Cycloids in Louis I. Kahn's Kimbell Art Museum at Fort Worth, Texas
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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 ScienceϩBusiness 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 F1Ǟ 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&3Ǟ 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 exam- ple of a strategic use of the mathemat- ical 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. ǟF5 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 begin- ning 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- ǟF6 cloid structure is easily recognized from Figure 3. Site plan of the Kimbell Art Museum. a distance. Each vault consists of 100 ϫ 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 ǟF4 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 ϫ 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). ǟF7 part between them has four vaults. The floor below, to the conservator’s studio. ǟF8 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 AU1Ǟ and a kitchen. All are arranged around the kind of light that I anticipate shapes and light reflectors (see figure © 2007 Springer ScienceϩBusiness 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 con- trol is now possible.