From Cartesian to Topological Geometry: Challenging Flatness in Architecture


This paper argues that recent topological expressions of surface in architecture have an intellectual lineage to late-modernism. Several key developments of that era challenged the limitations of a simplified Cartesian understanding of form. This paper commences with a discussion of Cartesian geometry in Modernism, relative to innovations in the building industry that promoted constancy, repetition and standardisation. Despite the same logic being transferred to digital working platforms where the architectural elements are normally designed as geometric prisms and their immediate derivatives, an interest in computation has enabled architecture to bypass orthogonal regularity. Concepts investigated during late-modernism—such as adaptability, disequilibrium, and smooth transitioning—are now central to performance and parametric-based design, examined through the interaction with data inputs, further pointing at critical updates of common software production tools.

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

Fig. 1

Source: Constantinos A. Doxiadis Archives. Constantinos and Emma Doxiadis Foundation

Fig. 2

Source: Yannis Zavoleas

Fig. 3

Source: Yannis Zavoleas

Fig. 4

Source: SAAI | Südwestdeutsches Archiv für Architektur und Ingenieurbau, Karlsruher Institut für Technologie

Fig. 5

Source: Yannis Zavoleas

Fig. 6

Source: Jenny Johnstone, Yannis Zavoleas, Peter Stevens, 2016


  1. Aish, Robert. 2013. First build your tools. In: Inside SmartGeomtery: Expanding the Architectural Possibilities of Computational Design, eds. Brady Peters and Terri Peters, 36–49. Chichester: Wiley.

    Google Scholar 

  2. Allen, Stan. 2001. Mat urbanism: The thick 2-D. In: Le Corbusier’s Venice Hospital, eds. Hashim Sarkis, Pablo Allard and Timothy Hyde, 118–126. New York: Prestel.

    Google Scholar 

  3. Banham, Reyner. 1955. The new brutalism. The Architectural Review, 118 (December): 354–361.

    Google Scholar 

  4. Cache, Bernard. 1995. Earth Moves: The Furnishing of Territories. Cambridge MA: The MIT Press.

    Google Scholar 

  5. Doxiadis, Constantinos A. 1963. Architecture in Transition. New York: Oxford University Press.

    Google Scholar 

  6. Doxiadis, Constantinos A. 1966. Between Dystopia and Utopia. London: Faber and Faber.

    Google Scholar 

  7. Drew, Philip. 1976. Frei Otto: Form and Structure. London: Crosby Lockwood Staples.

    Google Scholar 

  8. Garber, R. 2014. BIM Design: Realising the Creative Potential of Building Information Modelling. Chichester: Wiley.

    Google Scholar 

  9. Higman, Barry W. 2017. Flatness. London: Reaktion.

    Google Scholar 

  10. Johnston, Pamela. 1996. The Function of the Oblique: The Architecture of Claude Parent and Paul Virilio 19631969. London: AA Publications.

    Google Scholar 

  11. Lima, Adriana. 2014. Topological surface and digital fabrication: A generative approach. In: 2nd International Conference of Bio-digital Architecture and Genetics (Barcelona), ed. Alberto Estevez, 362–373.

  12. Lucan, Jacques. 1996. Introduction. In: The Function of the Oblique: The Architecture of Claude Parent and Paul Virilio 19631969, ed. Pamela Johnston, 5–9. London: AA Publications.

    Google Scholar 

  13. Lynn, Greg. 1993. Architectural curvilinearity: The folded, the pliant and the supple. In: Folding in Architecture. Architectural Design 102, ed. Greg Lynn, 8–15. West Sussex: Wiley.

  14. Lynn, Greg. 1999. Animate Form. New York: Princeton Architectural Press.

    Google Scholar 

  15. Migayrou, Frederic. 2002. Particularities of the minimum. In: Archilab’s futurehouse: Radical Experiments in Living Space, eds. Marie-Ange Brayer and Béatrice Simonot, 15–18. New York: Thames and Hudson.

    Google Scholar 

  16. Parent, Claude. P. 1996. A critical modernity. In: The Function of the Oblique: The Architecture of Claude Parent and Paul Virilio 1963-1969, ed. Pamela Johnston, 14–15. London: AA Publications.

  17. Rajchman, John. 1998. Constructions. Cambridge MA: The MIT Press

    Google Scholar 

  18. Sarkis, Hashim. 2001. Le Corbusier’s Venice Hospital. New York: Prestel.

    Google Scholar 

  19. Spuybroek, Lars. 2008. The Architecture of Continuity. Essays and Conversations. Rotterdam: V2 Publishing.

  20. Smithson, Alison. 1968. Team 10 Primer. Cambridge MA: The MIT Press.

    Google Scholar 

  21. Tzonis, Alexander. 2001. Le Corbusier: The Poetics of Machine and Metaphor. London: Thames and Hudson.

    Google Scholar 

  22. Tzonis, Alexander and Liane Lefaivre. 1999. Beyond monuments, beyond zip-a-tone, into space/time. In: Free University Berlin. Architectural Association Exemplary Projects 3. London: AA Publishing.

  23. Virilio, Paul. 1996. Architecture principe. In: The Function of the Oblique: The Architecture of Claude Parent and Paul Virilio 1963-1969, ed. Pamela Johnston, 11-14. London: AA Publications.

    Google Scholar 

  24. Virilio, Paul. 1998. The overexposed city. In: Architecture Theory Since 1968, ed. Michael K. Hays, 540–550. Cambridge MA: The MIT Press.

    Google Scholar 

  25. Vyzoviti, Sophia. 2003. Folding Architecture: Spatial, Structural and Organizational Diagrams. Amsterdam: BIS Publishers.

    Google Scholar 

  26. Vyzoviti, Sophia. 2013. The depth of surface: An evolutionary paradigm for digital architecture. In: International Theory of Architecture Conference ARCHTHE2013: Creativity, Autonomy, Function in Architecture (Istanbul), 284–292.

    Google Scholar 

  27. Wigley, Mark. 2001. Network fever. In: Grey Room 4: 83–122.

    Article  Google Scholar 

  28. Winston, Brian. 1998. Media, Technology, and Society: A History from the Telegraph to the Internet. London and New York: Routledge.

    Google Scholar 

  29. Zavoleas, Yannis. 2013. Digit mat(t)ers. Processes of normalization in architectural design. In: Surface: Digital Materiality and the New Relation Between Depth and Surface, EAAE Transactions on Architectural Education no 48, eds. Nikolas Patsavos and Yannis Zavoleas, 221–229. Athens and Brussels: Futura and EAAE.

  30. Zavoleas, Yannis. 2016. Rethinking architectural vocabulary. Organic resolution via integrated BIM platforms. In: Fifty Years Later: Revisiting the Role of Architectural Science in Design and Practice: 50 th International Conference of the Architectural Science Association (Adelaide), eds. Jian Zuo, Lyrian Daniel and Veronica Soebarto, 1–10.

  31. Zavoleas, Yannis and Tournikiotis, Panayotis. 2014. Archetypes in-formation. Strategies of transition in architecture and urban design. In: Cities in Transformation. Research and Design: Ideas, Methods, Techniques, Tools, Case Studies. Vol.II, eds. Marco Bovati, Michele Caja, Giancarlo Floridi and Martina Landsberger, 1411–1417. Padova: Il Poligrafo.

Download references

Author information



Corresponding author

Correspondence to Yannis Zavoleas.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zavoleas, Y., Taylor, M. From Cartesian to Topological Geometry: Challenging Flatness in Architecture. Nexus Netw J 21, 5–18 (2019).

Download citation


  • Topology
  • Flatness
  • Cartesian geometry
  • Soft architecture
  • Form-finding
  • BIM