Surface Mathematics

Definition

Surface mathematics names the practice of "defining interior volumes using surface mathematics to develop autogenic design topographies at the interior scale and context" (Anderson & Lovell-Anderson 2018) — the mathematical definition of surfaces as the generative medium for interior volume. The concept rests on the chapter's ontological ladder, after March & Stiny: "The formula for developing autogenic systems of design starts simply: shape < surface < volume. In the most primitive sense, shape is the two-dimensional primitives such as points and lines; surface is an interconnected set of points and lines that have length and breadth; and volume is the three-dimensional space enclosed by a collection of surfaces." The interior volume, on this account, does not exist prior to its surfaces — it is produced by "a collection of surfaces" whose geometry is computed: "the logical basis for defining and transforming interior volumes via visible surface mathematics."

The mathematics in question is concrete and manipulable: isosurface constraints and parametric equations (K3DSurf), trigonometric approximations (the gyroid's cos(x)·sin(y) + cos(y)·sin(z) + cos(z)·sin(x) = 0), the alteration of Cartesian values to "manipulate existing surfaces," extraction and mapping of edge curves onto a bounding cube's faces with vertices preserved at corners, subdivision of surfaces into modules, and local minimization ("The optimized geometry is a locally minimized surface that contains six vertices and six curved edges, where each edge is in contact with a unique face, and each vertex is in contact with a unique corner"). Geometric transformations of such surfaces "are often simulated through a process of cell aggregation to generate volume." Surface mathematics is thus the wiki's most explicit formalization of the surface as a design medium — the point at which Dubbeldam's "surface registrations of force fields" (thing-shapes) and Taylor's responsive surfaces (interactive-architecture) acquire an actual mathematical apparatus.

Key Thinkers

• Jonathon R. Anderson & Laura Lovell-Anderson — the concept's interior-scale theorists: surface mathematics as the medium "to develop autogenic design topographies at the interior scale and context" (2018).
• Lionel March & George Stiny — "Spatial Systems in Architecture and Design: Some History and Logic" (Environment and Planning B, 1985), the source of the shape < surface < volume formula and of the computational theory of design: "designs and their meanings [can] be viewed as the results of computations carried out according to rules of composition and correlative rules of description" (via notes 3, 6, 14). Not yet paged (see log).
• Janine Yeung — the case-study practitioner: the gyroid generated in K3DSurf through isosurface constraints and parametric equations; her observation that "there is often an ambiguity between invisible and defined boundaries in mathematical surfaces" (2016, p. 12).

Related Concepts

• autogenic-design — the process the mathematics serves: self-generated topographies whose "surface parameter" is the primary variable.
• the-gyroid — the exemplar surface: a mathematically defined minimal surface aggregated into an interior topography.
• computational-design — the computational substrate: isosurface and parametric-equation manipulation as the interior-scale branch of algorithmic form-finding.
• emergence — the generation logic: surfaces self-organizing into "a distinctive three-dimensional pattern" through tiling "in the mathematical logic of self-organization."
• boundaries — the representational problem: mathematical surfaces blur the line between the "invisible and defined boundaries" (Yeung) — the computed surface does not always know what it encloses.
• Shape Grammars — the rule formalism: "simulation of simple shape grammars in visual scripting" as the framework's middle term.
• thing-shapes — the phenomenological ancestor: Dubbeldam's spatiotemporal shapes with "material" qualities anticipated the surface-first ontology the mathematics now executes.
• Topology — the formal-historical cousin and warning: Voordouw's topological surfaces are also surface-first ("topology studies the surface of geometry – it has no space") but lack precisely the volume-enclosing capacity surface mathematics assumes ("volume is the three-dimensional space enclosed by a collection of surfaces") — the two practices diverge exactly at whether the defined surface is required to enclose. See topological-interiority.

Source Support

Sources in the wiki that discuss this concept:

• Transforming Interior Volumes (Anderson & Lovell-Anderson 2018) — the source that names the practice: the shape < surface < volume ladder, the isosurface/parametric-equation toolkit, cell aggregation, local minimization, and "visible surface mathematics" as the interior volume's "logical basis."
• Topology and interiority: Folding space inside (Voordouw 2018) — the contrast case: the topological surface as the surface-first practice without the volume-commitment — "If the surface is given thickness, it gains a form but not a 'space.' Space is now a result of form, not a means to form"; "the continuity of the surface prevents the (programmatic) definition of the interior and the porosity prevents its (spatial) enclosure." The comparison isolates what surface mathematics actually claims: not merely to define surfaces computationally, but to have them enclose — "volume is the three-dimensional space enclosed by a collection of surfaces." Voordouw's chapter is the empirical caution: computation alone guarantees form, not the enclosure that makes an interior. See topology.

Open Questions

• The ladder shape < surface < volume makes volume derivative of surface — a reversal of the tectonic tradition in which surfaces are the boundaries of pre-existing volumes. Is surface mathematics an ontology (volume is enclosed surfaces) or a representational convention (we model by surfaces because software does)? Compare Manack's sectioning, where "the multiple sections recede in the service of articulating… the singular space" (digital-fabrication).
• Yeung's "ambiguity between invisible and defined boundaries in mathematical surfaces" suggests the computed surface can fail as an enclosure: at what point does an autogenic topography stop being an interior volume and become an unbounded field? Is the interior discipline's object ("volume") stable under its own mathematics?
• March & Stiny (1985) grounded design-as-computation in spatial systems; the chapter transposes that ground to interior volume. What is lost or gained when the unit of computational design shifts from the plan (the spatial system) to the surface (the enclosing topology)?
• Is surface mathematics the interior's own formal theory — the long-missing computable counterpart to Rowe & Slutzky's phenomenal transparency (phenomenal-transparency) — or does it dissolve the interior back into pure geometry, erasing the inhabitant the discipline claims (interiority)?