Skip to main content
SpringerLink
Log in

A Mathematical Formalization of Making Grammars

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

Abstract

Making grammars model makers as perceiving their environment and acting to transform it. Shape grammars, which model designers as manipulating shapes, are then a special case of making grammars where perceiving is limited to seeing and doing is limited to drawing. This paper develops set-theoretical and graph-theoretical formalizations of making grammars. Existing set-theoretical formalizations of shape grammars show that designing is visual computing by demonstrating that a shape grammar can be devised to simulate any Turing machine. However, there are no mathematical arguments showing how making grammars describe computation. Using a previously published example of a shape grammar and its corresponding making grammar, this paper combines ideas from ecological psychology with methods from set theory and graph theory to show that, for each Turing machine, a making grammar can be devised which simulates it.

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

Access this article

Log in via an institution

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

Image: author based on Knight and Stiny 2015:4; Charm 2010; and Rfc1394 2005

Fig. 7
Fig. 8

Similar content being viewed by others

Availability of Data and Material

Not applicable.

Code Availability

Not applicable.

References

  • Agarwal, Manish, and Jonathan Cagan. 2000. “On the Use of Shape Grammars as Expert Systems for Geometry-Based Engineering Design.” Artificial Intelligence for Engineering Design, Analysis and Manufacturing 14 (5): 431–39.

    Article  Google Scholar 

  • Aksamija, Ajla, Kui Yue, Hyunjoo Kim, Francois Grobler, and Ramesh Krishnamurti. 2010. “Integration of Knowledge-Based and Generative Systems for Building Characterization and Prediction.” Artificial Intelligence for Engineering Design, Analysis and Manufacturing, no. 24: 3–16.

    Article  Google Scholar 

  • Benrós, D., Duarte J.P., Hanna S. 2012. “An Alternative Palladian Shape Grammar: A Subdivision Grammar for Palladian Villas.” In Beyond Codes and Pixels, 415–24. Beyond Codes and Pixels. Proceedings of the 17th International Conference on Computer-Aided Architectural Design Research in Asia, CAADRIA 2012. Chennai: CADRIA.

  • Charm. 2010. Pencil. https://en.wikipedia.org/wiki/File:Pencil-db.jpg.

  • Copeland, B. Jack, ed. 2004. The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma. Oxford: Oxford University Press.

    Google Scholar 

  • Duarte, José Pinto. 2001. “Customizing Mass Housing: A Discursive Grammar for Siza’s Malagueira Houses.” Thesis, Massachusetts Institute of Technology.

  • Dutta, Arindam. 2007. The Bureaucracy of Beauty: Design in the Age of Its Global Reproducibility. New York: Routledge.

    Google Scholar 

  • Gibbs, Raymond W, ed. 2008. The Cambridge Handbook of Metaphor and Thought. Cambridge: Cambridge University Press.

    Google Scholar 

  • Gibson, James Jerome. 1979. The Ecological Approach to Visual Perception. Boston: Houghton Mifflin.

    Google Scholar 

  • Granadeiro, Vasco, José Pinto Duarte, João R. Correia, and Vítor M.S. Leal. 2013. “Building Envelope Shape Design in Early Stages of the Design Process: Integrating Architectural Design Systems and Energy Simulation.” Automation in Construction, no. 32: 196–209.

    Article  Google Scholar 

  • Grasl, Thomas, and Athanassios Economou. 2010. “Palladian Graphs: Using a Graph Grammar to Automate the Palladian Grammar.” In Future Cities, 275–83. Zurich: Education and research in Computer Aided Architectural Design in Europe.

  • Ingold, Tim. 2013. Making: Anthropology, Archaeology, Art and Architecture. Milton Park, Abingdon, Oxon: Routledge.

    Book  Google Scholar 

  • Jowers, Iestyn, Chris Earl, and George Stiny. 2019. “Shapes, Structures and Shape Grammar Implementation.” Computer-Aided Design 111 (June): 80–92.

    Article  MathSciNet  Google Scholar 

  • Jowers, Iestyn, Chris Earl, and George Stiny. 2017. “Shape Computations Without Compositions.” In Computer-Aided Architectural Design. Future Trajectories, edited by Gülen Çağdaş, Mine Özkar, Leman Figen Gül, and Ethem Gürer, 348–65. Communications in Computer and Information Science. Singapore: Springer

  • Jungnickel, Dieter. 2005. Graphs, Networks and Algorithms. 2nd ed. Algorithms and Computation in Mathematics. Berlin, Heidelberg: Springer.

    Google Scholar 

  • Kamath, Ayodh Vasant. 2020. “Making Grammars for Material and Tectonic Complexity: An Example of a Thin-Tile Vault.” Design Studies 69 (C).

  • Knappett, Carl. 2004. “The Affordances of Things: A Post-Gibsonian Perspective on the Relationality of Mind and Matter.” In Rethinking Materiality: The Engagement of Mind with the Material World, 43–51. Cambridge: McDonald Institute for Archaeological Research.

    Google Scholar 

  • Knight, Terry. 2015. “Shapes and Other Things.” Nexus Network Journal 17 (3): 963–80.

    Article  MathSciNet  Google Scholar 

  • Knight, Terry. 2018. “Craft, Performance, and Grammars.” In Computational Studies on Cultural Variation and Heredity, edited by Ji-Hyun Lee, 205–24. Singapore: Springer Singapore.

    Chapter  Google Scholar 

  • Knight, Terry, and George Stiny. 2015. “Making Grammars: From Computing with Shapes to Computing with Things.” Design Studies, Special Issue: Computational Making, 41 (Part A): 8–28.

    Google Scholar 

  • Koutamanis, Alexander. 2000. “Reversal of a Shape Grammar for Indexing and Retrieval.” In Artificial Intelligence in Design ’00, edited by John S. Gero, 225–46. Dordrecht: Springer Science & Business Media.

    Chapter  Google Scholar 

  • Krstic, Djordje. 2019. “Grammars for Making Revisited.” In Design Computing and Cognition ’18, edited by John S. Gero, 479–96. Springer International Publishing.

  • Labrou, Katerina, and Sotirios D. Kotsopoulos. 2023. “Making Grammars for Computational Lacemaking.” In Design Computing and Cognition’22, edited by John S. Gero, 587–604. Cham: Springer International Publishing.

    Chapter  Google Scholar 

  • Lakoff, George, and Mark Johnson. 1980. Metaphors We Live By. Chicago: The University of Chicago Press.

    Google Scholar 

  • Mahankali, Ranjeeth, Brian R Johnson, and Alex T Anderson. 2018. “Deep Learning in Design Workflows: The Elusive Design Pixel.” International Journal of Architectural Computing 16 (4): 328–40.

    Article  Google Scholar 

  • Meadows, Donella. 2008. Thinking in Systems: A Primer. White River Junction: Chelsea Green.

  • Ortony, Andrew, ed. 1993. Metaphor and Thought. 2nd ed. Cambridge: Cambridge University Press.

    Google Scholar 

  • Prigogine, Ilya, and Isabelle Stengers. 1984. Order out of Chaos: Man’s New Dialogue with Nature. New York: Bantam Books.

    Google Scholar 

  • Rfc1394. 2005. A sheet of paper. https://commons.wikimedia.org/wiki/File:Paper_450x450.jpg.

  • Srinivasan, Ravi, and Kiel Moe. 2015. The Hierarchy of Energy in Architecture: Emergy Analysis. New York: Routledge.

    Book  Google Scholar 

  • Stiny, George. 1980. “Introduction to Shape and Shape Grammars.” Environment and Planning. B 7: 343–51.

    Article  Google Scholar 

  • Stiny, George. 1990. “What Is a Design.” Environment and Planning B: Planning and Design 17: 97–103.

    Article  Google Scholar 

  • Stiny, George. 1996. “Useless Rules.” Environment and Planning B: Planning and Design 23 (2): 235–37.

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Stiny, George, and William J Mitchell. 1978. “The Palladian Grammar.” Environment and Planning B: Planning and Design 5 (1): 5–18.

    Article  Google Scholar 

  • Stiny, George, and James Gips. 1971. “Shape Grammars and the Generative Specification of Painting and Sculpture.” In Segmentation of Buildings for 3d Generalisation. in: Proceedings of the Workshop on Generalisation and Multiple Representation. Leicester.

  • Stiny, George, and Lionel March. 1981. “Design Machines.” Environment and Planning B: Urban Analytics and City Science 8 (3).

  • Stiny, George. 1975. Pictorial and Formal Aspects of Shape and Shape Grammars. Basel; Stuttgart: Birkhäuser.

  • Strobbe, Tiemen, Sara Eloy, Pieter Pauwels, Ruben Verstraeten, Ronald De Meyer, and Jan Van Campenhout. 2016. “A Graph-Theoretic Implementation of the Rabo-De-Bacalhau Transformation Grammar.” Artificial Intelligence for Engineering Design, Analysis and Manufacturing 30 (2): 138–58.

    Article  Google Scholar 

  • Trudeau, Richard J. 1993. Introduction to Graph Theory. New York: Dover Publications.

    Google Scholar 

  • Turnbull, David. 1993. “The Ad Hoc Collective Work of Building Gothic Cathedrals with Templates, String, and Geometry.” Science, Technology, & Human Values 18 (3): 315–40.

    Article  MathSciNet  Google Scholar 

  • Turnbull, David. 2000. Masons, Tricksters and Cartographers: Comparative Studies in the Sociology of Scientific and Indigenous Knowledge. London: Routledge.

    Google Scholar 

  • Vardouli, Theodora. 2015. “Making Use: Attitudes to Human-Artifact Engagements.” Design Studies, Special Issue: Computational Making, 41 (November): 137–61.

    Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. O.P. Jindal Global University, Sonipat, Haryana, India

    Ayodh Vasant Kamath

Authors
  1. Ayodh Vasant Kamath
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ayodh Vasant Kamath.

Ethics declarations

Conflict of Interest

Not applicable.

Ethical Approval

Not applicable.

Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

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

Kamath, A.V. A Mathematical Formalization of Making Grammars. Nexus Netw J 25, 985–998 (2023). https://doi.org/10.1007/s00004-023-00731-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00004-023-00731-2

Keywords

Search

Navigation