Abstract
This present survey, based on personal experiences, focuses on the role that mathematical thought plays in furthering generic knowledge of architectural form, and is organised around the three categories identified by Froebel’s proposal for the study of form—quantitative, qualitative and relational . Quantitative studies measure the various material dimensions of form: length, area, volume. The quantitative relationships between building forms, heights, areas of sites, road widths, open areas are examined in early “land use and built form studies”. Qualitative studies of form emphasize spatial transformations, the modern group theory of symmetry, and topological relations. Distinct architectural forms may well be equivalent under symmetry, or topology. An example is given of an application of Polya’s enumeration theorem. Relational studies of form were said, by Froebel, to be “forms of life”. Relations of dimensions, symmetries and topology are now addressed by advances in shape computation.
First published as: Lionel March , “Architecture and Mathematics Since 1960”, pp. 9–33 in Nexus IV: Architecture and Mathematics, Kim Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books, 2002.
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Notes
- 1.
First in architecture at Cambridge University, for two years in urban studies at Harvard and MIT, back at Cambridge as Director of the Centre for Land Use and Built Form Studies, then in systems engineering at the University of Waterloo, in design technology at The Open University, later at the Royal College of Art, London, and—for the last two decades—in architecture and urban design at the University of California, Los Angeles, with a 6-month stop-over as a consultant at The National University of Singapore.
- 2.
For example, I co-authored a paper with the graph theorists Frank Harary in the USA and R. W. Robinson in Australia in 1978 entirely by correspondence. I met Harary some years later at a cocktail party in Cambridge, but Robinson, never.
- 3.
Professor of Architecture, University of Cambridge. Best known in Portugal for his work at the Gulbenkian Foundation in Lisbon, but more widely for the Royal Festival Hall, London.
- 4.
Friedrich Froebel (1782–1852) pioneered the modern study of form. He employed the three Aristotelian categories of quantity, quality and relation in structuring the educational content of his “gifts” which ran the gamut from solid, plane, line to point in descending order of concreteness, and ascending order of abstraction. His categories were “forms of knowledge” in which quantitative aspects are studied; “forms of beauty” in which the qualities of spatial transformations and symmetries come into play; and “forms of life” in which the forms are related to actual objects—a house, a bath, a chair, and so on. I use these categories to organize this chapter.
- 5.
Engineer Sr. Luis Lobato was an enthusiastic protagonist of this work.
- 6.
- 7.
The unit of distance taken in this example is half a block length. The mathematical question of mean distances is addressed in Baglivo and Graver (1983: 98–111). An architectural investigation of built forms is found in Tabor (1971). On the assumption that all trips are equally likely, it should be noted that the compact city-section favors short trips over the other two configurations, whereas the annular form has an even distribution. In practice, trips are not likely to be equiprobable, and this needs to be factored in according to expectations.
- 8.
For a practical application of these ideas to the Central Region of Chile, see Echenique (1994).
- 9.
“Ruralize the urban, urbanize the rural…fill the earth.” I was first introduced to the works of Ildefons Cerdà by Dr. Marcial Echenique who had joined LUBFS from Barcelona to direct the urban systems study. A useful summary is Soria y Puig (1999).
- 10.
The Sonoran Desert covers about 222,700 km2. in California and Arizona in the United States, and Baja California and Sonora in Mexico. It is the subject of a case study by the Nature Conservancy in AAAS (2000a: 188–89). In 1995 the regional population was 5.5 million, growing at the rate of 3.0 % per year.
- 11.
Froebel had trained as an architect in Frankfurt, but became an assistant to Christian Samuel Weiss, one of the founders of the modern science of crystallography.
- 12.
Joseph-Louis Lagrange (1736–1813) was the founding Professor of Mathematics at L’Ecole Polytecnique, Paris, during the tenure of the architect J. N. L Durand in the stereotomy department under Abbé Haüy—Weiss’s French rival in the architecture of crystal forms.
- 13.
There are just seven non-infinite spherical groups in space, two each associated with the Platonic duals, the cube/octahedron and the dodecahedron/icosahedron, and three related to the self-dual tetrahedron.
- 14.
The interior of La Rotonda has the subsymmetry D2.
- 15.
This concept arose initially from Newman (1964). Newman had directed the effort to build the world’s first programmable, electronic computer at the University of Manchester, 21 June 1948. He is especially regarded for having recruited Alan Turing —of Turing Machine fame—to his staff.
- 16.
- 17.
See, for example, Steadman , P. (1983). Architectural Morphology: An Introduction to the Geometry of Building Plans. London: Pion Ltd.
- 18.
“Meng Ker introduced me to the book—(March 1972)—and from there we did some morphological studies together and he did his thesis on Woodlands, one neighbourhood of Woodlands. And he redesigned a neighbourhood, the typical HDB [Housing Development Board] neighbourhood which was 10, 12, storeys at that time. He redesigned it to 5, 6 storeys, with one-third of the dwellings having gardens. So, to answer your question, are there alternatives to HDB, there are plenty of alternatives” (Bay 1998).
- 19.
See Pólya et al. (1983), where the lively “voice” of the nonagenarian Pólya can be heard still teaching at Stanford University, California.
- 20.
For an excellent introduction to Polyá’s work in the design context, I recommend (Economou 1999).
- 21.
- 22.
The work of Dr. Mário Júlio Teixeira Krüger, University of Coimbra, relates the space syntax tradition with the LUBFS approach. Dr. Nuno Portas was an important link with the Cambridge school following the 1974 Portuguese Revolution, as was Manuel Solá-Morales Rubó in Barcelona, who steered through the 1975 Spanish edition of Urban Space and Structures.
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Acknowledgment
I dedicate this chapter to the memory of Dr. John Ashby (1928–1999), a biochemist, and who, as a publisher with Pion Limited, London, had the foresight and courage to nurture and promote the journal Environment and Planning B.
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March, L. (2015). Mathematics and Architecture Since 1960. In: Williams, K., Ostwald, M. (eds) Architecture and Mathematics from Antiquity to the Future. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00143-2_38
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