (DI and CJ). The base vertexes of the equilateral triangle also meet precisely with two space reference plane lines. A square (ABCD) with 6 modules is drawn in which the side of the square (AD) equals the base of the equilateral triangle (DC). The pinnacle of the triangle forms the height of the chimney (E). The top is truncated 1 ft. each side of the point E for tall window bays. The sides of the truncated Fig.8. A. van Dekker in 1947. Progressive Development of the portions are all 2 ft. Scheme Using 30°, 60°, and 90° in the Final Scheme

12 JAABE vol.12 no.1 May 2013 Jin-Ho Park Lines CF, DG, and EH subdivide the side of the spirals, used triangles with 4:5:6 proportions. De equilateral triangle in half, respectively. The FG line Grook and De Bazel employed √2:1 triangle.31 Later, indicates the height of the ceiling fascia, forming De Klerk employed the Egyptian triangle in his 1910 the ceiling height of the house 6 ft.-8 in. from the competition project, "Reincarnatie," using square floor level, an exceptional ceiling height. Typically gridlines overlapped by 45° rotational grids. in Schindler's houses, the ceiling height is 8 ft. and the door height is 6 ft. 8 in. The truncated roof forms 4. Conclusion another equilateral triangle (EFG). Sides EF, FG, and Schindler used two basic geometric shapes in the GE of the roof triangle measure 12 ft. (3 modules.). Packard House, the square and the equilateral triangle. Here again, all measurements use Schindler's 4-ft. unit Where the triangle is a pro-generative device used module and its sub-modules (12 in., 16 in., and 24 in.). to determine the shape of the house, the square is a Although the triangular schema naturally involves a lot proportional tool used for scalar measures of facades of angles with irrational dimensions, Schindler labels and plans. Thus, the square grid applied to a three- all dimensions using whole numbers. dimensional building takes the form of a regular The three-dimensional roof shape is further dissected proportioning system, although it remains invisible in from a simple equilateral triangular prism (Fig.11.). the building itself. Four bottom corner points of the prism are truncated Based on the present analysis, it is apparent that according to the hypotenuse of the equilateral triangle. many descriptions regarding the Packard House are Here Sarnitz's observation of the 45° angles was incorrect. For example, David Gebhard's claim that wrong. The roof is placed on top of the living room the exterior geometry of the house is "arbitrary" is where one end of the truncated prism meets the top of inaccurate, because it has been demonstrated that the kitchen, and the other, the top of the living room all geometric decisions were rationally determined. alcove. The vertical module of the house, which is 2 Ester McCoy's description of the house as "Y-shaped" ft. (1/2 module), regulates all vertical heights of details appears too vague, since a very specific form has including furniture, window and door mullions, and the been seen to underlie the layout of the house. Giella's width of the copper roofing. The geometric roof shape observation that Schindler's triangular motif was is reflected in Schindler's later projects, particularly the "derived from Wright's own contemporary experiment Tischler House of 1949-50. in polygonal and triangular forms" is implausible, since Wright's designs of the same period do not exhibit any significant use of a triangular grid; Wright used the triangular grid much later.33 It is truly a shame that the Packard House was not preserved. Fortunately, the original idea expressed in the house will remain appreciated, underscoring Fig.11. The Truncated Roof Geometry is Based on the Schindler's metaphysical conception of immaterial Equilateral Triangle. The Roof Cover with Thin Sheets of eternity. In lecture notes following his 1912 Manifesto, Copper and the Tall Window Bays to the Front are Regulated by the 2 ft. Sub-Module in a discussion of monumentality, Schindler wrote: "Essential parts [built] for eternity" [do] not mean In architecture, the use of equilateral triangles both "durable material but the eternal conception of room." in the plan and façade is rare. However, this unusual In other words, "The timelessness of architecture is approach was used by Dutch architects, and might in [its] form and its expression."34 For Schindler, the have been known to Vienna and German circles – and quest of the architectural process is to build something therefore to Schindler.28 Although systematic methods tangible out of a mental conception. Through his vary, its history can be traced to the medieval idea legacy, a genesis of triangular geometry in architecture of the triangular grid used in the planning of space.23 lives on. Hendrik Petrus Berlage's use of the "Egyptian" triangle in the elevational treatment of his celebrated Acknowledgements Amsterdam Stock Exchange Building in 1898 remains This work was supported by an INHA University a prime example of a contemporary application of the Research Grant. The early draft of this article was triangular grid. Berlage's 1908 essay, "The Foundations presented at the NNJ Conference Mexico City. The and Development of Architecture," provides an author thanks Kim Williams for reading early drafts of excellent summary of geometrical and proportional this article. understanding at the turn of the century, both in terms of historical analysis and design synthesis.30 References His theory and works are well known in Viennese 1) Alofsin, A. (1993) Frank Lloyd Wright: The Lost Years, 1910-22. architectural circles.31 Other architects approached Chicago and London: The University of Chicago Press. 2) Berlage, H. P. (1996) Thoughts on Style 1886-1909. Whyte I. B. the use of triangular designs in various ways. J. L. (Intro.), Whyte I. B. and Wit W. (Trs.) Santa Monica: The Getty M. Lauwerik, noted for his labyrinth of rectangular Center.

JAABE vol.12 no.1 May 2013 Jin-Ho Park 13 3) Eaton, L. (1998) Fractal Geometry in the Late Work of Frank Theory of Space Architecture and Its Theoretic Application to Space Lloyd Wright: The Palmer House. NEXUS II: Architecture and Development of 1945," The Journal of Architecture, 11/1(2006): Mathematics, Fucecchio: Edizioni Dell'Erba. 37-54. It is commonly known that Schindler's great influential 4) Frank, S. (1984) J.L.M. Lauwericks and the Dutch School of figure, Adolf Loos, considered Vitruvius' book as his "Bible." See Proportion," AA Files 7. Kokoschka, O. 1971, Ma Vie, French translation (Paris, 1986), p. 5) Garland Architectural Archives (1993) The Architectural Drawings 76, and also his preface to Ludwig Munz and Gustav Kunstler, of R.M. Schindler: The Architectural Drawing Collection, Adolf Loos: Pioneer of Modern Architecture (London: Thames University Art Museum, University of California, Santa Barbara, and Hudson, 1966). This information derives from Panayotis David Gebhard, ed. New York: Garland Publication. Tournikioti's book, Adolf Loos, Chapter one, note 18, (New York: 6) Gebhard, D (1980) RM Schindler, Santa Barbara and Salt Lake Princeton Architectural Press, 1994). City: Peregrine Smith, Inc. 3 See Lionel March, "Proportion is an Alive and Expressive Tool," 7) Gebhard, D. (1993) The Architectural Drawings of R.M. Schindler, in Lionel March and Judith Sheine (Eds.), R.M. Schindler: Eds. New York: Garland Pub. Composition and Construction, (London: Academy Editions, 8) Giella, B. (1985) R. M. Schindler's Thirties Style: Its Character 1994): 88-101; and Jin-Ho Park, "Subsymmetry analysis of (1931-1937) and International Sources (1906-1937) (Ph.D. diss.,) architectural designs: some examples," Environment and Planning New York: Columbia University. B: Planning and Design, 27 (2000): 121-136. 9) Kokoschka, O. (1971) Ma Vie, French translation (Paris, 1986). 4 Another of Schindler's masterpieces, the 1931 Wolfe House in 10) Mallgrave, H. F. (Ed.) (1993) Otto Wagner, Santa Monica: The Avalon was demolished in 2001. Getty Center. 5 Esther McCoy, Five California Architects, (Los Angeles: 11) March, L. and Steadman, P. (1971) The Geometry of Environment, Hennessey + Ingalls, Inc, 1987): 163. Cambridge: The MIT Press. 6 August Sarnitz, A. R.M. Schindler, architect (1887-1953): a 12) March, L. (1994) Proportion is an Alive and Expressive Tool," pupil of Otto Wagner, between international style and space in Lionel March and Judith Sheine (Eds.), R.M. Schindler: architecture, (New York: Rizzoli, New York, 1988): 26. Composition and Construction, London: Academy Editions. 7 Barbara Giella, R. M. Schindler's Thirties Style: Its Character 13) March, L (1995) Sources of characteristic spatial relation in Frank (1931-1937) and International Sources (1906-1937) (Ph.D. diss., Lloyd Wright's decorative designs, in The Phoenix Papers, Volume Columbia University, 1985): 243-246. 2, Natural Pattern of Structure, L. Johnson (Eds.) Tempe, AZ: The 8 Judith Sheine, R. M. Schindler, (London and New York: Phaidon Herberger Center for Design Excellence. Press, 2001): 123. 14) McCoy, E. (1987) Five California Architects, Los Angeles: 9 David Gebhard, RM Schindler (Santa Barbara and Salt Lake City: Hennessey + Ingalls, Inc. Peregrine Smith, Inc., 1980): 37-38. 15) Munz, L. and Kunstler, G. (1966) Adolf Loos: Pioneer of Modern 10 The site is located on Gainsborough Drive, east of San Gabriel Architecture, London: Thames and Hudson. Blvd, Pasadena, California. 16) Park, J. (2006) R. M. Schindler's Theory of Space Architecture 11 Elizabeth T. T. Smith, E, et al. The Architecture of R.M. Schindler. and Its Theoretic Application to Space Development of 1945," The (Los Angeles: The Museum of Contemporary Art, 2001). Journal of Architecture, 11/1, pp.37-54. 12 Garland Architectural Archives, The Architectural Drawings of 17) Park, J. (2000) Subsymmetry analysis of architectural designs: R.M. Schindler, the Architectural Drawing Collection, University some examples," Environment and Planning B: Planning and Art Museum, University of California, Santa Barbara, edited by Design, 27(1), pp.121-136. David Gebhard, (New York: Garland Pub., 1993): Drawing no. 18) Sarnitz, A. (1998) R.M. Schindler, architect (1887-1953): a pupil 2519. of Otto Wagner, between international style and space architecture, 13 Schindler wrote, "Miter treatment of gables and flattened ridge to New York: Rizzoli. adapt the high room, used as interior space form, to a snowless 19) Schindler, R. (1946) Reference Frames in Space. Architect and climate." Engineer, 165: 10, 40, pp.44-45. 14 Gebhard, 1980, pp.62. 20) Schindler, R. (1934) Space Architecture. Dune Forum, February. 15 See Garland no. 2522. 21) Schindler, R. (1935) Space Architecture. California Arts and 16 Rudolph M. Schindler (1949), Answer to questionnaire from the Architecture, January. school of Architecture, University of Southern California, R.M. 22) Sheine, J (2001) R. M. Schindler, London and New York: Phaidon Schindler Papers. The Architectural Drawing Collection. The Press. University of California, Santa Barbara. The strong relationship 23) Smith, E, et al. (2001) The Architecture of R.M. Schindler, Los between lot and house-form stems from his 1916 Church School Angeles: The Museum of Contemporary Art. lecture notes. 24) Storrer, W. (1993) The Frank Lloyd Wright Companion. Chicago 17 Rudolph M. Schindler (1946). "Reference Frames in Space," 1941. Architect and Engineer, 165: 10, 40, pp.44-45. 25) Tournikiotis, P. (1994) Adolf Loos, New York: Princeton 18 Schindler's early admiration of Wright's architecture was shown Architectural Press. in his paper 'Space Architecture' (1934 and 1935). He wrote 26) Wright, F. L. (1928) In the Cause of Architecture – 1.The Logic of that 'shortly after my revelation in the mountains, a librarian in the Plan. The Architectural Record. Vienna handed me a portfolio – the work of Frank Lloyd Wright. 27) Vitruvius (1931) De Architectura Granger, F. (Trs.) Cambridge: Immediately I realized – here was a man who had taken hold Harvard University Press. of this new medium. Here was "space architecture"'. In 1918, Schindler joined Frank Lloyd Wright whose Wasmuth portfolio Notes (1910) was significant in Schindler's early development. Schindler 1 Schindler Memorial Exhibition Statement housed in the Schindler worked in Wright's Chicago and Taliesin offices originally on the archive at the University of California, Santa Barbara (UCSB). Imperial Hotel project for Tokyo. Rudolph M. Schindler, "Space 2 Schindler is indebted to Vitruviu's De Architectura on the subject. Architecture," Dune Forum (Feb. 1934): 44-46; and California Arts and Architecture (Jan. 1935): 18-19. Frank Granger, Trs. Vitruvius on Architecture (Cambridge: Harvard 19 University Press, 1931). Schindler also summarized Vitruviu's Schindler (1946). In Wright's early work triangular planning is not Book I, Chapter I in his 1916 lecture notes, and added his own evident. According to Leonard K. Eaton, Wright's Desert Camp opinion: "An architect must have the theory of all these knowledge; in 1927 is known as the first building executed with a triangular the practice is only of his own." The lecture notes are housed in planning module. See Leonard K. Eaton, "Fractal Geometry in the the Schindler Archive. Refer to Jin-Ho Park, "R. M. Schindler's Late Work of Frank Lloyd Wright: The Palmer House," NEXUS II: Architecture and Mathematics, (Fucecchio: Edizioni Dell'Erba,

14 JAABE vol.12 no.1 May 2013 Jin-Ho Park 1998). Perhaps the St Mark Tower of 1929 is one of the best his designs with variations. See Anthony Alofsin, Frank Lloyd known examples of triangular grids. Wright's use of triangular Wright: The Lost Years, 1910-22 (Chicago and London: The planning grids is more prevalent in his later work, for example, University of Chicago Press, 1993). the Palmer House of 1950-51. See Lionel March L and Philip 32 See Hendrik Petrus Berlage: Thoughts on Style, (1996): 14. Steadman, The Geometry of Environment, (Cambridge: The MIT 33 In Wright's early work triangular planning is not evident, although Press, 1971). a hexagonal grid is clearly shown in the Hanna House of 1936. 20 In the case of Wright, he introduced various types of grid such as Leonard K. Eaton (1998) presents an analysis of Wright's use of the rhomboid, hexagonal, circular grid, and so on. For example, the triangle. According to him, Wright's Desert Camp is known as he provided the use of the hexagonal grid in the Hanna House the first building executed with a triangular planning module in with a particular reason: "we call it Honey-comb House because 1927. Wright's use of triangular planning grids is more prevalent the structure was fashioned upon a hexagonal unit system. in his later work, for example, the Palmer House of 1950-51. The hexagonal is better suited to human movement than the 34 Schindler, 1916 lecture notes, Lecture VI: "Planning." rectangle." It seems that Wright used the hexagonal system as the appropriateness of functional ease. 21 The reason that he missed out the grid is noted in his words, "... even though I have been forced, frequently, to make duplicate conventionally measured plans without grid line ..." (Schindler, 1946). 22 Refer to Frank Lloyd Wright, "In the Cause of Architecture – 1. The Logic of the Plan," The Architectural Record, (1928). 23 William A. Storrer summarized Wright's use of various unit modules in his book, The Frank Lloyd Wright Companion, (Chicago and London: The University of Chicago Press, 1993). 24 The Beach House for Olga Zacsek was built in 1936/38 but demolished. 25 McCoy (1987). 26 Lionel March, "Sources of characteristic spatial relation in Frank Lloyd Wright's decorative designs," in The Phoenix Papers, Volume 2, Natural Pattern of Structure, editor L. Johnson. (Tempe, AZ: The Herberger Center for Design Excellence, 1995): 12-48. 27 Gebhard, (1980): 58. 28 Much of my discussion is indebted to Suzanne Frank's article, "J.L.M. Lauwericks and the Dutch School of Proportion," AA Files 7, (1984), and also a book on Henriks Petrus Berlage: Thoughts on Style 1886-1909, Introduction by Iain Boyd Whyte, Translation by Iain Boyd Whyte and Wim De Wit, (Santa Monica: The Getty Center, 1996). In particular, the Dutch influence on Peter Behrens' architecture has been discussed by Asche and Suzanne. The influence might be from either Berlage or Lauweriks. Suzanne Frank argues Behrens' use of grids in the Oldenberg Pavilions of 1905 might be from Lauweriks in his pedagogical collaboration with Lauweriks at Düsseldorf; rather, Kurt Asche argues that it may be from Berlage. However, Suzanne's argument is persuasive, since Asche doesn't provide any critical clue. Le Corbusier in France showed interest in the Dutch system as well. 29 It has been known that the triangle figure was introduced by Viollet- le-Duc in his Entretiens through Cuypers. See Suzanne, (1984). 30 There are many striking similarities between Schindler and Berlage: their use of a systematic method as a fundamental principle and the emphasis of spatial form as their primary concern. Also, Berlage pointed out the deviation of the system where Schindler acknowledged it too: "you should not be slave of such a system… Such a system let us say it quite candidly, can succeed only when you know at which point to abandon it…" See Henriks Petrus Berlage: Thoughts on Style 1886-1909, Introduction by Iain Boyd Whyte, Translation by Iain Boyd Whyte and Wim De Wit, (Santa Monica: The Getty Center, 1996): 212. 31 The linkage between Wagner in Austria and Berlage in the Netherlands has been elaborated by Lain Boyd Whyte's article, "Modernist Dioscuri? Otto Wagner and Hendrik Petrus Berlage." He comparatively analyzed their theoretical and practical similarities and differences between two figures. In particular he exemplified the 1884 competition for the Stock Exchange in Amsterdam competition project which both Wagner and Berlage entered, and Berlage's much later design was completed in 1903. See Harry Francis Mallgrave (Ed.), Otto Wagner, (Santa Monica: The Getty Center, 1993): 156-197. Frank Lloyd Wright also gained an appreciation of geometry and mathematics in architecture. He applied an isosceles triangle, which has 4 as a base and 5 as the height, called the Egyptian triangle in the competition project for the Stock Exchange in Amsterdam. The system had been used in

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