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Grammars for Making Revisited

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Design Computing and Cognition '18 (DCC 2018)

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Abstract

The domain of shape grammars has been recently extended to include the field of making. This paper examines what makes abstract shapes look like concrete spatial entities (things) and what changes to the shape grammar formalism are needed to support calculations with things. Several new grammars capable of handling things are developed. Algebras supporting these are briefly addressed as well.

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Notes

  1. 1.

    See also other papers in Computational Making, Design Studies Special Issue, November 2015, Vol. 41 Part A.

  2. 2.

    Advances in Implemented Shape Grammars: Solutions and Applications AIEDAM Special Issue, Spring 2018, Vol. 32, No. 2.

  3. 3.

    Formally: x ≤ u + v ⇒x ≤ u | x ≤ v; x ≤ u − v ⇒ x ≤ u & ¬(x ≤ v); x ≤ u · v ⇒ x ≤ u & x ≤ v and x ≤ u ⊕ v ⇒ [x ≤ u & ¬(x ≤ v)] | [x ≤ v & ¬(x ≤ u)], where &, |, ¬, and ⇒ stand for and, or, negation, and implication, respectively.

  4. 4.

    Based on Boolean identity (u − v) · w = u · (v − w), validity of which is shown by x ≤ (u − v) · w ⇒ x ≤ (u − v) & x ≤ w ⇒ x ≤ u & ¬(x ≤ v) & x ≤ w ⇒ x ≤ u & x ≤ w − v ⇒ x ≤  u · (w − v) so that (u − v) · w ≤ u · (w − v); x ≤  u · (w − v)  ⇒ x ≤ u & x ≤ (w − v) ⇒ x ≤ u & x ≤ w & ¬(x ≤ v) ⇒ x ≤ u − v & x ≤ w ⇒ x ≤ (u − v) · w so that u · (w − v) ≤ (u − v) · w and finally (u − v) · w = u · (w − v).

  5. 5.

    Indeed, rules a → b and g(a) → g(b), where g is a transformation, are equivalent. If the former applies to shape c under transformation t such that t(a) ≤ c then the latter applies under tg−1. That is, tg−1(g(a)) = t(a) ≤ c and [c − tg−1(g(a))] + tg−1(g(b)) = (c − t(a)) + t(b) = c′.

  6. 6.

    (x, x) ∈ R, (x, y) ∈ R ⇒ (y, x) ∈ R, (x, y) ∈ R & (y, z) ∈ R ⇒ (x, z) ∈ R, respectively, where R is a relation and x, y, and z are elements of the set on which R is defined.

  7. 7.

    For b − a < u < b, no combination of c, t(a − b) and t(u) can produce t(a) and t(b), and no combination of c, t(a) and t(b) can produce t(u) so that Ar(3) ≠ Ar(10). For u = b − a and u = b, no combination of c, t(a − b) and t(b − a) can produce t(a) and t(b), no combination of c, t(a − b) and t(b) can produce t(a), and combining t(a) and t(b) can produce t(a − b) and t(b − a) so that Ar(10) ⊂ Ar(3).

References

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Author information

Authors and Affiliations

  1. BioSig Technologies, Los Angeles, USA

    Djordje Krstic

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  1. Djordje Krstic
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Correspondence to Djordje Krstic .

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Editors and Affiliations

  1. Department of Computer Science and School of Architecture, University of North Carolina at Charlotte, Charlotte, NC, USA

    John S. Gero

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Krstic, D. (2019). Grammars for Making Revisited. In: Gero, J. (eds) Design Computing and Cognition '18. DCC 2018. Springer, Cham. https://doi.org/10.1007/978-3-030-05363-5_26

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  • DOI: https://doi.org/10.1007/978-3-030-05363-5_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05362-8

  • Online ISBN: 978-3-030-05363-5

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