6 NOTES

1.: For a comprehensive, ‘‘rich and definitive account of the life and work of the brilliant maverick from the western plains,’’ according to Helen Searing (whose 1996 Goff-designed house in Kansas City, Kansas, includes variations on extruded truncated hexagonal prisms), refer to David G. De Long, Bruce Goff. Toward Absolute Architecture, MIT Press, Cambridge, MA 1988.
2.: Refer to Sidney K. Robinson, ‘‘Bruce Goff and Music,’’ in The Architecture of Bruce Goff, 1904–1982, Art Institute of Chicago/Prestel, 1995.
3.: Examples of Goff’s buildings utilizing forms other than polyhedra include: the Ford/Robinson House, Aurora, Illinois, 1948 (torus, cone); the Bavinger House, 1950 (helix, cylinder); the Garvey House 1 project, 1952 (sphere, torus, helix); Sid Lodge, Crested Butte, Montana, 1965 (16-sided, polygonal pyramid approaching cone); and the Struckus House, Woodland Hills, California, 1983 (cylinder). In 1978 the Bavinger House was recognized with a ‘‘Twenty-Five Year Award’’ from the American Institute of Architects as an ‘‘extraordinary work of imagination …a vivid expression of American values.’’
4.: For example, the undulating and ragged ‘‘free forms’’ of the Don Leidig House project, 1946, and the Ledbetter House, Norman, Oklahoma, 1947. Tension structures also occur in Goff’s output, e.g., the aforementioned Bavinger and Lei-dig Houses, the Plunkett Guest House project, the Shin’enKan Museum in Los Angeles, etc. Tension structures employ cables deployed in straight-line configurations. Both two-and three-dimensional arrays of cables occur in Goff’s work, in the latter case resulting in subtle geometric volumes—cylindrical, conical, pyramidal, etc. Tension arrays were often used ornamentally by Goff, frequently as ‘‘curve stitching,’’ i.e., the overlapping of a group of straight lines such that their intersections result in polygonal or curved shapes.
5.: Many books illustrate and analyze these figures, e.g., Anthony Pugh, Polyhedra, A Visual Approach, University of California Press, 1976; Magnus J. Wenninger, Polyhedron Models, Cambridge University Press, 1971. The latter covers both regular and semi-regular polyhedra, or more generally, the 75 ‘‘uniform’’ polyhedra, including stellations.
6.: Crystallographers study centers, planes, and axes of symmetry and the relative orientation of x, y, and z axes in minerals, using terminology such as ‘‘monoclinic,’’ ‘‘brachypinacoid,’’ and ‘‘orthorhombic.’’ A recent concern of crystallographers is the hypercrystal, which is an irregularly organized, quasiperiodic structure. See Chapter 15 for a discussion of quasicrystals.
7.: Robert Lawlor, Sacred Geometry, Philosophy and Practice, Crossroad Publishing, New York, 1982, p. 104. Here is another source on polyhedra, ‘‘gnomonic expansion,’’ the spiral, the golden section, metaphor, and universal order, etc., with many good illustrations and diagrams.
8.: John Sergeant, Architectural Design Profiles 16, p. 55.
9.: D’Arcy Thompson, On Growth and Form, John Tyler Bonner, ed., Cambridge University Press, abridged Ed., 1966, p. 122. Bonner, a biomorphologist, is the author of Morphogenesis, An Essay on Development, Atheneum, 1963, which discusses three basic determinants of form in the biological world: growth, differentiation, and morphogenetic movement. As architectural metaphors, growth is cell (module) iteration or duplication; differentiation represents the functional variations of spaces such as living room, dining room, entry, and storage; and morphogenetic movement represents uniform characteristics carried throughout different parts of the same building, e.g., structural members, mechanical lines, and materials assemblies.
10.: The startling architecture of the zoological world is revealed in Karl von Frisch, Animal Architecture, Harcourt Brace Jovanovich, 1974. This book covers habitations and devices actually built by animals, not the anatomy of the animals themselves.
11.: A steel frame sculpture objectifies this abstraction in Venturi and Rauch’s Franklin Court, Philadelphia, 1972–1976. It is illustrated in Charles Jencks, The Language of Post-Modern Architecture, Rizzoli, 1984, p. 89.
12.: For a capsule visual history of architectural form covering 111 ‘‘styles,’’ including a profusion of recent entries, in 25 pages, see Chap. 1 of Ernest Burden, Elements of Architectural Design, A Visual Resource, Van Nostrand Reinhold, 1995. For a capsule history of Goff’s work, see the graphic, multipage pull-out section—pp. 7–14—in I’Architecture d’Aujourd Hui, No. 227, June 1983.
13.: Islamic ornamental patterns are illustrated and analyzed in Syed Jan Abas and Amer Shaker Salman, Symmetries of Islamic Geometrical Patterns, World Science, River Edge, NJ, 1995. The 17 unique symmetry groups are used as a basis for this study. Another source of clear graphic analysis of intertwining Islamic tessellation (not only in architecture, but in planning, poetry, and music!) is Issam E. Said and Ayse Parman, Geometric Concepts in Islamic Art, World of Islam Festival Publishing Co., London, 1976.

1
2Grids related to the Islamic can be found in Kaiyama Kyusaburo, The Book of Japanese Design—Bansho Shin Hinagata, Bijutsu Oyo, Crown, New York, 1969. Two-dimensional patterns attempt to rise off the paper into the third dimension—some quite similar to Goff designs in the Price Studio. The three-dimensional metal curtain walls of Goff’s First National Bank are related to these patterns.

14.: ‘‘The Architecture of Bruce Goff,’’ Architectural Design, May 1957, p. 151. Author and historian Park was the University of Oklahoma architectural school librarian during Goff’s chairmanship. Goff, incidentally, explaining the bland conformity throughout the profession, said, ‘‘We all read the same magazines.’’
15.: Francine du Plessix and Cleve Gray, ‘‘Bruce Goff, Visionary Architect,’’ Art in America, Winter 1965.
16.: See John C. Lilly, Programming and Metaprogramming in the Human Bio-Computer, Julian Press, 1972.
17.: Mentioned in an interview with Johnson in Vanity Fair, June 1993, p. 157.
18.: An increasing number of three-dimensional graphics programs are available for computers, e.g., Truespace. One can start with a polygon, drag (extrude) it to make a prism, shrink one end, see it in three dimensions, and rotate it—here is an embryo polyhedral unit of the Price Studio. Or one could call up a preformed ‘‘organic,’’ i.e., a single geometric volume and transform it by pulling it through a ‘‘deformation lattice’’ or ‘‘energy field.’’ Then one can apply a surface texture, shades, and shadows, etc. Artifice contains a ‘‘solid modeling program’’ in which a three-dimensional background ‘‘space’’ appears with the cursor attached to a ‘‘wireframe,’’ e.g., lines representing the x, y, and z axes, in perspective. Typical commands are ‘‘linear duplicate,’’ ‘‘wallify’’ and the like. Not to mention the ubiquitous Autocad and animation programs such as Lightwave and Video Toaster. Information can be obtained from Computer Chronicles Newsletter, 1–800–800–9520. See Chapter 12 for a discussion of computers and polyhedra.
19.: Introduction to A Portfolio of the Work of Bruce Goff, designed and compiled by William Murphy and Louis Muller, Architectural League of New York and American Federation of Arts, 1970. For an elaboration of these ideas, see Herb Greene, Mind and Image, Ait Essay on Art and Architecture, University Press of Kentucky, 1976, and Building to Last, Architecture as Ongoing Art, with Nanine Hillard Greene, Architectural Book Publishing Co., 1981. The Greenes go on to say (in the latter, p. 77), ‘‘Nature seems patient of as many systems of geometries as we can discover. At this time who can say which warrants the highest metaphysical status.’’
20.: Suzanne K. Langer, Feeling and Form, A Theory of Art Developed from Philosophy in a New Key, Scribner’s, New York, 1953, p. 99.
21.: Le Corbusier’s treatises: Modulorl irll, Harvard University Press, 1980. For a discussion of the golden mean or golden section, the ratio 4>> i-e., 1:1.618…as inherent in certain polyhedra, e.g., the icosahedron and dodecahedron, see H. E. Hunt-ley, The Divine Proportion, A Study in Mathematical Beauty, Dover, 1970.
22.: This taxonomic series could be continued with a similar grouping for curvilinear forms—first, extrusions of two-dimensional curves, then curves beyond the flat plane, i.e., spheres, etc., singly and in clusters—continuing further with combinations of spherical surfaces and polyhedral forms and finally ‘‘free forms’’ and beyond. A transition from the polyhedral to the curvilinear can be seen in polygons with increasing numbers of edges—but this is for another study. A number of Goff designs were curvilinear in the first go round and later revised to polygonal to fit the budget. Scaling down occurs with the Price Studio designs, but away from complex polyhedra rather than away from the curvilinear.

3
4In his article ‘‘Bruce Goff, The Strict Geometrist,’’ in Architectural Design Profiles 16 (a special issue on Goff), John Sergeant divides Goff’s work into crystalline (with three categories—rectangular, diagonal, and triangular) and curvilinear (four categories—circular, radial, spiral, and processional).

23.: For a discussion of depth cues, see James J. Gibson, Perception of the Visual World, Riverside Press, Cambridge, 1950.
24.: As a student of Goff in the early 1950s, I felt that Goff was neither particularly interested in nor impressed by Fuller’s ideas, unlike a few of his students, myself included. Goff was, rather, politely respectful and appreciative of Fuller’s work, as he was with that of technologists such as ’Ibrroja, Candela, Nervi, and Maillart, to name a few.

5
6The Louis Kahn/Ann lyng high-rise space frame office tower, part of Kahn’s Plan for Central Philadelphia, was well publicized at this time (mid-1950s). As a classroom assignment this writer attempted to design a ‘‘High-Rise Monastery for Downtown Houston,’’ incorporating living, circulation, and service spaces within a story-high tet-oct grid (i.e., a space frame whose structural members were composed of a ‘‘close packing’’ of alternating tetrahedra and octahedra).
7We also designed a Chinese restaurant using the Fuller/Kenneth Snelson concept of discontinuous compression in the form of a sphere, or, more accurately, a tensegrity-modulated icosidodecahedron, as a structural/envelope matrix. In both cases almost the entire time allotted for the projects was taken up just trying to figure out the projective geometrical drawing of these forms, which today would be a snap on the computer. A pentagonal prism core/mast of concrete containing vertical circulation, kitchens, mechanical equipment, etc., served and supported the tensegrity bubble, which enveloped and supported the dining platforms. Goff grasped the nature of the beast and was most helpful in overcoming problems for the utilization of these forms in an architectural setting.
8Just as a reminder of the typically strong effect Buckminster Fuller had on students across the country during the 1950s—another Goff student, New York architect Robert Tieger, says, ‘‘Fuller gave a talk at the O.U. Student Union. He talked on and on. The janitor wanted to turn the lights out at 10pm but Fuller went on ‘til about 12. It was the greatest thing I’d ever listened to.’’

25.: De Long, Bruce Goff.
26.: Michael Field and Martin Golubitsky, Symmetry in Chaos, A Search for Pattern in Mathematics, Art and Nature, Oxford University Press, 1992, p. 160.
27.: The variety of space-filling polyhedral units and their manipulation via twisting, tipping, and other metamorphoses are explored in Donald G. Wood, ‘‘Space Enclosure Systems, The Variables of Packing Cell Design,’’ Bulletin 205, Engineering Experimental Station, Ohio State University, 1968.
28.: De Long, Bruce Goff, p. 304.
29.: Ibid.
30.: Ibid., pp. 232–233.
31.: Kathy Nicol, letter to author, August 25, 1995.
32.: Architectural Design, May 1957, p. 173.
33.: De Long, Bruce Goff, pp. 152–153.
34.: John Sergeant, ‘‘Bruce Goff, The Strict Geometrist,’’ Architectural Design Profiles 16, Vol. 48, No. 10, 1978, p. 58.
35.: Ibid. This article provides thumbnail illustrations of some two dozen floor plans on a single page (!) for comparison. Sergeant’s ‘‘crystalline’’ categorization of Goff’s work is mentioned elsewhere in this chapter.
36.: The Lacey Hotel was, incidentally, a forerunner of a rhombic grid of glass cladding—recalled by Goff’s Crystal Chapel—although Wright’s diamonds were arranged in an overlapping shingle-like pattern, whereas Goffs were set flush within a supporting grid.
37.: As illustrated in George T. Kerr, ‘‘Synthetic Zeolites,’’ Scientific American, July 1989. Zeolites are minerals composed of silicon or aluminum and oxygen molecules, which form truncated octahedra. A cross section through the center of a truncated octahedron is an octagon. This may provide a clue to the subtle yet complex sense of space imparted in the Jones House—that complex, repeating, spatial symmetries can be suggested by a few pieces of the grid.
38.: For the full text of the letter, see De Long, Bruce Goff, p. 126.
39.: This preceded by some years a scene from the sci-fi film Zardoz, in which Sean Connery is trapped within a magic gemstone. Note Mrs. Warriner’s crystalline comment near the end of this chapter.
40.: De Long, Bruce Goff, p. 305. The quote within is from Herman George Schef-fauer, ‘‘Dynamic Architecture,’’ Dial, Vol. 70, March 1921, pp. 323–328.
41.: Ibid., p.13.
42.: For further descriptions of functions, see De Long, ibid., and Jeffrey Cook, The Architecture of Bruce Goff, Harper and Row, 1978, pp. 41–45.
43.: Sergeant, ‘‘Bruce Goff.’’
44.: De Long, Bruce Goff, p. 13.
45.: James J. Gibson, Perception of the Visual World, Riverside Press, Cambridge, 1950.
46.: See Dennis Sharp, Modern Architecture and Expressionism, Braziller, New York, 1966. An untoward side effect of some of these projects is the chilling recall of the Nietszchean mountain-climbing mystique in German films of the 1920s such as The Blue Light by Leni Riefenstahl, who later became a propagandist for Hitler.
47.: Ibid.
48.: Both illustrated in De Long, Bruce Goff.
49.: The Sierpinski arrowhead, a sponge-like tetrahedral fractal, will probably be used sooner or later as a basis for high-rise office or apartment buildings, if it hasn’t already. The figure is illustrated in John Briggs, Fractals, The Patterns of Chaos, Discovering a New Aesthetic of Art, Science and Nature, Simon and Schuster, 1992, p. 68.
50.: Reminiscent of the roof trusses of Goffs very low budget Hopewell Baptist Church, Edmund, Oklahoma, 1949. In this dodecagonal pyramid the subtlety of the basic ‘‘tepee’’ is increased by using over a dozen variations of the angle of the sloping roof—in walls, bell armature, truss chords, etc.
51.: Safdie, ‘‘Pyramid City,’’ p. 90; Christian W. Thomsen, Visionary Architecture, From Babylon to Virtual Reality, Prestel Verlag, Munich, New York, 1994, p. 164. Several others in this book—Bruno Taut’s 1918 ‘‘The Crystal Mountain,’’ p. 81, and Wassili Luckhardt’s 1919 ‘‘Project for a Sacred Building’’ in concrete and colored glass, p. 85.
52.: For Johnson’s comment about Goff’s Crystal Chapel, as well as high praise from other architects in support of the Crystal Chapel over a traditional period piece, see De Long, Bruce Goff, p. 104.
53.: This has a spherical/helical counterpart in Goff’s first Garvey House.
54.: At Expo ’67 in Montreal, multistoried truncated tetrahedra were constructed from space trusses for (appropriately) the Space Frame Exhibition Hall by architects Affleck, Desbarats, Dimikopolous, Lebensold, and Sise; structural engineers de Stein and Associates, Eskenazi, and Barass.
55.: Architectural Design, Vol. 28, May 1957, p. 163. More on the Wilson House can be found in Sergeant, ‘‘Bruce Goff,’’ p. 22.
56.: Progressive Architecture, Dec. 1962.
57.: L. Lines, Solid Geometry, Dover, 1965. Lines illustrates close packing in polyhe-dra such as hexagonal prisms, ‘‘rhomboidal dodecahedra,’’ ‘‘rhombo-hexagonal dodecahedra,’’ and truncated octahedra—in order to prove properties relating to vertices, centers, etc.
58.: See Haresh Lalvani, Transpolyhedra, Dual Transformations by Explosion—Implosion, Lalvani, New York, 1977. See also Chapter 14.
59.: Claude Bragdon, Projective Ornament, Dover, New York, 1992, p. 28.
60.: ‘‘Supercube’’ was a term used by Lester Walker, New York architect, teacher, and author, for a cubic device, totally different from the Wilson module, which he designed, patented, and built in the 1960s—a prefabricated, unfolding, expanding, ‘‘exploding’’ multipurpose studio furnishing module with a miscellany of built-in functions such as bed, desk, storage, and fighting.
61.: De Long, Bruce Goff, pp. 154, 325
62.: Architectural Design Profiles 16, p. 27.
63.: Anthony Pugh, Polyhedra, A Visual Approach, University of California Press, 1976, p.43.
64.: Laura Warriner, telephone conversation with author, 1994.
65.: Annemarie van Roessel, ‘‘Bruce Goff’s Built Works,’’ in The Architecture of Bruce Goff, 1904–1982, Design for the Continuous Present, Pauline Saliga, Mary Wbolever, and Sidney K. Robinson, eds., Art Institute of Chicago and Prestel Verlag, 1995—the catalog issued in conjunction with the 1995 exhibition of Goff’s architecture and paintings at the Chicago Art Institute. The Institute holds Goff’s archives.

6.1 BIBLIOGRAPHY

9Abas, S. J., and Salman, A. S.: Symmetries of Islamic Geometrical Patterns, World Science, River Edge, NJ, 1995.

10Baukunst und Werkform, No. 7, Frankfurt, 1953.

11 Bonner, J. T: Morphogenesis, An Essay on Development, Atheneum, 1963.

12 Bragdon, C.: Projective Ornament, Dover, New York, 1992.

13 Briggs, J.: Fractals, The Patterns of Chaos, Discovering a New Aesthetic of Art, Science and Nature, Simon and Schuster, New York, 1992.

14 Burden, E.: Elements of Architectural Design, A Visual Resource, Van Nostrand Reinhold, 1995.

15 Cook, J.: The Architecture of Bruce Goff, Harper and Row, New York, 1978.

16 Coxeter, H. S. M.: Introduction to Geometry, Wiley, New York, 1961.

17 De Long, D. G.: Bruce Goff, Toward an Absolute Architecture, MIT Press, Cambridge, MA, 1988.

18 Field, M., and Golubitsky, M.: Symmetry in Chaos, A Search for Pattern in Mathematics, Art and Nature, Oxford University Press, 1992.

19 Frisch, K. v.: Animal Architecture, Harcourt Brace Jovanovich, 1974.

20 Fuller, R. B.: Ideas and Integrities, A Spontaneous Autobiographical Disclosure, Robert W. Marks, ed., Collier Books, Macmillan, 1963.

21 Ghyka, M.: Geometrical Composition and Design, Alec Tiranti, London, 1952.

22 Gibson, J. J.: Perception of the Visual World, Riverside Press, Cambridge, 1950.

23Goff, Bruce, Architect (boxed folio), distributed by Prairie School Press, Chicago, 1978.

24 Gray, C., and du Plessix, E: ‘‘Bruce Goff, Visionary Architect,’’ in Art in America, Winter 1965.

25 Greene, H.: Mind and Image, An Essay on Art and Architecture, University Press of Kentucky, 1976.

26 Greene, H.: Introduction to A Portfolio of the Work of Bruce Goff, designed and compiled by Louis Muller and William Murphy, Architectural League of New York and American Federation of Arts, 1970.

27 Greene, H., with Greene, N. H.: Building to Last, Architecture as Ongoing Art, Architectural Book Publishing Co., 1981.

28 Hambidge, J.: The Elements of Dynamic Symmetry, Yale University Press, 1919, reprinted 1948.

29 Holden, A.: Shapes, Space and Symmetry, Columbia University Press, 1971.

30 Huntley, H. E.: The Divine Proportion, A Study in Mathematical Beauty, Dover, 1970.

31 Jencks, C.: The Language of Post-Modern Architecture, Rizzoli, 1984.

32 Johnson, P.: Vanity Fair, June 1993.

33 Kerr, G.T.: ‘‘Synthetic Zeolites,’’ Scientific American, July 1989.

34 Kyusaburo, K.: The Book of Japanese Design—Bansho Shin Hinagata, Bijutsu Oyo, Crown, New York, 1969.

35 Lalvani, H.: Transpolyhedra, Dual Transformations by Explosion-Implosion, Lalvani, New York, 1977.

36 Langer, S. K.: Feeling and Form, A Theory of Art Developedfrom Philosophy in a New Key, Scribner’s, New York, 1953.

37 Lawlor, R.: Sacred Geometry, Philosophy and Practice, Crossroad Publishing, New York, 1982.

38 Le Corbusier: Modtilor I ir II, Harvard University Press, 1980.

39 Lilly, J. C.: Programming and Metaprogramming in the Human Bio-Computer, Julian Press, 1972.

40 Lines, L.: Solid Geometry, Dover, 1965.

41 Mohri, T.: Bruce Goff in Architecture, Kenchiku Planning Center Co., Tokyo, 1970.

42 Park, B. A.: ‘‘The Architecture of Bruce Goff,’’ in Architectural Design, Vol. 28, May 1957.

43 Parman, A., and Said, I. E.: Geometric Coticepts in Islamic Art, World of Islam Festival Publishing Co., London, 1976.

44Progressive Architecture, Dec. 1962.

45 Pugh, A.: Polyhedra, A Visual Approach, University of California Press, 1976.

46 Robinson, S. K.: ‘‘Bruce Goff and Music,’’ in The Architecture of Bruce Goff, 1904–1982, Art Institute of Chicago/Prestel, 1995, catalog to the exhibition.

47 Scheffauer, H. G.: ‘‘Dynamic Architecture,’’ Dial,Vol. 70, March 1921.

48 Sergeant, J.: ‘‘Bruce Goff, The Strict Geometrist,’’ in Architectural Design Profiles 16, Vol. 48, No. 10, 1978.

49 Sharp, D.: Modem Architecture and Expressionism, Braziller, New York, 1966.

50 Thompson, D’A. W.: On Growth and Form, Abridged ed., John Tyler Bonner, ed., Cambridge University Press, 1966.

51 Thomsen, C.W.: Visionary Architecture, From Babylon to Virtual Reality, Prestel Verlag, Munich, New York, 1994.

52 van Roessel, A.: ‘‘Bruce Goff’s Built Works,’’ in The Architecture of Bruce Goff, 1904–1982, Design for the Continuous Present, Pauline Saliga, Mary Woolever, and Sidney K. Robinson, eds., Art Institute of Chicago and Prestel Verlag, 1995.

53 Wenninger, M. J.: Polyhedron Models, Cambridge University Press, 1971.

54 Wood, D. G.: ‘‘Space Enclosure Systems, The Variables of Packing Cell Design,’’ Bulletin 205, Engineering Experimental Station, Ohio State University, 1968.