Cycloids in Louis I. Kahn's Kimbell Art Museum at Fort Worth, Texas

The Mathematical Tourist Dirk Huylebrouck, Editor

ouis I. Kahn (1901–1974), one of crete buildings. Kahn’s work shows his Cycloids in the most outstanding modern great reverence for making accessible L American architects, is noted for the value of clear structural systems, ma- the use of simple geometric forms and terials and light. Pushing the structural Louis I. spatial compositions in massive con- systems and materials to their bound- Kahn’s Kimbell Art Museum at Fort Worth, Texas

JIN-HO PARK, YONGSUN JOO, AND JAE-GUEN YANG

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the café where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected] Figure 1. Exterior views on the Kimbell Art Museum.

© 2007 Springer ScienceBusiness Media, Inc., Volume 29, Number 2, 2007 1 aries is one of the essential components plan, with cycloid vaults, and that was would be gentle. The stepped en- of his architectural approach. Kahn also the design that was executed. trance court passes between the por- sought to optimize natural light for the The imposing structure opened four ticos and their pools with a fountain occupants of the building. years later, in October 4, 1972. The mu- around which one sits, on axis de- Born on the island of Saaremaa, Es- seum incorporates the slope of the site signed to be the source of the por- tonia, in 1901, Kahn emigrated to the and has entrances at both the lower and tico pools. The west lawn gives the United States and practiced there until upper levels. Public access for pedes- building perspective. his death in 1974. Among his best- trians is offered on the west side, at the Kahn continues: known buildings is the Kimbell Art Mu- upper level, whereas the parking lots The south garden is at a level 10 feet seum in Fort Worth, Texas (1966–1972), are located on the east side, at the lower below the garden entrance ap- one of the most-admired architectural level (accessed from Arch Adams proached by gradual stepped lawns pieces of the time. Other masterpieces Street). There is a central axis passing sloped to be a place to sit to watch are the Salk Institute in California (USA), through the front and the rear en- the performance of a play music or the Indian Institute of Management in trances, the lobby and the stairs; two dance, the building with its arched Ahmedabad (India), and the National water pools outside the porticos are silhouette acting as the backdrop of Assembly at Dacca (Bangladesh). symmetric too, along that axis. This ax- a stage. When not so in use it will F1 The Kimbell museum sits on the ial approach is typical of Kahn’s de- seem only as a garden where sculp- gentle slope of the Amon Carter Square signs. On June 25, 1969, he wrote, in a ture acquired from time to time Park, about two miles from downtown letter to Mrs. Kimbell [Johnson 1975]: would be. The north garden, though Fort Worth. The structure, covering a Two open porticos flank the en- mostly utilitarian, is designed with gross area of 120,000 square feet, lies trance court of terrace. In front of ample trees to shield and balance in a trapezoid-shaped area of 9.5 acres, each portico is a reflecting pool the south and north sides of the bounded by Camp Bowie Blvd., Arch which drops its water in a continu- building.” Adams St., and West Lancaster Ave. ous sheet about 70 feet long in a The series of 16 structural units with There are a number of cultural institu- basic two feet below. The sound their vaulted roof are supported inde- tions in the vicinity: the Amon Carter Museum of Western Art, the Fort Worth Museum of Science and History, and, directly opposite the Kimbell, along Arch Adams St., the new Modern Art F2&3 Museum of Fort Worth. The Kimbell Art Museum is home to one of the world’s finest art collections, with acquisitions ranging from antiquity to the twentieth century [Loud, 1987]. Among the interesting architectural fea- tures of the building are the unique vault roofs, which exhibit a rare example of a strategic use of the mathematical properties of the cycloid [Gast, 1998; Benedikt, 1991]. An art museum with a mathematical property: could a mathematical tourist ask for more?

A Man, a Plan: Kahn Kahn started the design for the Kimbell in 1967, arriving at his final scheme around September 1968. August Komendant, Kahn’s structural engineer, describes this period as “very difficult and agonizing.” The design process seemed to have been an ordeal for Kahn: the first scheme was a square plan, with a circulatory street, an arcade around the complex, and polygonal ceilings. The second plan was an H- shape, with two connected structures covered by a walkway and curved Figure 2. Partial map of Fort Worth, Texas, showing the location of the Kimbell Art roofs. The final layout was a C-type Museum.

2 THE MATHEMATICAL INTELLIGENCER their proportions, their foliation, or their sky reflections on surfaces, or on water will give. F5

The Cycloid Choice The most distinctive design element of the museum is the series of sixteen parallel cycloid vaults. Kahn did not arrive at the cycloid roof at the beginning of the design process; it was determined after a series of trials in collaboration with Komendant and Marshall D. Meyers, project manager for the museum: Kahn’s presentation in June had pro- posed a semicircular ceiling, which Brown had found ostentatious and thus, dictated by its geometry, too high. Kahn’s response in September had been a flattened arch, which met Brown’s requirements but was not particularly graceful. Meyers had explored the possibilities of a flat roof with quarter-round edges, el- lipses, and segmented circles, none of which proved up to the task.” [Leslie, 2005] Rather than using the equations of a cycloid, the design was made using a simple drawing of the curve generated by a point on a rolling circle, as shown in Figure 6. The rhythmic sequential cy- F6 cloid structure is easily recognized from Figure 3. Site plan of the Kimbell Art Museum. a distance. Each vault consists of 100 23 foot shells poured in place by post- tensioning cables and reinforced con- pendently and clustered with preserva- a large open block dedicated to staff fa- crete, and sheathed by a calcium-lead tion of their independent structural cilities including offices, shops, storage, layer. The roofs are about six feet apart, character. At the same time, each vault conservation and photography studios, separated by flat-roofed channels for air F4 is an inseparable element of the whole, and workplaces for shipping and re- conditioning and electrical distribution 2nd making the order in the composition ceiving. The three courts are all open ducts. The height of a vault from the col clear: the Kahn approach. to the sky and allow natural light into upper level is 20 feet and the height These structural elements are the museum. The largest square court from the lower level is 40 feet. Each grouped on a tripartite scheme, with on the northern side borders the café, concrete cycloid vault is supported by northern, central, and southern parts. while two smaller square courts are lo- four 2 2 foot corner concrete piers, Whereas each of the north and the cated on the south: one is two stories thus providing column free spaces (Fig- south parts has six vaults, the central high so that the light penetrates to the ure 7). F7 part between them has four vaults. The floor below, to the conservator’s studio. F8 permanent collection, an auditorium, a Kahn provides an account of his intent Seeing Cycloid Light 1 café and the galleries occupy the north- for having separate gardens and nam- Each vault has a longitudinal 2 /2 foot ern part. The central part, with only four ing each court separately [Latour 1991]: narrow slit at the roof, over the entire vaults, provides an entrance court in Added to the sky light from the slit length of the structure, covered by which even the trees are arranged in an over the exhibit rooms, I cut across acrylic Plexiglas. Perforated metal re- orderly fashion. It includes the main en- the vaults, at a right angle, a coun- flectors are installed under the light slit trance hall, the lobby, a bookstore, the terpoint of courts, open to the sky, of the cycloid vault. Kahn’s series of museum shop, and the library. The of calculated dimensions and char- preliminary sketches show an evolu- southern part holds galleries for tem- acter, marking them Green Court, tion, in which he experimented with porary exhibitions, the reception hall, Yellow Court, Blue Court, named for various relationships between roof AU1 and a kitchen. All are arranged around the kind of light that I anticipate shapes and light reflectors (see figure

© 2007 Springer ScienceBusiness Media, Inc., Volume 29, Number 2, 2007 3 the light, sufficiently so that the in- jurious effects of the light are con- trolled to whatever degree of control is now possible. [Johnson 1975]. F10

Unusual Computational Complications The key to the museum design is the structural novelty of the cycloid roof, with a span of 100 ft, supported by four corner columns. The monolithic structure is only four inches thick, so thin it looked “fragile,” even to Kahn himself. 1 The roof, with its longitudinal 2 /2 foot narrow slit at the apex of each vault, is braced by cross concrete struts every 10 feet, transferring the load along the vault, longitudinally and vertically, to the lower corner of the vault and to the four corner columns. As shown in Fig- ure 11, the forces of the longitudinal F11 load are absorbed in two ways by three post-tensioning cables, a technique com- monly employed to improve the load- bearing properties of concrete by in- corporating high-tensile steel stretched cables into the structure. The diaphragms at the end absorb the forces of the vertical load. Traditionally, a barrel roof can be supported along the longitudinal edges or on the curved ends. When a barrel is supported along the longitudinal edges, it behaves like a row of arches, one next to the other, developing hor- izontal forces pushing outward and ab- Figure 4. Gallery level (top) and lower level floor plan (bottom) of the museum (re- sorbed by buttress walls. When a bar- drawn by the author). rel vault is supported on its curved ends, it behaves like a beam conveying the load to the walls at the end and to F9 9). The cycloid vaults and the lighting Kimbell’s first director, Richard Brown, the ground (Salvadori, 1980). However, reflectors were carefully designed to was especially concerned that “natural in the Kimbell Art Museum design, the control the quality and intensity of day- light should play a vital part in illumi- structure of the roof is pushed beyond light in the museum. Kahn writes [John- nation.” It may be conventional knowl- the conventional knowledge of typical son, 1975]: edge that natural light distracts the vault structures. Kahn’s design employs The scheme of enclosure of the mu- visual experience of viewers in a mu- an unconventional mixture of techniques seum is a succession of cycloid seum, but Kahn succeeded in using using those complicated post-tensioning vaults each of a single span [100] feet natural light as the main source for cables and diaphragms. It seems Komen- long and [23] feet wide, each form- lighting the museum galleries. Natural dant’s contribution to the cycloid struc- ing the rooms with a narrow slit to light penetrates through the slit in the ture for the museum was essential the sky, with a mirrored shape to vault and reflects off the mirrored shape (Komendant, 1975; Leslie, 2005). spread natural light on the side of of the convex perforated metal reflec- Mathematicians have long admired the vault. This light will give a glow tors back to the side of the interior vault the distinctive properties of the cycloid, of silver to the room without touch- and then diffuses to the exhibition and, more generally of the cycloid fam- ing the objects directly, yet give the room. Kahn describes ‘the luminosity ily, including epicycloids, hypocycloids, comforting feeling of knowing the of solver’ of the perforated metal re- epitrochoids, hypotrochoids, or car- time of day. flectors with cycloid vaults as follows: dioids. The length of one arch of a cy- The natural light was an integral part . . . rather a new way of calling cloid is four times the diameter of the of the program for the museum design something; it is rather a new word rolling circle, while the area under the from the beginning [Loud 1987]. The entirely. It is actually a modifier of arch is three times the area of the circle.

4 THE MATHEMATICAL INTELLIGENCER Figure 5. South courtyard with garden outdoor water fountain. AU2

Figure 6. The generation of the cycloid (after Louis I. Kahn).

Figure 8. Detailed photo showing the roof structure of the museum (left), and porticos with a side access path and reflecting pool (right).

Figure 9. A series of Kahn’s preliminary sketches for the roof, made in 1967, and how the light should infiltrate (Redrawn by the author).

Figure 10. A gallery, showing the transparent ceiling reflector (left), and a café and gallery areas, showing the north garden court with Maillol’s “L’Air” (right).

6 THE MATHEMATICAL INTELLIGENCER coid roof design and the natural light fixture of the museum in only one other building, the Wolfson Center for Me- chanical and Transportation Engineer- ing (1968–1977) on the university cam- pus of Tel Aviv.

ACKNOWLEDGMENT We thank Kazi Ashiraf for allowing the use of his Kimbell Museum photographs in this article.

REFERENCES The Kimbell Art Museum website: http://www. kimbellart.org/ Benedikt, Michael. 1991. Deconstructing The Kimbell. New York: Sites Books. Brownlee, David B., and De Long, David G. 1991. Louis I. Kahn: In the Realm of Archi- tecture. New York: Rizzoli International Pub- lications. Buttiker, Urs. 1994. Louis I. Kahn: Light and Space. New York: Whitney Library of Design. Gast, Klau-Peter. 1998. Louis I. Kahn: The Ideal Order. Basel, Switzerland: Birkhauser. Johnson, Nell E. 1975. Light is the Theme: Louis I. Kahn and The Kimbell Art Museum. Fort Worth: Kimbell Art Museum. Komendant, August E. 1975. 18 Years with Ar- chitect Louis I. Kahn. Englewood, NJ: Aloray Publisher. Latour, Alessandra. Intro and ed., 1991. Louis I. Kahn: Writings, Lectures, Interviews. New York: Rizzoli Inc. Leslie, Thomas. 2005. Louis I. Kahn: Building Art, Building Science. New York: George Figure 11. Structural details of the cycloid shell (after August Komendant), (a) Par- Braziller, Inc. tial side elevation of the cycloid roof, with post-tensioning cables; (b) Partial plan, Loud, Patricia Cummings. 1987. In Pursuit of where post-tensioning cables are installed; (c) End of a roof with a glass separa- Quality; The Kimbell Art Museum: An Illus- tion and walls (A-A ), and a cross-section (B-B ). trated History of the Art and Architecture. Kimbell Art Museum, Fort Worth, TX. The cycloid shape of the Kimbell Art space form embodying a mathematical Ronner, Heinz, and Jhaverti, Sharad. 1987. Museum is perhaps the most singular curve into a physical building. Kahn Louis I. Kahn: Complete Work 1935–1974. example ever applied in building used different types of arcs such as cir- Basel, Switzerland: Birkhauser. design. The visitor feels the full effect cle, semi-circle, and circle sectors in Salvadori, Mario. 1980. Why Buildings Stand of a quintessentially dignified cycloid other buildings, but he repeated his cyl- Up. New York: W.W. Norton & Company.

Jin-Ho Park, Yongsun Joo, Jae-Guen Yang, Inha University Kyodo Architects & Associates Inha University Department of Architecture 9-9 NIK Bldg., Sanei-cho, Shinjuku-ku Department of Architecture and Architectural Engineering Tokyo 160-0008 Japan and Architectural Engineering 253 Yongyun-dong, Nam-gu email: [email protected] 253 Yongyun-dong, Nam-gu Incheon 402-751 Korea Incheon 402-751 Korea e-mail: [email protected] email: [email protected]

AU2 Is this the blue, yellow, or green court or something else?