Skip to main content
Log in

Skeletons, Shapes, and the Shift from Surface to Structure in Architectural Geometry

  • Research
  • Published:
Nexus Network Journal Aims and scope Submit manuscript

Abstract

Architects who digitally manipulate geometry confront a rift between what is being displayed on the screen (metric shapes) and what is being computed (their non-metric skeletons). This article critically reads this relationship between surface appearance and abstract structure against a historical backdrop of changing attitudes toward the visual world in postwar architectural and mathematical cultures. First, it examines skeletal (graph theoretic) representations of floor plan geometry advanced in centres of architectural research at the University of Cambridge and the Open University. Then, it interprets this work’s technical and discursive outputs in the context of contemporaneous disciplinary and pedagogical debates around geometry in British mathematics. By positioning skeletons (graphs) in a genealogy of ambivalence toward concrete appearance, this article seeks to activate critical historical perspectives on descriptions of geometry currently reified in computer software.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Alexander, Christopher. 1964. Notes on the Synthesis of Form. Cambridge, Mass.: Harvard University Press.

    Google Scholar 

  • Ayzenberg, Vladislav, and Stella F. Lourenco. 2019. Skeletal Descriptions of Shape Provide Unique Perceptual Information for Object Recognition. Scientific Reports 9 (1): 9359.

    Article  Google Scholar 

  • Bærentzen, J. A., M. K. Misztal, and K. Welnicka. 2012. Converting Skeletal Structures to Quad-dominant Meshes. Computers and Graphics 36 (5): 555–561.

    Article  Google Scholar 

  • Berge, Claude. 1962. The Theory of Graphs and Its Applications. Alison Doig, trans. New York: Wiley; London: Methuen & Co., Ltd.

  • Bloch, C. J. and Ramesh Krishnamurti. 1978. The Counting of Rectangular Dissections. Environment and Planning B: Planning and Design, 5(2), 207–214.

    Article  Google Scholar 

  • Brooks, Rowland L., Cedric A. B. Smith, Arthur H. Stone, William T. Tutte. 1940. The Dissection of Rectangles into Squares. Duke Mathematical Journal 7: 312–340.

    Article  MathSciNet  Google Scholar 

  • Cameron, Peter J. 2011. Aftermath: A Personal View of Combinatorics. In Combinatorics, Ancient and Modern, eds. J. J. Watkins and R. J. Wilson, 355–66. Oxford University Press.

  • Chermayeff, Serge. 1961. Let Us Not Make Shapes: Let Us Solve Problems. Box 21. Serge Ivan Chermayeff Architectural Records and Papers, 1909-1980. Dept. of Drawings & Archives, Avery Architectural and Fine Arts Library, Columbia University.

  • Combes, L. 1976. Packing Rectangles into Rectangular Arrangements. Environment and Planning B 3: 3–32.

    Article  Google Scholar 

  • Dieudonné, J. A. 1961. Professor Dieudonné’s Address. In New Thinking in School Mathematics, 31–48. Paris: Organisation for European Economic Co-operation, and Office for Scientific and Technical Personnel.

  • Earl, C. F. 1977. A Note on the Generation of Rectangular Dissections. Environment and Planning B 4: 241–246.

    Article  Google Scholar 

  • Earl, C. F., 1978. Joints in Two- and Three-dimensional Rectangular Dissections. Environment and Planning B 5: 179–187.

    Article  Google Scholar 

  • Ferreirós, J., and J. J. Gray, eds. 2006. Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford ; New York: Oxford University Press.

    MATH  Google Scholar 

  • Flemming, Ulrich. 1978. Representation and Generation of Rectangular Dissections. In Proceedings of the 15th Design Automation Conference (DAC ‘78). Piscataway, NJ: IEEE Press, 138-144.

  • Goh, W. B. 2008. Strategies For Shape Matching Using Skeletons. Computer Vision and image Understanding 110 (3): 326-345.

    Article  Google Scholar 

  • Gray, Jeremy. 2006. Modern Mathematics as a Cultural Phenomenon. In Architecture of Modern Mathematics: Essays in History and Philosophy, eds. J. Ferreirós and J. J. Gray, 371–96. Oxford and New York: Oxford University Press.

    MATH  Google Scholar 

  • Gray, Jeremy. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton, N.J.: Princeton University Press.

    Book  Google Scholar 

  • Grattan-Guinness, I. 2009. Mathematics Ho! Which Modern Mathematics Was Modernist? The Mathematical Intelligencer 31 (4): 3–11.

    Article  MathSciNet  Google Scholar 

  • Harary, Frank. 1971. Aesthetic Tree Patterns in Graph Theory. Leonardo 4 (3): 227–31.

    Article  Google Scholar 

  • Hubert, Evelyne, and Marie-Paule Cani. 2012. Convolution Surfaces Based on Polygonal Curve Skeletons. Advances in Mathematics Mechanization 47 (6): 680–99.

    MathSciNet  MATH  Google Scholar 

  • Ingold, T. 2013. Of Blocks and Knots: Architecture as Weaving. The Architectural Review, 26–27.

  • König, Dénes. 1990. Theory of Finite and Infinite Graphs. Translated by Richard McCoart. Boston-Basel-Berlin: Birkhäuser.

  • Mashaal, Maurice. 2006. Bourbaki: A Secret Society of Mathematicians. Trans. Anna Pierrehumbert. Providence, RI: American Mathematical Society.

  • March, Lionel. 2015. Architecture and Mathematics Since 1960. In Architecture and Mathematics from Antiquity to the Future, Kim Williams and Michael J. Ostwald, eds., vol. II, 553-578. Basel: Birkhäuser.

  • March, Lionel, and Philip Steadman. 1974. The Geometry of Environment: An Introduction to Spatial Organization in Design. Cambridge, Mass.: The MIT Press.

    Google Scholar 

  • March, Lionel and Philip Steadman. 1978. From Descriptive Geometry to Configurational Engineering. In Proceedings. International Conference on Descriptive Geometry, G. K. Hilliard, ed., 21–24. Washington, DC: American Society for Engineering Education.

  • Martin, Leslie. 1967. Architect’s Approach to Architecture. RIBA Journal 74 (5): 191–200.

    Google Scholar 

  • Mitchell, W. J., J. P. Steadman and Robin S, Liggett. 1976. Synthesis and Optimization of Small Rectangular Floor Plans. Environment and Planning B: Planning and Design 3 (1): 37–70.

    Article  Google Scholar 

  • Nuffield Mathematics Project. 1969. Nuffield Mathematics: Environmental Geometry. London: Chambers and Murray.

    Google Scholar 

  • Perzylo, Alexander, Nikhil Somani, Markus Rickert, and Alois Knoll. 2015. An Ontology for CAD Data and Geometric Constraints as a Link between Product Models and Semantic Robot Task Descriptions. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 4197–4203. Hamburg, Germany. https://doi.org/10.1109/iros.2015.7353971.

  • Phillips, Christopher J. 2014. The New Math: A Political History. Chicago: University of Chicago Press.

    Book  Google Scholar 

  • Quadling, Douglas. 1972. The Mathematics of SMP. In The School Mathematics Project: The First Ten Years, ed. Bryan Thwaites, 217–21. London: Cambridge University Press.

  • Retsin, Gilles (ed.). 2019. Discrete: Reappraising the Digital in Architecture. Architectural Design 89 (2). John Wiley & Sons, Ltd.

  • Steadman, Philip. 1970. The Automatic Generation of Minimum-Standard House Plans. LUBFS Working Paper 23 (March).

  • Steadman Philip. 1983. Architectural Morphology: An Introduction to the Geometry of Building Plans. London: Pion Ltd.

    Google Scholar 

  • Steadman, P., Brown, F., and P. Rickaby. 1991. Studies in the Morphology of the English Building Stock. Environment and Planning B: Planning and Design, 18(1): 85–98.

    Article  Google Scholar 

  • Steadman Philip. 2016. Interview by Theodora Vardouli. Email. Published in Vardouli Theodora, Graphing Theory: New Mathematics, Design, and the Participatory Turn, Ph.D. dissertation, Massachusetts Institute of Technology, 2017.

  • Steingart, Alma. 2011. Conditional Inequalities : American Pure and Applied Mathematics, 1940-1975. Ph.D. United States—Cambridge, Mass.: Massachusetts Institute of Technology.

  • Steingart, Alma. 2020. The Axiomatic Aesthetic. In Computer Architectures: Constructing the Common Ground, eds. Theodora Vardouli and Olga Touloumi, London: Routledge.

    Google Scholar 

  • Stiny, George. 1980. Introduction to Shape and Shape Grammars. Environment and Planning B: Planning and Design 7 (3): 343–51.

    Article  Google Scholar 

  • Stiny, George. 1982. Letter to the Editor: Spatial Relations and Grammars. Environment and Planning B 9: 113–14.

    Article  Google Scholar 

  • Stiny, George. 2006. Shape: Talking about Seeing and Doing. Cambridge, Mass.; London: MIT Press.

  • Stiny, George, and James Gips. 1972. Shape Grammars and the Generative Specification of Painting and Sculpture. In Proceedings of IFIP Congress 1971. Amsterdam: North Holland Publishing Co.

  • Strang, Gilbert. 2008. Lecture 13: Kirchhoff's Current Law (Computational Science and Engineering I). MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/video-lectures/lecture-13-kirchhoffs-current-law/

  • Tagliasacchi, A., T. Delame, M. Spagnuolo, N. Amenta, and A. Telea. 2016. 3D Skeletons: A State-of-the-Art Report. In Eurographics 35(2), ed. J. Madeira, G. Patow, and T. Romao.

  • Tutte, W. T. 1990. Commentary. In König, Dénes. Theory of Finite and Infinite Graphs. Translated by Richard McCoart. Boston-Basel-Berlin: Birkhäuser.

  • Wheatley, Tim. 1972. S.M.P. and Curriculum Development. In The School Mathematics Project: The First Ten Years, ed. Bryan Thwaites, 222–24. London: Cambridge University Press.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Theodora Vardouli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vardouli, T. Skeletons, Shapes, and the Shift from Surface to Structure in Architectural Geometry. Nexus Netw J 22, 487–505 (2020). https://doi.org/10.1007/s00004-020-00478-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00004-020-00478-0

Keywords

Navigation