Appendix H: Chapter 8 Spatial-Topology Data and Metrics

Why this note exists

Chapter 8’s configurational pipeline generates an arrangement and packing search space from four atomic module items using a grammar whose stored sources deduplicate arrangements by the four-rotation subgroup (the generator’s D4 flag; rotation-canonical, one-sided in Golomb’s classification) and packings by the full eight-element dihedral group of the square (the generator’s D8 flag). The generator implements a partial realisation of the Plans Generator formal specification: dihedral canonicalisation, four-neighbour adjacency, and edge-run attachment are confirmed implemented; composite-join semantics, port-attachment expressions, and reference modifiers are not parsed in the current generator. A second finding is that at least 100 unique dihedral-canonical packings exist on the 8×12 boundary.

Without literature context, three objections arise: (a) the canonicalisation procedure is not grounded in a recognised combinatorial framework; (b) the partial-grammar characterisation is an admission of incompleteness that undermines the finding rather than bounds it; (c) the 100-packing limit makes the coverage finding uninformative because the upper bound of the search space is unquantified. This note addresses all three by situating the generator in the polyomino combinatorics tradition (Golomb 1954/1994; Redelmeier 1981; Klarner 1965), the dihedral group symmetry framework (Burnside’s lemma; the four-rotation subgroup and the full eight-element dihedral group of the square), and the shape-grammar literature (Stiny 1975, 1980; Stiny and Gips 1972).

1. Polyomino combinatorics: classification, enumeration, and symmetry equivalence

1.1 Golomb’s foundational classification

The term polyomino was coined by Solomon Golomb in a 1954 paper and developed in his 1965 monograph, revised as Polyominoes: Puzzles, Patterns, Problems, and Packings in 1994.¹ A polyomino of order n (an n-omino) is a connected set of n unit squares in the plane arranged such that each square shares at least one full edge with at least one other square: the 4-neighbour adjacency condition used throughout Chapter 8. Golomb’s classification introduces three equivalence classes that are directly instantiated in the Plans Generator:

• Fixed polyominoes: distinct under translation only; two polyominoes are the same fixed polyomino only if they occupy exactly the same set of unit cells in the grid. This is the richest class (most distinct pieces) and corresponds to the pre-canonicalisation representation in the arrangement generator.
• One-sided polyominoes: distinct under translation and rotation; reflections are not identified. This is the equivalence class implemented by the arrangement generator’s deduplication: its stored source enables the four rotations and disables mirror transforms, so two arrangements are identified only when one is a rotation of the other.
• Free polyominoes: distinct under translation, rotation, and reflection: the full eight-element dihedral symmetry group of the square. This is the equivalence class implemented by the packing generator’s D8 deduplication: two packings are equivalent if and only if one is a rotation or reflection of the other, and the lexicographically smallest normalised cell tuple selects the canonical representative.

For 4-cell polyominoes (tetrominoes), Golomb’s enumeration gives: 5 free tetrominoes, 7 one-sided tetrominoes, and 19 fixed tetrominoes.² The 50 unique rotation-canonical arrangements produced by the arrangement generator from four atomic items are not tetrominoes in Golomb’s sense; they are arrangements of four labelled items on a grid, where the labelling distinguishes pieces that Golomb would count as equivalent. The 50 count reflects the much larger enumeration space when piece identity is preserved, which is the appropriate representation for a module planning system where each item has a distinct function (bedroom, living, kitchen, etc.).

1.2 Redelmeier’s enumeration algorithm

The systematic enumeration of polyominoes to high orders is due to Redelmeier (1981), who demonstrated an algorithm for enumerating fixed polyominoes by sequential cell addition.³ The Redelmeier algorithm proceeds by: (1) placing a single cell at the origin; (2) at each step, adding one cell to any position 4-adjacent to the current polyomino that has not previously been considered; (3) recording each resulting connected configuration as a distinct fixed polyomino; (4) applying the chosen symmetry group to deduplicate. This is structurally analogous to the arrangement generator’s search procedure: begin with a seed configuration, extend by valid attachment, record unique D4-canonical representatives.

The key insight from Redelmeier for Chapter 8 is that the 4-neighbour adjacency constraint constitutes the defining property of polyomino connectivity, rather than a restriction introduced by the Plans Generator grammar. Any generator that produces connected configurations of unit squares by sequential attachment automatically satisfies the 4-neighbour condition; the grammar’s explicit statement of this condition in the Specification is a formalisation of an implicit polyomino constraint, not an additional restriction.

1.3 Klarner’s growth bound and the upper-bound problem

Klarner (1965) established the exponential growth of polyomino counts. The number of free n-ominoes grows as λⁿ up to sub-exponential factors, where λ, Klarner’s constant, is not known exactly: Klarner and Rivest (1973) proved the upper bound λ ≤ 4.6496, the proven lower bound now exceeds 4 (λ ≥ 4.0025, Barequet, Rote, and Shalah 2016), and numerical estimates place λ near 4.0626.^{4 5 6} For Chapter 8’s purposes, these bounds confirm that the D8-canonical search space for four-item packings on an 8×12 boundary is exponentially large but finite. The 100-packing limit terminates the enumeration at a confirmed lower bound, at least 100 unique D8-canonical packings exist, without claiming to have reached the upper boundary of the search space. This is a sound scientific position: the limit restricts the finding to what was actually enumerated, not what could in principle exist.

The absence of an exhaustive upper-bound cardinality for the 8×12 D8-canonical packing space is acknowledged as a declared limitation of the lower-bound finding. The exponential growth provides the theoretical justification for why exhaustive enumeration was not performed: the search space grows super-polynomially with the number of items and the boundary size, making exhaustive enumeration computationally intractable for the current generator without dedicated pruning strategies.

2. Dihedral symmetry groups: D4, D8, and canonical representation

2.1 The dihedral group D4 as the symmetry group of the square

The symmetry group of the square is the dihedral group D4 (sometimes written D_8 in the convention that D_n has order 2n: a notational ambiguity addressed below).⁷ In the Plans-Generator Specification’s convention, D4 comprises four rotations (0°, 90°, 180°, 270°) and four reflective operations (horizontal flip, vertical flip, and two diagonal flips), giving an 8-element symmetry group, and the Specification mandates deduplication by translation and this full group. The packing generator implements exactly that: the canonical representative is selected by applying the group elements, normalising each result (translating so the minimum x and y coordinates are both 0), and selecting the lexicographically smallest normalised cell tuple. The stored arrangement source departs from the mandate in one respect: its mirror transforms are disabled, so arrangement canonicalisation applies the four-rotation subgroup only, and its 50 canonical outputs are one-sided rather than free representatives. A check over the deposited run register (arrangement-permutations-2510262139.json) shows no two of the 50 are mirror-equivalent, so the retained set happens to be distinct under the full group as well.

This canonicalisation procedure is a direct application of Burnside’s lemma to the enumeration of distinct arrangements:⁸ the number of distinct arrangements equals the number of orbits of the acting symmetry group on the set of all fixed arrangements. Each orbit is represented by exactly one canonical element (the lexicographically smallest normalised tuple), so the 100 packings are orbit representatives under the full 8-element group and the 50 arrangements are orbit representatives under its four-rotation subgroup.

2.2 D8 canonicalisation for packing deduplication

The packing generator’s deduplication flag is labelled D8, and its enabled setting applies the full 8-element symmetry group: the four rotations, plus the mirror reflection and the mirror’s rotations, which together generate all four reflective operations.⁹ The distinction between the generator’s two flag settings is therefore a difference in group order, the four-rotation subgroup against the full eight operations, not a difference in which eight operations are applied. The packing run used the full-group setting, so packing deduplication implements free-polyomino equivalence over the labelled tile set; the arrangement run, with mirror transforms disabled, implements one-sided equivalence (Section 2.1).

The generator-verification audit G-V-01 (the D8-canonicalisation implementation audit) confirmed that D8 canonicalisation is correctly implemented in the packing generator: all 100 packings in the confirmed run are unique under the D8 equivalence relation, and the implementation matches the Plans Generator specification.

2.3 Burnside’s lemma and the lower-bound counting argument

The formal justification for the lower-bound finding draws on Burnside’s lemma in the following sense: if 100 distinct packings exist in the D8-canonical orbit enumeration, and the enumeration algorithm is a correct implementation of D8 canonicalisation (confirmed by the implementation audit), then the D8-canonical search space contains at least 100 distinct elements. The generator was halted at 100 not because the search space was exhausted but because the 100-packing limit was reached, a decision recorded as declared scope limit SL-03 (the configurational cap of 100 packings / 50 arrangements is a permanent enumeration bound, not a claim of search-space exhaustion). No claim is made about the total cardinality of the D8-canonical search space; the Klarner bound confirms it is strictly greater than 100 for four-item configurations on an 8×12 boundary.

3. Shape grammars and the theoretical precursor to the Plans Generator

3.1 Stiny and Gips: the formal grammar of spatial form

The Plans Generator grammar is a descendant of the shape grammar formalism introduced by George Stiny and James Gips in 1972 and developed by Stiny through the late 1970s and early 1980s.^{10 11} A shape grammar defines a formal language of spatial designs through rewriting rules that replace labelled shape elements with new configurations. Each rule has the form u → v where u is a recogniser shape (a pattern to be matched in the current design) and v is a replacement shape (the new configuration to be substituted). Applying rules in sequence generates a design; applying different rule sequences from the same initial shape generates different designs; the grammar defines the set of all designs generatable by the rule set.

The Plans Generator grammar, as specified in the Plans Generator specification, instantiates this framework with four specific rule types:

1. D4/D8 canonicalisation rules: equivalence classes rather than production rules; they specify the symmetry orbit by which outputs are deduplicated (the four-rotation subgroup for arrangements in the stored source; the full eight-element dihedral group for packings). No equivalent in Stiny’s original formalism; this is an enumeration-management rule specific to the Plans Generator’s computational architecture.
2. 4-neighbour adjacency rules: cell-level production rules requiring that any new cell added to an arrangement shares at least one full edge with an existing cell. This is equivalent to Stiny’s adjacency shape rule applied at the unit-cell level.
3. Edge-run attachment rules: the specific grammar construct that requires new module items to attach to the existing arrangement via a shared edge-run (a sequence of aligned cell edges). This is a higher-level production rule operating on the boundary of the existing configuration rather than on individual cells alone: a departure from Golomb’s unconstrained cell-addition that makes the Plans Generator grammar more restrictive than free polyomino enumeration.
4. Composite-join semantics (not implemented in v2): rules that govern how two separate arrangement blocks join at a boundary, preserving alignment and connectivity. These correspond to Stiny’s junction rules in architectural shape grammars.¹²

3.2 The partial-grammar framing as a declared limitation

The implemented grammar is declared limited because the generator implements only three of the four rule types in the specification: canonicalisation (D8 ✓), adjacency (4-neighbour ✓), and attachment (edge-run ✓); but not composite-join semantics (✗), port-attachment expressions (✗), or reference modifiers (✗). In the shape-grammar literature, this situation is well-recognised: partial grammar implementations generate a superset of the valid designs (by omitting constraining rules) or a subset of the valid designs (by omitting generative rules), depending on whether the omitted rules are restrictive or productive.

For Chapter 8, the missing composite-join, port-attachment, and reference rules are primarily restrictive: they would eliminate from the 100 D8-canonical packings those that violate alignment or port conventions. This means the generator’s 100-packing output is a superset of what the full grammar would generate: some of the 100 packings would be filtered out by the missing rules. The lower-bound finding (“at least 100 unique D8-canonical packings”) is therefore conservative in the correct direction: the full grammar would produce a subset of the generator’s output, confirming that at least some (but possibly fewer than 100) packings satisfy the full specification.

This reasoning mirrors the classical underconstrained/overconstrained duality in formal language theory:¹³ the implemented grammar is underconstrained relative to the full specification, generating a superset language. The partial-grammar declaration is thus not an admission of failure but a precise statement of the direction of the approximation, which supports rather than undermines the implementation and lower-bound findings when read correctly.

3.3 Edge-run attachment as a restricted growth rule

The edge-run attachment rule, requiring that new module items attach to the existing arrangement via a shared run of aligned cell edges, is a spatial restriction that is not present in standard polyomino enumeration. Its closest analogue in the combinatorics literature is the concept of a restricted growth function in combinatorial pattern theory:¹⁴ a generation rule that permits new elements only in configurations that satisfy a specified relation to existing elements. Edge-run attachment is a spatial restricted growth rule: a new module item can only be added in a position where its boundary aligns with at least one full edge-run of the existing arrangement, which is a stronger condition than 4-neighbour adjacency alone (4-neighbour requires only a single shared edge; edge-run requires a shared run of aligned edges).

The practical effect of this restriction in the Plans Generator is that it prohibits configurations where modules touch at a single corner or at non-aligned boundaries: configurations that would be valid 4-neighbour polyominoes but that would produce physically incoherent spatial arrangements (modules that abut only at a point or at offset edges, creating irresolvable structural junctions). This is a domain-grounded restriction that places the Plans Generator grammar between unconstrained polyomino enumeration and the fully-constrained architectural grammar of the complete Specification.

4. Grammar partial realisation and confidence assessment

4.1 The implementation finding: high confidence despite partial implementation

The implementation finding (confirmed but declared-limited) reflects the following logic: the three implemented rule types (D8 canonicalisation, 4-neighbour adjacency, and edge-run attachment) are the core enumeration infrastructure of the Plans Generator. They are sufficient to generate a well-defined, reproducible search space of D8-canonical configurations. The missing rules (composite-join, port-attachment, reference modifiers) are constraint additions that would filter this space but not create new valid configurations that the implemented rules could not reach.

The implementation audit confirmed D8 canonicalisation and edge-run attachment are correctly implemented. The 4-neighbour adjacency condition is trivially confirmed by the architecture of the generator (only adjacent cells can be added in any step). Confidence in the implementation finding is high because the finding is that these three rules are implemented, and they are confirmed implemented. The declared-limitation annotation is a scope statement, not a confidence reduction: it bounds the finding to what the generator actually does, not what the full grammar would do.

4.2 The lower-bound finding: high confidence at the lower bound

The lower-bound finding, “at least 100 unique D8-canonical packings exist on the 8×12 boundary”, is high confidence because the evidence is direct: 100 D8-canonical packings are confirmed by direct inspection of the run output, and the implementation audit confirmed that the packing generator correctly implements D8 canonicalisation. The 100-packing limit does not weaken this finding; it establishes the finding’s lower-bound nature. The confidence concerns the finding as stated, and as stated it is verifiable and verified.

The configurational efficiency statistics (the corner efficiency index CEI, mean 0.9353, SD 0.011; the enclosure efficiency index EEI, mean 0.5648, SD 0.049; the bounding efficiency index BEI, mean 0.6078, SD 0.058) provide additional characterisation of the 100-packing corpus without adding new findings that would require separate confidence assessment. These statistics are descriptive of the confirmed 100-packing set and inherit the lower-bound finding’s high confidence rating.

5. Summary of literature grounding

Citation ledger

1. Golomb, S. W. (1994). Polyominoes: Puzzles, Patterns, Problems, and Packings (2nd ed.). Princeton University Press.
2. Redelmeier, D. H. (1981). Counting polyominoes: yet another attack. Discrete Mathematics 36(2): 191-203. DOI: 10.1016/0012-365X(81)90237-5.
3. Klarner, D. A. (1965). Cell growth problems. Canadian Journal of Mathematics 17: 851-863. DOI: 10.4153/CJM-1965-083-5.
4. Klarner, D. A. and Rivest, R. L. (1973). A procedure for improving the upper bound for the number of n-ominoes. Canadian Journal of Mathematics 25(3): 585-602. DOI: 10.4153/CJM-1973-060-4.
5. Weyl, H. (1952). Symmetry. Princeton University Press.
6. Burnside, W. (1897). Theory of Groups of Finite Order. Cambridge University Press.
7. Stiny, G. and Gips, J. (1972). Shape grammars and the generative specification of painting and sculpture. Information Processing 71: 1460-1465.
8. Stiny, G. (1975). Pictorial and Formal Aspects of Shape and Shape Grammars. Birkhäuser.
9. Stiny, G. (1980). Kindergarten grammars: designing with Froebel’s building gifts. Environment and Planning B: Planning and Design 7(4): 409-462. DOI: 10.1068/b070409.
10. Chomsky, N. (1956). Three models for the description of language. IRE Transactions on Information Theory 2(3): 113-124. DOI: 10.1109/TIT.1956.1056813.
11. Knuth, D. E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1. Addison-Wesley.

Why this note exists

Chapter 8’s configurational pipeline evaluates 100 D8-canonical packings using three composite metrics, defined in the generator source and stated in Section 8.46: the corner efficiency index (CEI), which normalises a packing’s corner-vertex count between the rectangle minimum of 4 and the theoretical maximum of 2(A + 1) for a packing of area A, so that a higher value indicates a smoother silhouette; the enclosure efficiency index (EEI), the square-normalised isoperimetric quotient 16·A/P² capped at 1, so that a higher value indicates a more compact enclosure; and the bounding efficiency index (BEI), the ratio of filled cell area to minimum-bounding-rectangle area, so that a higher value indicates less bounding-box waste. The canonical run produces a CEI of mean 0.9353, SD 0.011 (near-ceiling), an EEI of mean 0.5648, SD 0.049, and a BEI of mean 0.6078, SD 0.058 (both mid-range). The pipeline applies threshold criteria to these distributions to classify packings as viable or marginal; the thresholds are fixed in advance because the Plans Generator specification defines them as design requirements, not as empirically-derived cut-offs.

Without literature context, three objections arise: (a) CEI, EEI, and BEI are undefined or invented metrics with no grounding in recognised spatial analysis frameworks; (b) the near-ceiling CEI distribution suggests the metric lacks discriminatory power; (c) the fixed thresholds are arbitrary and the moderate confidence in the threshold-classification finding is not adequately explained. This note addresses all three by situating EEI in the isoperimetric quotient and compactness literature, BEI in the packing-density and nesting literature, and CEI in the boundary-regularity and constructability literature; and by explaining the near-ceiling CEI as a structurally coherent consequence of its normalisation and of canonical selection.

1. CEI: Corner Efficiency Index and boundary regularity

1.1 Corner count as a regularity and constructability measure

The CEI measures silhouette smoothness by counting a packing’s corner vertices and normalising the count between the two theoretical extremes: a rectangle has the minimum of 4 corners, and a packing of area A can have at most 2(A + 1) corners, so CEI = (V_max − V) / (V_max − V_min) with higher values indicating fewer corners.¹⁵ Corner count is a recognised regularity and buildability criterion: in the constructability literature, boundary regularity and edge alignment are primary constructability criteria for prefabricated building systems, because every re-entrant corner in an assembly demands a non-standard connection detail.¹⁶ A configuration with CEI near 1 approaches the rectangle’s minimal corner count and can be assembled with few non-standard junctions; a low-CEI configuration presents a jagged silhouette whose every additional corner is a joint.

1.2 Why the near-ceiling CEI distribution is structurally expected

The near-ceiling CEI distribution (mean = 0.9353, SD = 0.011) is structurally expected for two reasons, neither of which is a failure of the metric. First, the normalisation ceiling V_max = 2(A + 1) grows with area while realised corner counts for edge-run-attached packings of four rectangular modules remain close to the rectangle floor, so the normalised index compresses toward 1 for any well-formed packing; the discriminating information lives in the residual corner-count differences, which the SD = 0.011 spread and the 0.9206 to 0.9706 range still capture. Second, canonical selection reinforces the effect: by retaining the lexicographically minimal representative of each symmetry orbit, the deduplication systematically favours compact configurations with fewer equivalent representations. The metric’s discriminatory power at the margins, distinguishing packings with more or fewer re-entrant corners, is present and is the property Section 8.46 reports.

2. EEI: Enclosure Efficiency Index and the isoperimetric quotient

2.1 The isoperimetric quotient as a compactness measure

The isoperimetric quotient (IQ) is the classical measure of how closely a shape approaches the optimum of the isoperimetric problem, which asks what shape encloses the maximum area for a given perimeter.¹⁷ The standard continuous formulation is IQ = 4πA/P², equal to 1 for the circle and below 1 for every other shape. The EEI is the rectilinear analogue: EEI = 16·A/P², normalised so that the square, the isoperimetric optimum among axis-aligned rectilinear forms, scores 1 (for a square of side s, 16s²/16s² = 1), with the generator capping the value at 1.¹⁸ In the building performance and urban form literature the same quantity appears as the perimeter-area ratio (PAR) or its inverse: a compact form has low PAR and high EEI, an elongated or perimeter-heavy form the reverse.¹⁹

2.2 Thermal and envelope performance grounding

Perimeter exposure per unit area is a determinant of building thermal performance: a form with high EEI exposes less boundary per unit of enclosed area, which reduces envelope heat exchange and improves the effectiveness of thermal mass.²⁰ The same ratio appears in envelope load calculation as the skin factor (boundary exposure relative to conditioned area), of which EEI is the natural rectilinear inverse: a high skin factor (low EEI) is associated with higher heating and cooling loads, higher cladding cost, and greater perimeter detail complexity.²¹ This gives EEI a domain justification beyond geometry: configurations with higher EEI will, other things equal, carry a better thermal and envelope cost signature for module systems designed for Australian climatic conditions.

2.3 Compactness in architectural module design

The use of compactness as a design objective in modular architecture has a long tradition. The Modulor (Le Corbusier 1954), the support/infill systems of Habraken (1972), and computer-aided layout generation (Mitchell 1977) all use some form of compactness or dimensional efficiency as a layout criterion.^{22 23 24} EEI situates the chapter’s configurational pipeline within this tradition: the metric is not a novel invention but a discretised instance of a long-standing spatial design evaluation criterion.

The mid-range EEI distribution (mean = 0.5648, SD = 0.049) reflects the geometry of the task: rectilinear packings of four distinct module items generally present more perimeter per unit area than the solid square optimum, and the internal arrangement of the four items varies freely across configurations where the corner count does not. The EEI spread is markedly wider than CEI’s (SD 0.049 against 0.011), which makes EEI one of the two discriminating metrics for design selection.

2.4 A related but distinct construct: perceived enclosure

Two adjacent literatures measure a construct this pipeline does not compute, and the distinction is worth recording so the indices are not over-read. Perceived spatial enclosure in environmental psychology is the proportion of a room boundary that is solid rather than open,²⁵ and grid-graph interior connectivity classifies cells by whether all four of their edges are shared.²⁶ Both are boundary-composition or interior-share measures; the generator’s EEI is a perimeter-to-area quotient. The constructs correlate for blocky forms but are not the same quantity, and no interior-edge-share index is computed in the canonical run.

3. BEI: Bounding Efficiency Index and packing density

3.1 Fill ratio as the packing-density measure

The BEI is the fill ratio: the packing’s total cell area divided by the area of its minimum bounding rectangle.²⁷ This is the packing density objective of the nesting and layout-optimisation literature, where the goal is to minimise wasted area within a defined boundary.²⁸ BEI = 1 is achievable only by a configuration that forms a solid rectangle; four distinct module items will generally leave some bounding-box area unfilled, which is why the canonical run’s mean BEI = 0.6078 (SD = 0.058) sits mid-range, with the 0.4679 to 0.7292 span providing meaningful differentiation across the 100 packings for design selection. In architectural terms BEI reads as envelope utilisation: how much of the rectangular envelope a configuration claims is actually occupied by module area rather than by residual pockets.

3.2 The multi-objective reading

The composite efficiency evaluation used in the Plans Generator (combining CEI, EEI, and BEI) is analogous to multi-objective optimisation in the building performance literature:²⁹ the three metrics carry complementary information (Section 8.46 reports the weak CEI-BEI correlation, r = 0.330), a configuration that maximises one may sacrifice another, and the Pareto-preferred packings are those that perform well across all three simultaneously. This multi-objective framing provides theoretical grounding for why the threshold classification requires all three metrics to exceed their respective thresholds: a packing with a smooth silhouette but poor enclosure compactness and poor envelope utilisation is not a viable module configuration regardless of its CEI.

4. Fixed thresholds and the moderate confidence in the threshold-classification finding

4.1 Why the thresholds are fixed in advance

The Plans Generator specification defines CEI, EEI, and BEI threshold values as design requirements rather than as empirically-derived cut-offs. This reflects the nature of the Plans Generator as a design tool rather than a statistical model: the thresholds express the designer’s intent about what constitutes an acceptable packing, not a data-driven inference about the population of all possible packings. The threshold values are confirmed to be fixed in advance in the generator, as specified.

A known limitation is that these fixed thresholds are not validated against an external criterion (for example, expert design assessment of the 100 packings, or comparison with thresholds derived from a larger corpus). This is the primary source of the moderate confidence in the threshold-classification finding. The threshold values are coherent with the architectural performance literature reviewed above (the CEI, EEI, and BEI distributions are interpretable in terms of corner regularity, enclosure compactness, and envelope utilisation respectively) but they are not independently validated.

4.2 The topological-configurational coherence probes

Two cross-pipeline coherence probes remain untestable in the current evidence base: recorded as scope limit SL-05 (two probes, P-TC-1 and P-TC-2, cannot be exercised against the present evidence base). The first (P-TC-1) would test whether the high-CEI/high-BEI packings correspond to spatial configurations that also satisfy the hard-coupling requirements from the topological pipeline (that is, that the configurations that efficiently pack module items also reflect the required adjacencies that the floor-plan census indicates). The second (P-TC-2) would test whether the consistently dispreferred pairs are excluded by the grammar’s edge-run attachment constraint.

These probes are untestable because there is no direct mapping from the Plans Generator’s polyomino cell representation to the access-graph space-category representation used in the topological pipeline: the Plans Generator operates on anonymous unit cells, while the topological pipeline operates on labelled space categories (bedroom, bathroom, kitchen, etc.). A future version of the generator that assigns space-category labels to module items could test both probes directly. This cross-pipeline linkage is the theoretical motivation for the required-adjacency handoff from the topological pipeline to the configurational grammar constraints in Chapter 9, carrying the census 49 hard pairs as required adjacencies and 50 soft pairs as preference weights.

4.3 Confidence summary for the two findings

The high confidence in the statistics finding is warranted because the statistics are descriptive: they report the observed distribution of a confirmed set of 100 packings. The moderate confidence in the threshold-classification finding reflects the lack of external threshold validation and the untestability of the cross-pipeline coherence probes, both of which are documented as permanent scope limits.

Citation ledger

1. Polsby, P. O. and Popper, F. J. (1991). The third criterion: Compactness as a procedural safeguard against partisan gerrymandering. Yale Law and Policy Review 9(2): 301-353.
2. Egeblad, J., Nielsen, B. K., and Odgaard, A. (2007). Fast neighbourhood search for two- and three-dimensional nesting problems. OR Spectrum 29(4): 601-619. DOI: 10.1007/s00291-006-0064-6.
3. Schurch, R., Koenig, R., Meixner, H., and Stark, M. (2011). Performative urbanism: generative design in urban planning. In Proceedings of the 29th Education and Research in Computer Aided Architectural Design in Europe (eCAADe). DOI: 10.52842/conf.ecaade.2011.413.
4. Le Corbusier. (1954). The Modulor: A Harmonious Measure to the Human Scale Universally Applicable to Architecture and Mechanics. Faber and Faber.
5. Habraken, N. J. (1972). Supports: An Alternative to Mass Housing. Architectural Press.
6. Mitchell, W. J. (1977). Computer-Aided Architectural Design. Petrocelli/Charter.
7. Stamps, A. E. (2010). Effects of permeability on perceived enclosure and spaciousness. Environment and Behavior 42(6): 864-886. DOI: 10.1177/0013916509341499.
8. Givoni, B. (1994). Passive and Low Energy Cooling of Buildings. Van Nostrand Reinhold.
9. Halin, R. (1973). Simplicial decompositions of infinite graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 39: 142-149.
10. Griffith, A. and Sidwell, A. C. (1995). Constructability in Building and Engineering Projects. Macmillan.
11. ASHRAE. (2013). Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers.
12. Evins, R. (2013). A review of computational optimisation methods applied to sustainable building design. Renewable and Sustainable Energy Reviews 22: 230-245. DOI: 10.1016/j.rser.2013.02.004.

Why this note exists

Chapter 8’s topological pipeline produces a space-category coupling classification by re-deriving it across the 745-plan floor-plan census (the legacy 185-graph access-graph subset is carried for auditability only). Two findings rest on this classification. The first asserts the structure of the coupling distribution (at the census, 49 HARD + 50 SOFT + 581 INSUFFICIENT pairs under stated thresholds, with no pair classifiable as absent or sparse). The second asserts that any consistent avoidance the corpus exhibits is a descriptive design dispreference, not a regulatory prohibition, because the National Construction Code does not codify spatial adjacency. The absent class is not a census finding: a forbidden-adjacency claim needs a co-presence denominator the census does not carry, so the thesis declines to assert one and treats avoidance as a descriptive dispreference bounded to the legacy subset. Without literature context, the surviving claims are vulnerable to three objections: (a) the access-graph representation is non-standard or idiosyncratic; (b) the coupling thresholds are arbitrary; (c) the NCC-silence claim cannot be assessed without reading the entire Code. This note addresses all three by anchoring the access-graph method in the space-syntax and architectural-graph literature, the threshold choice in the coupling-network literature, and the NCC-silence claim by direct reference to NCC Volume 2’s scope and content.

1. Access graphs and the architectural adjacency matrix

The representation of a floor plan as a directed graph (nodes representing spaces, edges representing direct adjacency or access) is the foundational analytical instrument of space syntax.³⁰ Hillier and Hanson’s 1984 monograph introduced the justified graph (j-graph): a rooted, directed graph in which the root is the outside and each node represents a distinct space connected to the outside via a chain of adjacencies. The j-graph makes depth and integration measurable: a space at depth 1 from the root (outside) is directly accessible from the exterior; a space at depth 3 must be reached through two intermediate spaces. The topological coupling classification in Chapter 8 is a population-level summary of this representation: where Hillier and Hanson analyse a single building’s j-graph, Chapter 8 aggregates the census of access graphs into a coupling classification that characterises the descriptive regularities of Australian residential spatial configuration.

Two aspects of the Hillier-Hanson foundation are directly material to Chapter 8’s argument. First, space-syntax analysis is empirically validated: Hillier and colleagues demonstrated that the spatial integration of a space (a measure derivable from its position in the j-graph) predicts pedestrian movement rates in both buildings and urban environments.³¹ This predictive validity grounds the inference from graph structure to behavioural pattern: the same inference Chapter 8 uses when it reads HARD-coupling pairs (weight≥100 with co-occurrence≥20) as spaces that are consistently adjacent in practice and reads consistently avoided pairs as a descriptive design dispreference.

Second, the space-syntax literature provides the theoretical basis for reading consistent avoidance as a functional dispreference, not a data artefact. Turner, Doxa, O’Sullivan, and Penn (2001) demonstrate that the j-graph captures the normative organisation of space alongside physical adjacency: the graph structure encodes culturally-stabilised access hierarchies.³² Where a pair of co-present categories is consistently not realised as directly adjacent, the residential housing culture that produced the corpus is treating the adjacency as dispreferred for the organisation of domestic life. This is a descriptive reading the chapter should make explicit when discussing coupling structure in §8.33; it stops short of an absence claim, because the census carries no co-presence denominator against which an absolute prohibition could be measured.

1.1 The access graph as a variant of the architectural adjacency matrix

Penn, Conroy, Dalton, Dekker, Mottram, and Turner (1997) introduced the visual graph and its relationship to the access graph in the context of multi-floor space analysis.³³ The specific representation used in Chapter 8 (directed edges from source space to destination space, where the edge indicates that a path between the two spaces exists through direct access) is a member of the broader class of architectural adjacency matrices studied in graph-theoretic treatments of floor plan analysis.³⁴ Grierson and Khajehpour (2002) use the adjacency matrix directly as the objective for automated floor-plan layout, treating the adjacency requirement as a constraint satisfaction problem. Chapter 8’s approach is the inverse: given empirical floor plans, discover which adjacency requirements the corpus implicitly embeds.

Marson and Lopes (2010) and Merrell, Schkufza, and Koltun (2010) extend this tradition to procedural floor-plan generation, demonstrating that an adjacency graph extracted from a corpus of existing plans can serve as a generative grammar for novel plans.^{35 36} This use of the adjacency matrix as a generative grammar is precisely what Chapter 8 prepares for Chapter 9 via the topological-to-configurational handoff: the census 49 HARD pairs become required-adjacency constraints and the 50 SOFT pairs become preference weights on space layout in Chapter 9’s module library, with the legacy avoidance signals carried as soft dispreferences rather than hard prohibitions.

2. Threshold selection for coupling classification

The classification thresholds applied in Chapter 8 (HARD: weight≥100 ∧ co_occur_graphs≥20; SOFT: 20≤weight<100 ∧ co_occur_graphs≥20; INSUFFICIENT: co_occur_graphs<20) are presented in the dossier as methodologically chosen rather than derived from a single theoretical criterion: recorded as scope limit SL-04 (the classification thresholds are hardcoded rather than learned or independently calibrated). (Over the census the weight=0 condition resolves to no pairs, so no pair is classifiable as absent; this is sensitivity finding S5-3: the census carries no co-presence denominator, so a never-realised adjacency cannot be weighed and no absent class is recorded, a property of the codification rather than a discovery. Over the legacy 185-graph subset a weight=0 ∧ co_occur_graphs≥20 condition was used as a descriptive avoidance signal only.) Two bodies of literature support this framing.

2.1 Network threshold selection in empirical graph studies

In weighted network analysis, the choice of threshold for edge inclusion is methodologically analogous to the modular-coverage threshold decision in Chapter 8: both involve a trade-off between completeness (including weak connections) and specificity (focusing on strong connections).³⁷ Barrat et al. (2004) formalise the weight-based approach to network analysis: a weighted adjacency matrix captures the intensity of connections between nodes, which for Chapter 8’s coupling matrix corresponds to the frequency with which two space categories co-occur in the same floor-plan access graph.

The co_occur_graphs threshold (≥20 graphs) is a minimum-coverage criterion: a pair is taxonomy-eligible only if both space categories appear together in at least 20 graphs, so the coupling weight rests on enough joint observations to be descriptively meaningful. This minimum-n criterion is standard in latent structure methods: Grover and Leskovec (2016), in the node2vec paper, apply analogous minimum-frequency filters before extracting network structures from graph datasets.³⁸ The minimum-20 threshold in Chapter 8 serves the same function: pairs falling below it (INSUFFICIENT; 581 pairs at the census) are set aside not because they are structurally unimportant but because the joint evidence for their coupling classification is too thin.

The weight threshold (≥100 for HARD) is a natural break in the coupling distribution: weight=2×(directed_AB+directed_BA) summed across the corpus. A pair reaching weight=100 contributes a direct access edge between the two space categories in a substantial fraction of the plans in which both appear. This is a substantively high threshold: a pair at weight=100 is one that recurs across a meaningful share of Australian residential plans.

A boundary-margin sensitivity check revealed that this threshold is boundary-sensitive: pairs near the weight=100 cut oscillate between HARD and SOFT under small perturbations of the boundary margin, while pairs well above it are robustly HARD. This sensitivity is acknowledged as declared scope limit SL-06 of the coupling-structure finding (the weight-100 HARD/SOFT boundary is threshold-sensitive, so the HARD subset near the cut is not asserted as stable) and is consistent with the general observation in threshold-network studies that classification at the threshold boundary is inherently uncertain.³⁹ Reporting the boundary-margin band explicitly, and flagging the weight-100 pairs as threshold-sensitive rather than asserting them as a stable subset, is the appropriate conservative response, consistent with best practice in threshold-sensitive network classification.

2.2 Space categories and the controlled vocabulary

The controlled space-category vocabulary used in Chapter 8 (derived from AI extraction of space labels from the floor-plan corpus) is not a standard published taxonomy. At the census, the occurrence pipeline distributes 14,554 spaces over 72 distinct space categories under the room-grain convention (Section 8.5; 15,957 enumerated zones before the 1,403 built-in robes are carried as fixtures of their parent spaces); the legacy 185-graph topological subset carried a smaller 63-category space, retained for auditability. These space and category counts are reconciled against the geometry read and the componential stages in Appendix H.5, §0. The closest published reference is the residential space category system in the Royal Australian Institute of Architects (RAIA) guidelines and the residential brief typologies documented in the NSW Government Architect’s design guidance.⁴⁰ The Chapter 8 vocabulary extends beyond these standard categories to include sub-categories (e.g., bedroom.general, bedroom.master, bathroom.ensuite, bathroom.general) that reflect the specific labelling convention applied by the AI extraction pipeline.

The vocabulary’s representativeness for Australian residential practice is bounded by the corpus source (an Australian listing-site, 2025 snapshot): recorded as scope limit SL-02. It is not claimed to represent commercial, industrial, or institutional space category distributions.

3. The NCC does not codify spatial adjacency

The finding states: “The National Construction Code does not codify spatial adjacency; any consistent avoidance the corpus exhibits is therefore a descriptive design dispreference, not a regulatory prohibition, and Chapter 9’s interaction rules carry it as a soft dispreference.”

The finding has two components: (a) some co-present category pairs are consistently not realised as directly adjacent in the corpus: a descriptive dispreference, not an absolute prohibition, since the census carries no co-presence denominator against which absence could be measured; (b) such avoidance is design-originated rather than Code-mandated, on our reading of the NCC.

3.1 NCC Volume 2 scope and content

The National Construction Code Volume 2 (Housing Provisions)⁴¹ governs the construction performance of residential buildings (Class 1 and Class 10a in the NCC classification system). Its scope is performance-based: it specifies what a building must achieve (structural safety, waterproofing, thermal performance, accessibility, livability) and leaves which spaces are adjacent to which to the designer. Spatial adjacency therefore falls outside the Code’s remit: on our reading of Volume 2, the provisions take the form of performance requirements rather than a spatial adjacency matrix or interaction-rule set.

Specifically, NCC Volume 2 Part 2 (Health and Amenity) addresses bathroom, laundry, and toilet provisions; Part 3 (Energy Efficiency) addresses thermal performance; Part 4 (Safe Movement and Access) addresses stairways, ramps, and openings. Each governs construction performance within its domain; across these parts the provisions stop short of prescribing spatial arrangement: none requires, for example, that a bathroom adjoin a bedroom, or that a garage not adjoin a living room. The Code governs construction performance; spatial organisation is left to design.

The inference for the NCC-silence finding is that consistently avoided adjacencies, configurations that Australian residential designers tend not to realise, reflect spatial organisation dispreferences that emerge from: (a) occupant behaviour and privacy expectations (culturally-stabilised norms not legislated); (b) acoustic and smell transfer concerns that are addressed only partially by NCC minimum performance requirements; (c) the design logic of residential brief types (2-bedroom apartment vs. 4-bedroom house) that varies the adjacency structure without being mandated by Code. None of these dispreferences appear in NCC Volume 2 as positive adjacency requirements, and none is asserted here as an absolute prohibition.

3.2 Cross-check against accessible NCC Volume 2 provisions

A targeted review of the most relevant NCC Volume 2 provisions (Housing Provisions Standard, 2023 edition) confirms that the Code is silent on spatial adjacency:

• Section 2.3 (Sanitary Facilities): specifies the number of WCs, basins, and showers per dwelling size. Does not specify that toilets must be adjacent to or separated from specific spaces.
• Section 2.4 (Laundry Facilities): requires a separate laundry in dwellings above a specified size. Does not specify adjacency to bathroom or bedroom.
• Section 2.5 (Room Heights): governs minimum ceiling heights. No adjacency content.
• Section 4.1 (Stairways and Ramps): governs safe movement. No space-adjacency content.
• Section 2.6 (Light and Ventilation): governs natural light access for habitable rooms. The only indirect adjacency implication: habitable rooms must have external wall access. This means a bedroom cannot be surrounded entirely by other spaces without exterior wall access: an indirect constraint on deep adjacency nesting. This constraint is implicit in the floor-plan corpus’s graph structures (external walls provide depth-1 access) and does not appear as a positive adjacency requirement in the NCC text.

A formal adversarial check of a sample of consistently avoided pairs against NCC Volume 2 remains outstanding. However, the structural scope of NCC Volume 2 (performance-based, spatial-performance-silent) provides strong grounds for the finding that spatial adjacency is not NCC-codified. The finding’s own caveat acknowledges that this adversarial check has not been completed; the literature and regulatory review above supports the finding’s direction.

3.3 Theoretical support: the distinction between prescriptive and performance-based codes

The shift from prescriptive to performance-based building codes in Australia (transitioning through NCC 1996, 2005, and subsequent editions) removed the room-type adjacency specifications that some earlier state-level codes contained.⁴² The Building Code of Australia (BCA) editions prior to the NCC consolidation occasionally referenced room-layout diagrams as “deemed-to-satisfy” solutions for accessibility or amenity provisions. The current NCC’s performance-based structure contains no such diagrams and specifies no spatial adjacency requirements beyond the indirect implications of the ventilation and access provisions noted above.

This regulatory context supports the direction of the NCC-silence finding: consistently avoided adjacencies are not codified because the Australian building regulatory system does not codify spatial organisation; it codifies spatial performance. Such avoidance represents a different type of signal: cultural, functional, and behavioural dispreferences that the design profession and the market have collectively stabilised without regulatory mandate, carried in the apparatus as soft preference weights rather than hard prohibitions.

4. Confidence assessment

Topological coupling structure: moderate confidence. The census coupling structure (49 HARD / 50 SOFT / 581 INSUFFICIENT, with no pair classifiable as absent or sparse) is grounded in the 745-plan census and a well-established weighted-network analysis method. A boundary-margin sensitivity check reveals threshold-boundary uncertainty: pairs well clear of the weight=100 cut carry higher confidence, while the weight-100 boundary pairs are threshold-sensitive. The finding is therefore declared limited at the threshold boundary.

NCC silence on spatial adjacency: descriptive. NCC Volume 2’s scope is performance-based and does not contain spatial adjacency requirements. A formal adversarial read of a sample of consistently avoided pairs against NCC text remains outstanding; the assessment is supported by the structural scope analysis above and by the NCC’s well-documented performance-based framework. Consistent avoidance is reported as a descriptive design dispreference, not an absolute prohibition.

5. Citation ledger for §8.33 prose

5.1 Space syntax and access graphs: primary sources

• Hillier, B. and Hanson, J. (1984). The Social Logic of Space. Cambridge University Press. Cited >5,000 times.
• Turner, A., Doxa, M., O’Sullivan, D., and Penn, A. (2001). From isovists to visibility graphs. Environment and Planning B 28(1): 103-121.
• Penn, A. et al. (1997). Intelligent architecture: new tools for three-dimensional analysis of space. 1st Space Syntax Symposium. Paper 30.
• Hillier, B., Penn, A., Hanson, J., Grajewski, T., and Xu, J. (1993). Natural movement. Environment and Planning B 20(1): 29-66.

5.2 Architectural adjacency matrices and procedural generation

• Grierson, D. and Khajehpour, S. (2002). Method for conceptual design applied to office buildings. Journal of Computing in Civil Engineering 16(2): 83-103.
• Marson, F. and Lopes, S. R. R. (2010). Automatic real-time generation of floor plans based on squarified treemaps. IJCGT.
• Merrell, P., Schkufza, E., and Koltun, V. (2010). Computer-generated residential building layouts. ACM TOG 29(6): Art. 181.

5.3 Weighted network analysis and threshold selection

• Barrat, A., Barthélemy, M., Pastor-Satorras, R., and Vespignani, A. (2004). The architecture of complex weighted networks. PNAS 101(11): 3747-3752.
• Opsahl, T., Colizza, V., Panzarasa, P., and Ramasco, J. J. (2008). Prominence and control: the weighted rich-club effect. Physical Review Letters 101: 168702.
• Grover, A. and Leskovec, J. (2016). node2vec: Scalable feature learning for networks. KDD 2016.

5.4 NCC Volume 2 (Australian regulatory)

• Australian Building Codes Board. (2023). National Construction Code Volume 2: Housing Provisions. ABCB. NCC 2022 (2024 amendments). https://ncc.abcb.gov.au/
• NSW Government Architect’s Office. (2020). Apartment Design Guide. NSW Government.
• Steinhardt, D. A. (2014). The Importance of Home: NCC evolution discussion.

Why this note exists

The empirical analysis in Chapter 8 identifies a 50-100 mm metric grid in the Australian residential product market (50 mm the dominant component grain, with 25 mm a common but secondary sub-module) across the validated rebuild corpus of 40,342 product dimensions. The finding is read descriptively over a census and warrants situating in the international tradition of modular coordination, both to show that it is not an artefact of one corpus and to state precisely how it relates to the standards that might explain it. This note establishes that context. Its central claim is deliberately modest and is the opposite of an adoption claim: the market’s dominant grain (50 mm, the ISO half-module M/2, on a grid whose dominant values are the standard sheet and door sizes) is consistent with ISO 2848:1984, but it is equally consistent with AS 1684 and with metricated-imperial sizing, and the census cannot single out one of these as causal. ISO 2848 is one mutually-consistent referent among several, not a standard the market has been shown to adopt; and the grain the market concentrates on (50 mm) is in fact coarser than the finest sub-modules ISO permits (25 mm and 12.5 mm). The note is therefore not a warrant for an adoption claim, there is none, but the context that lets the descriptive finding be read against the tradition without over-reading it.

1. The ISO 2848 framework and the definition of 25 mm as M/4

International Standard ISO 2848:1984 (Building construction — Modular coordination — Principles and rules) specifies the foundational framework for dimensional coordination in construction.⁴³ The standard is paired with ISO 1006:1983 (Basic module), which fixes the basic module at M = 100 mm: in imperial countries, equivalent to 4 inches = 101.6 mm.⁴⁴ Building components, assembly dimensions, and construction grids are to be dimensioned in multiples of M.

The standard also formalises sub-modular increments below the basic module for components that require finer-grain dimensional control:

• Sub-module 0.5M = 50 mm
• Sub-module 0.25M = M/4 = 25 mm

The 25 mm sub-module is an officially recognised dimensional increment within the ISO modular-coordination framework.⁴⁵⁴⁶ Its intended use is precisely for the class of products the Bunnings corpus contains: plate thicknesses, profile heights, panel thicknesses, hardware reach dimensions, fixing centres: components where 100 mm granularity would over-quantise the design space.

The implication for Chapter 8’s finding must be stated carefully, because the relationship is consistency rather than confirmation. The market’s dominant component grain is 50 mm, the ISO half-module M/2, and 25 mm, the M/4 sub-module, survives as a real but secondary increment. Both values are recognised ISO sub-modules, so the grid the corpus exhibits is consistent with the ISO sub-module hierarchy. But consistency is not adoption: the same values are AS 1684 increments and emerge from metricated-imperial sizing, so the corpus cannot show that ISO in particular shaped the grid. What the data do establish is narrower and worth stating plainly: the market concentrates on the coarser ISO sub-module (50 mm), not on the finest the standard permits (25 mm or 12.5 mm), so where the earlier reading inferred a market “implicitly adopting” the M/4 sub-module, the validated evidence shows the M/4 value present but secondary to the M/2 grain.

1.1 The multimodules and their correspondence to the Bunnings meso-tier

The ISO 2848 multimodule series (3M, 6M, 12M, 15M, 30M, and 60M) defines the coarser dimensional grid used for room dimensions, structural spans, and large components.⁴⁷ In millimetres: 300 mm, 600 mm, 1200 mm, 1500 mm, 3000 mm, 6000 mm.

The validated rebuild corpus’s top-ranked frequency values (see tbl_frequency_threshold_derivation.md) are: 600 mm (n = 1,838, rank 1), 1,200 mm (n = 1,668, rank 2), 900 mm (n = 1,604, rank 3), 2,400 mm (n = 1,081, rank 4), and 450 mm (n = 940, rank 5). Four of the top five are exact ISO multimodules (6M, 12M, 9M, 24M) and the fifth (450 mm) is 4.5M. But these same values are the AS 1684 framing grid and the metricated-imperial sheet sizes (2,400 mm ≈ 8 ft, 1,200 mm ≈ 4 ft, 600 mm ≈ 2 ft), so the coarse grid is multiply determined: it is consistent with the ISO multimodule series rather than identical to it in any way that isolates ISO as the cause.

The same caution applies at both scales. The coarse grid (600 / 900 / 1,200 / 2,400 mm) and the fine grain (50 mm dominant, 25 mm sub-module) are each consistent with the ISO system (multimodules are multiples of M, sub-modules are fractions of M) but each is equally consistent with the Australian framing standard and with metricated-imperial sizing. The corpus shows a metric grid coherent across scales; it does not show which standard produced it, and the earlier claim that the market converges on the ISO system “at both scales” over-read a consistency as an adoption.

2. Historical precedent: modular coordination predates ISO

The ISO 2848 framework is the modern codification of an argument that stretches back to Vitruvius: that a single dimensional unit, repeated, proportioned and subdivided, is the most efficient organising principle for building. Three precedents from the literature ground this lineage and inform the interpretation of the Bunnings finding as more than a one-corpus artefact.

2.1 Japanese kiwari and the ken module

The Japanese kiwarijutsu (木割術) tradition, formalised in the kiwari-sho manuals of the Muromachi and Edo periods (approximately 1392-1868), is the most historically-deep system of procedural dimensional generation in architectural practice.⁴⁸⁴⁹⁵⁰ The ken (a structural bay unit whose regional values ranged from 1757 mm (Edo-tatami) to 1970 mm (Kyōma/Western Kyoto), with the most widespread value being 1818 mm (Inakama/Eastern, ≈ 6 shaku)⁵¹) is coordinated with the shaku sub-unit of 303 mm, the sun (30.3 mm), and the bu (3.03 mm).

Two primary-source studies refine the interpretation of this tradition in ways directly relevant to the chapter’s argument. Cruz-Saito, Nishida, and Bonnin (2007), in a detailed French-language ethnographic study of tatami and Japanese spatiality, explicitly warn that Western modernists (Gropius cited by name) adopted the tatami as an icon of “normalised modulation” while in fact the mat expresses a dimensional complexity the Western module concept does not capture: the tatami’s size is locally re-calibrated to room and family and cannot be reduced to a single fixed value.⁵² Okamoto and Naito (1984) document the tatami-shikiyō-hinagata manuals, early-modern Japanese pattern-books specifying floor-mat layouts by room type, as a formal record of the procedural-generation logic that linked the ken-shaku module to room dimensions via published rules.⁵³ Together these primary sources support the chapter’s analogy: Chapter 9’s procedural module-library generation on the dwelling’s grid, a 50 mm micro grain with a 150 mm meso grid above it, is the algorithmic counterpart of the kiwari rule-book’s procedural generation of room sizes from a ken module.

Two observations from this tradition bear directly on the chapter’s argument. First, the Japanese system was procedural: every derived dimension (column section, rafter depth, beam span, tatami size) was generated algorithmically from the ken through published proportional ratios: the nearest pre-modern analogue to the module-library derivation in Chapter 9. Second, the system permitted regional variation around a shared conceptual module (1757 mm to 1970 mm: a ±6% band around 1850 mm). The Ch8 empirical finding of a dominant 50-100 mm metric grid does not preclude regional drift in the Australian market around those values; it describes the central tendency of the captured corpus. The kiwari precedent shows that an empirical modular grid with regional variation is the historical norm, not the exception.

2.2 The Roman castra and the Vitruvian column-radius module

The Roman legionary camp (castra) codified a standardised spatial plan in which the positions of every major element (praetorium, via principalis, barracks) were repeatable across imperial territory.⁵⁴ Vitruvius, in De architectura (first century BCE), made the module explicit at component scale: the radius of a column base is the module from which the entire proportional system of the temple is derived: the earliest formal system of proportional design documented in Western architecture.⁵⁵

The chapter’s metric-grid finding occupies the same conceptual place as Vitruvius’s column-radius: a small dimensional unit, here the 50 mm micro grain, whose consistent application generates a coherent system of larger dimensions. The difference is methodological. Vitruvius deduced his module from a proportional theory of the classical orders; the Bunnings analysis induces its grid descriptively, by lift over a null, from a corpus of 40,342 product dimensions. Both arrive at the same structural intuition, that a small unit carries a coherent modular hierarchy, via opposing epistemic routes, the inductive one reported here making no claim beyond the grid the corpus exhibits.

2.3 SCSD, Ehrenkrantz, and the failure of one-corpus modularity

The School Construction Systems Development programme (SCSD) in 1960s California, led by Ezra Ehrenkrantz, developed a coordinated dimensional system for school buildings based on a 5-foot (1524 mm) planning grid with sub-modular components for structural, mechanical, and lighting subsystems.⁵⁶⁵⁷ SCSD was successful in its own right, the dimensional coordination allowed multiple subsystem vendors to compete at the component level, but did not generalise beyond schools, because the 5-foot grid was not aligned with residential or commercial practice. The chapter’s finding acknowledges the same boundary: the metric grid it reports (50 mm dominant, 25 mm sub-module) is a property of this domain and this corpus, not a universal optimum.

The ISO 2848 framework is precisely the response to the SCSD boundary problem. By establishing 100 mm (M) as the universal basic module and permitting sub-modules (50 mm, 25 mm) and multimodules (300 mm, 600 mm, 1200 mm) to serve domain-specific applications, ISO creates a dimensional coordination system that spans residential, commercial, industrial, and institutional construction without the narrowness of any single corpus. The Bunnings corpus’s dominant grid is consistent with that universal framework, its values are recognised ISO sub-modules and multimodules, so the Ch8 finding shows the grid the market exhibits to be coherent with the framework, while making no claim that the framework in particular produced it.

2.4 UK and international adoption: Brookes, Osborne, and Milton

The UK’s post-war programme of dimensional coordination, reaching back to the Hertfordshire Schools of 1955, provides the direct lineage for ISO 2848’s residential and educational dimensional standards. Brookes’s 2005 retrospective traces this history and the successive attempts by the UK government and agencies (including the Property Services Agency’s PSA Method of Building) to mandate dimensional coordination, together with the resistance from contractors who preferred the flexibility of non-coordinated dimensions.⁵⁸ Osborne’s 1959 paper, one of the earliest primary sources in the English-language modular-coordination literature, offers a concise formal definition that still holds: a module is “a unit of size … which, operating in three dimensions, unifies the work of designer, manufacturer and builder … [postulating] the employment in building of dimensionally related units, components and fittings which are made and shaped to fit together, and which architect and builder know from the start will fit together, without cutting, packing or fitting.”⁵⁹ Osborne’s phrase “without cutting, packing or fitting” is exactly the benefit a coordinated dimensional standard delivers to the Bunnings corpus: products dimensioned on a shared metric grid fit together at installation without on-site adjustment.

Milton’s 2018 interim report provides the most recent catalogue of “international, regional (multi-national), and national standards dealing with the principles and practical application of modular and dimensional coordination in building, including joints and tolerances”: a survey that can be cross-referenced when § 8.10’s cross-jurisdictional transferability argument is drafted, since the same review documents the parallel standards in imperial and soft-metric regimes (where M/4 ≈ 25.4 mm = 1 inch).⁶⁰ Baldauf’s dissertation on Brazilian modular coordination (2004) provides a useful parallel case: Brazil, like Australia, operates ISO-aligned metric standards but has struggled to achieve full market-level coordination in the absence of strong code mandates: a situation the metric-grid finding suggests may apply in Australia too (emergent coordination on a shared grid in the product corpus, with no formal ISO compliance audit and no evidence that ISO in particular is the coordinating standard).⁶¹ The mid-1960s Spanish theoretical programme (Yraola 1966a, 1966b, 1966c) documents the continental European adoption of modular coordination in the pre-ISO era: providing a reference point for the specific design-theoretic arguments that led to the 100 mm basic module being adopted over alternatives such as 120 mm or 200 mm.^{62 6364}

3. Why a retail corpus exhibits a coherent metric grid

The method applied in Chapter 8, sweep all candidate base modules with no normative floor and score each by its lift over a reporting-convention null, finds a 50-100 mm metric grid rather than a smear of arbitrary dimensions. That a national retail corpus should exhibit a coherent metric grid at all is explained by three mechanisms documented in the literature on power-law and modularity dynamics. None of these mechanisms is specific to ISO 2848: each predicts convergence on a shared grid, and in the Australian market ISO 2848, AS 1684, and metricated-imperial sizing all point at the same coarse values, so the mechanisms explain the existence of the grid without isolating its cause.

3.1 Power-law dynamics in coordinated systems

The Pareto principle (the 80/20 rule) describes an emergent property of systems governed by preferential attachment, multiplicative growth, or scale-invariant selection pressures, rather than an arbitrary ratio.⁶⁵ In a market where each product’s commercial viability depends on compatibility with existing dimensional conventions, a preferential-attachment dynamic operates: products dimensioned on modules already in wide use are more likely to be specified, stocked, and installed. The module with the most support attracts more support; the result is a power-law-like convergence on a small number of dominant module values.⁶⁶

Where the dimensional conventions in force in the jurisdiction (ISO 2848, AS 1684, and the metricated-imperial sheet trade, all mutually consistent at the values they share) concentrate on a metric grid, the preferential-attachment dynamic will pull product dimensions onto that grid. The chapter’s 50-100 mm grid is the empirical signature of exactly this dynamic; because the converging conventions agree at the coarse values, the dynamic explains why the grid exists without identifying any one standard as the attractor.

3.1a Corpus-based module analysis: a research lineage

The broad approach used in Chapter 8, analysing a candidate set of base modules against a product corpus to identify the grain a market expresses, has an established lineage in the modular-product-family design literature. The present method differs from that tradition in two ways worth stating, so the precedent is not mistaken for the method: Chapter 8 scores candidates by descriptive lift over a reporting-convention null with no normative floor, not by a fitness or quality-loss function to be optimised, and it makes no optimisation or selection claim; it reports which grain the corpus exhibits. With that distinction made, four heavily-cited primary sources locate the corpus-based module-analysis approach in a twenty-year research tradition:

• Rai and Allada (2003) introduce an agent-based Pareto-optimisation framework for modular product-family design, pairing a Pareto-frontier search over module configurations with a post-optimal quality-loss function analysis. Their method sits adjacent to Chapter 8’s at a smaller scale, and the contrast is the point: where they select a module set that maximises a coverage-like objective under a quality-loss constraint, Chapter 8 does not select or optimise but measures, by lift over a reporting-convention null, which grain the corpus already exhibits.⁶⁷
• Chakravarty and Balakrishnan (2001) formalise the trade-off between manufacturing cost, development cost, and market share in modular product design. They show that the number of variants and the choice of module variation values are jointly determined by the cost-share curve’s knee: structurally the same knee argument, though Chapter 8 makes no optimisation claim: its descriptive reading finds the 50 mm grain carrying the highest lift over the reporting-convention null, with 25 mm surviving as the finest sub-module beneath it rather than as a selected optimum.⁶⁸
• Yigit and Allahverdi (2003) frame optimal module-instance selection as an integer nonlinear programming problem with a quality-loss objective, demonstrating that the Pareto-optimal module set is a small subset of the candidate space: consistent with the unconstrained sweep (m over the [2, 300] mm range) concentrating on a single dominant grain, 50 mm on the validated rebuild, with 25 mm present but secondary.⁶⁹
• Song and Kusiak (2009) explicitly mine Pareto-optimal modules from historical product sales data for delayed product differentiation: arguably the closest methodological analogue to Chapter 8’s Bunnings-corpus mining approach. Their framework estimates demand probability from sales data and selects modules to maximise a joint commonality-and-differentiation objective; the 50 mm dominant grain can be read as the commonality-maximising grain of the Bunnings corpus, with 25 mm a secondary sub-module, read descriptively rather than as an optimised selection.⁷⁰

These four heavily-cited papers establish that analysing a candidate set of modules against product data is a canonical move in product-family design. Chapter 8’s contribution is its application to a large retail corpus rather than a single manufacturer’s product line, and its descriptive posture: where that literature optimises a module set against a cost or quality-loss objective, Chapter 8 only measures, by lift over a null, which grain the corpus already exhibits, and draws no optimality claim. This both locates the work within a mature research tradition and marks where it departs from it.

Two further supporting citations frame the method’s broader context: Wei, Liu, Lu, and Wuest (2015) on multi-principle module identification for product platforms,⁷¹ and Goswami (2018) on integrative product-line redesign under competitive-market constraints.⁷² Both extend the Pareto-selection framework to multi-objective and redesign contexts, indicating that the method travels well to adjacent problem classes: which supports the chapter’s external-validity argument that the Bunnings method could transfer to other jurisdictions or corpora (noted as open research in §8.63 and §11 discussion).

3.2 Network effects of a shared modular API

Modularity is not only a geometric or ergonomic property; it is a network-effect property.⁷³ Each additional product dimensioned on the shared metric grid increases the value of every other grid-dimensioned product in the ecosystem: stacking, nesting, fastening, alignment, and coordination all improve. The Bunnings corpus’s concentration on a 50-100 mm grid is therefore the built signature of a market where that grid has crossed the network-effect threshold at which conformance is cheaper than deviation. The theoretical prediction (that where shared dimensional conventions are locally effective, products converge on a common grid) matches the empirical result; the prediction is about convergence on a grid, and does not identify ISO 2848 as the specific convention at work.

3.3 Modularity as socio-technical contract

The final mechanism, documented in contemporary modular-systems literature, is that modular coordination is a governance regime as much as a technical one.⁷⁴ Three institutional elements are required for a modular system to sustain: (1) specification of what may vary (component internals, material, finish); (2) specification of what is fixed (the interface dimension); (3) a governance regime that rewards conformance. The Bunnings ecosystem supplies all three: product diversity is preserved across the corpus’s many product types; the metric grid is the fixed interface (across the validated corpus the 50 mm grain carries the highest lift of any standard grain over the reporting-convention null in every cohort, with 100 mm next and 25 mm a secondary sub-module); and conformance is rewarded through reduced product-integration friction at point of sale.

4. The Australian regulatory context

The National Construction Code (NCC), published by the Australian Building Codes Board, is the performance-based building standard that governs residential and commercial construction throughout Australia.⁷⁵ The NCC references AS 1684 (Residential timber-framed construction) for timber framing, AS 2870 (Residential slabs and footings) for foundations, and associated standards for components such as plasterboard (AS/NZS 2588), glass (AS 1288), and fixings. These documents make no formal ISO 2848 requirement. They are nevertheless structured around dimensional conventions (450 mm and 600 mm stud spacings, 1200 × 2400 mm plasterboard sheets, 900 mm door leaves, 1800 mm window-sill heights) most of which are directly ISO multimodules (600 mm = 6M, 1200 × 2400 mm = 12M × 24M, 900 mm = 9M, 1800 mm = 18M); the 450 mm stud spacing is a multiple of the M/2 = 50 mm sub-module (9 × 50 mm), not an integer multimodule of the 100 mm basic module.⁷⁶ The Australian residential construction system is in practice ISO-coordinated even where it is not ISO-mandated.

Three caveats are important for the Ch8 argument.

First, the Australian modular construction market, as distinct from the broader traditional construction market, remains niche, at 5-8% of output as of 2024, with strong concentrations in Victoria (manufacturing base) and Queensland (government procurement).⁷⁷ The metric-grid finding therefore applies to the broader product-supply market (which Bunnings anchors) rather than the modular-construction-manufacturing subset.

Second, using Bunnings as a corpus proxy is a novel methodological choice: comparable dimensional-analysis studies typically draw on standards documents, manufacturer specifications, or project datasets rather than a national hardware retailer’s product catalogue. Adopting the catalogue is at once the methodological contribution Chapter 8 makes and a limitation whose representativeness must be argued explicitly in Section 8.10. Commentary on construction-industry data maturity (the Housing Industry Association and the Australian Building Codes Board both note the “data fog” around prefabrication terminology⁷⁸) makes the Bunnings-proxy method defensible as the best available alternative in a low-data-maturity national market.

Third, AS 1684’s specific dimensional provisions (stud centres, wall-plate sections, bearer depths) operate on 25 mm and 50 mm increments within the broader 3M/6M/12M framework.⁷⁹ Chapman’s 1981 primary source confirms the 600 mm stud-spacing convention (= 6M = 24 × M/4) that has governed Australian timber framing for decades.⁸⁰ Jiang, Ottenhaus, and Gattas (2023) demonstrate a current-practice parametric extension: their parametric framework for AS 1684 span tables reduces the over-specification of timber products by exploiting the systematic dimensional relationships embedded in the standard: an explicit computational-design use of the modular coordination the chapter identifies empirically.⁸¹ That AS 1684 itself works in 25 mm and 50 mm increments mirrors the validated finding exactly, a 50 mm dominant grain with 25 mm as its sub-module, so the metric-grid finding is coherent with, rather than merely adjacent to, the specific Australian residential standard that governs the target construction domain, without that coherence implying the market follows ISO 2848 in particular.

Fourth, recent Australian prefabrication research (Navaratnam, Rahardjo, and Godakandage 2025) provides an evidence-based assessment of the potential for upscaling prefabricated timber modular buildings in Australia: a parallel study that, like Chapter 8, combines empirical audit of current practice with design-system implications.⁸² Dewsbury, Tooker, and Fay (2013) document the Australian residential thermal-performance regulations adopted under the National Construction Code, tracing their impact on wall- and roof-material dimensions: the regulatory coupling that shapes which ISO multimodules become effectively required in Australian construction practice.⁸³ Clayton (2010) adds a component-level view: his paper on steel wall studs with service holes notes that the 30 mm service-hole convention (which aligns with the finer M/10 = 10 mm sub-module, 30 = 3 × 10 mm, rather than the 25 mm sub-module) reflects a further layer of dimensional coordination between framing and services that is assumed without formal codification.⁸⁴

5. Implications for the chapter’s claims

Integrating the literature context above, the bearing on the chapter’s validated claims (printed here at their authority-ID anchors, CL-8-01, CL-8-02, and CL-8-03) is as follows:

1. CL-8-01: §8.21, the metric-grid finding. The chapter reports a 50-100 mm metric grid, with 50 mm the dominant component grain and 25 mm a secondary sub-module, recovered by lift over a reporting-convention null with no normative floor. The literature situates this as a market exhibiting the shared metric grid that ISO 2848, AS 1684, and metricated-imperial sizing jointly express, but it is framed throughout as consistency, not adoption: the grid is multiply determined, and the analysis does not isolate ISO as its cause. This is the honest replacement for the earlier “25 mm Pareto-optimal / market adopted ISO” framing, which was an artefact of an ISO-anchored search floor plus an un-baselined coverage score and is not reproduced by the validated rebuild.

2. CL-8-02: §8.26, the dimension-value library. The corpus’s most frequent values (600, 1,200, 900, 2,400, 450 mm) are ISO multimodules, but equally AS 1684 framing values and metricated-imperial sheet sizes. They are reported as the heavy-tailed head of the value distribution that grounds the required/common/rare library partition, consistent with the preferred series rather than evidence that the market validates ISO’s series in particular.

3. SL-01: §8.63, the corpus scope limit. The representativeness argument is stated as a plain scope limit: the Bunnings corpus is one retailer, one snapshot, Australia, title-stated nominal dimensions only, ~52 per cent product coverage, read descriptively over a census. It is not an ISO compliance audit and licenses no inference that ISO 2848 is followed in Australia: only that the dominant grid in this retail corpus is consistent with the ISO sub-module and multimodule values. A cross-jurisdictional extension to imperial-unit markets (basic module 4 in = 101.6 mm; sub-modules near 25.4 mm) is an open research question flagged here and returned to in Chapter 11.

6. References supporting Chapter 8 integration

The sources supporting the §8.21, §8.26, and §8.63 prose are grouped by theme below; full bibliographic details appear in the footnotes throughout this note.

International standards

• ISO 2848:1984: Modular coordination principles and rules
• ISO 1006:1983: Basic module (M = 100 mm)
• ISO 1040 (referenced in 2848): Multimodules

6.2 Modular-coordination theory and history

• Osborne (1959), Modular Co-Ordination as Related To Sanitary Accommodation and Fittings
• Yraola (1966a/b/c), three seminal Spanish papers on ISO modular-coordination theory
• Brookes (2005), Theory and Practice of Modular Coordination: UK history from the Hertfordshire Schools, 1955
• Baldauf (2004), Contribuição à implementação da coordenação modular da construção no Brasil
• Milton (2018), International and National Standards on Dimensional Coordination, Modular Coordination, Tolerances and Joints

6.3 Corpus-based module analysis: research lineage (the present method departs from its optimisation framing)

• Chakravarty and Balakrishnan (2001), Achieving product variety through optimal choice of module variations
• Rai and Allada (2003), Modular product family design: Agent-based Pareto-optimization …
• Yigit and Allahverdi (2003), Optimal selection of module instances for modular products …
• Song and Kusiak (2009), Mining Pareto-optimal modules for delayed product differentiation
• Wei, Liu, Lu and Wuest (2015), A multi-principle module identification method …
• Goswami (2018), An integrative product line redesign approach …

6.4 Japanese kiwari and ken tradition

• Engel (1964), The Japanese House: A Tradition for Contemporary Architecture
• Okamoto and Naito (1984), Types of the Architectural Manuals “Tatami-Shikiyō-Hinagata”
• Cruz-Saito, Nishida and Bonnin (2007), Le tatami et la spatialité japonaise …, Ebisu 37
• Sustainability 15(7): 5800 (2023): kiwari measurement study
• Building Research Institute Japan, Research Paper 131: traditional wooden structure dimensional analysis

6.5 Pareto and power-law dynamics: theoretical anchor

• Newman (2005), Power laws, Pareto distributions and Zipf’s law, Contemp. Phys. 46

6.6 Australian context

• AS 1684 series: residential timber-framed construction
• AS 2870: residential slabs and footings
• National Construction Code (ABCB, current edition)
• Chapman (1981), Timber Wall Framing …
• Dewsbury, Tooker and Fay (2013), thermal performance of Australian lightweight residential construction
• Clayton (2010), Design of Steel Wall Studs with Service Holes
• Jiang, Ottenhaus and Gattas (2023), A parametric design framework for timber framing span tables
• Navaratnam, Rahardjo and Godakandage (2025), An evidence-based assessment of the potential for upscaling prefabricated timber modular buildings
• Housing Industry Association guidance on AS 1684: industry practice

6.7 Research method: module-lift over a reporting-convention null on the Bunnings corpus

• Rai and Allada (2003); Chakravarty and Balakrishnan (2001); Yigit and Allahverdi (2003); Song and Kusiak (2009): the research lineage of corpus-based module analysis (from which the present method departs by scoring descriptive lift over a null rather than optimising a fitness)
• Newman (2005): theoretical anchor for the power-law-emergence argument
• CL-8-03: robustness of the 50 mm dominant grain across three cohort cuts and a matching-tolerance sweep over the unconstrained search range: descriptive recurrences over the complete census, with no interval estimate or significance test attached

Why this note exists

The geometry read of the 745-plan census (the canonical thesis-wide space count of 12,849 spaces, on which the dimensional and generator pipelines rest) yields a set of quantitative design references that the empirical chapter and the Chapter 9 generator draw on. This note collects them: observed room-size targets, the dimensional grid, fixture packing densities, the kit-reuse ratio, and the context-conditioned size rules. Every figure is a descriptive statistic over a complete enumeration: an observed band a designer reads and dials within, not an estimate or a fitted target. Over a complete population the median and its interquartile range are the population’s own values, not estimates of an unobserved one, so no measure of sampling variability attaches.⁸⁵ Because printed dimensions are present for 7,287 of the 12,849 spaces (56.7 per cent: structurally, the habitable rooms a buyer evaluates), the size tables describe that dimensioned subset, while counts (kit reuse, packing density, served-to-serving balance) use the complete inventory and are the more robust figures.

This collection differs in kind from the two reference traditions it sits beside. Unlike a normative dimensional handbook (Neufert’s Architects’ Data; the ISO 2848 grid of Appendix H.4), whose figures are prescribed targets, these are measured central tendencies and spreads read off what Australian project builders actually drew. And unlike Alexander’s pattern language, whose patterns are qualitative and author-asserted,⁸⁶ each reference here carries a numerical scope and an observed spread.

0. Census enumeration reconciled: the three space counts and four category counts

Chapter 8 reports the same 745-plan census through three pipelines that each count spaces on a different basis, and the space taxonomy is reported at four reference points: the census category count, its two-stage componential consolidation, and a legacy topological subset retained for auditability. The figures appear in different Appendix H modules for different purposes, so this table gathers them in one place and states, where one count derives from another, the transform that carries it across. It reconciles the structure of the enumeration; the per-figure numeric derivations it references (in particular the occurrence-to-geometry delta and the category reduction) are the subject of a separate data-quality reconciliation and are not re-derived here.

Space counts. Three pipeline reads of one census:

Category counts. Four reference points:

Of the 12,849 canonical spaces, printed dimensions are present for the dimensioned subset used by the size tables below (Section 1); the count-based figures (kit reuse, packing density, served-to-serving balance) use the complete 12,849-space inventory. The precise size of the dimensioned subset, and the small residual discrepancies between the count bases recorded across the Appendix H modules, are registered to the same data-quality reconciliation rather than adjudicated in this note.

1. Room-size targets by category

For each category the median floor area is reported with its interquartile range (IQR) and quartile coefficient of dispersion (QCD, a scale-free spread measure), with the median long and short side. The QCD separates the categories a designer may treat as standardised (sized to a fixed object) from those that absorb residual area.

Categories with fewer than twenty dimensioned instances are reported in the data file but flagged small-n. The most standardised categories are the garage (QCD 0.087), main bedroom (0.145) and media room (0.154): each sized to a fixed object; the most variable are the store (0.637), porch (0.568) and alfresco (0.408): the leftover and outdoor spaces that absorb whatever area remains.

1.1 Context-conditioned size targets (the generator’s sampling vocabulary)

Room size is conditioned on context; the strongest conditional levers, read directly off the census, are:

• The main bedroom grows with the dwelling; the secondary bedroom does not. Median main-bedroom area rises 12.3 → 13.9 → 14.3 → 16.8 m² across two-, three-, four- and five-plus-bedroom dwellings, while the secondary bedroom holds near 10.8 m². The pooled bedroom median appears to fall as bedroom count rises, but that is a composition effect, larger dwellings hold a higher share of (smaller) secondary bedrooms, not a shrinking of any one room.
• Living scales with its graph role, not the bedroom count. A living space that is a circulation hub has a median area of 23.2 m² against 15.1 m² for a leaf living space.
• The garage is bimodal: a single garage clusters near 6.0 × 3.2 m (n = 140), a double near 6.0 × 5.8 m (n = 425), doubles outnumbering singles about three to one.

These conditional bands are the vocabulary the Chapter 9 generator samples (full cell table in the geometry-census analysis bundle).

2. The dimensional grid

Of 14,574 printed room dimensions, 99.6 per cent are multiples of 5 mm, 99.5 per cent of 10 mm, 89.4 per cent of 25 mm (equivalently 89.4 per cent of 50 mm: only four printed values are multiples of 25 but not 50), 87.3 per cent of 100 mm, and 35-36 per cent of 150 and 300 mm. Australian project-home room dimensions therefore sit on a 50/100 mm grid: the same dominant metric grain the product corpus exhibits (the 50 mm micro grain of Appendix H.4), and coarser only than its 25 mm sub-module. The single most common exact dimension is 3,000 × 3,000 mm, and 3,000 mm is the modal short side; as rooms enlarge both sides grow, but the long side grows faster than the short (a factor of about 2.2 against 1.8 across long-side quintiles), so the room lengthens faster than it widens and the aspect ratio rises.

These percentages describe how project builders currently draw rooms; the grain they reveal, 50/100 mm, is a description of the stock, not the thesis’s prescription. The grid the thesis prescribes for dimensioning room modules is the coarser 150 mm meso grid, and the two are not in tension. The 100 mm figure is the resolution at which the existing stock is drawn; 150 mm is the grain at which room modules compose with least combinatorial misfit, derived as a calibrated design choice in the integration of Section 8.50. On the composability axis a coarser grid is the better one: snapping the census to 150 mm cuts the distinct room-module catalogue from 1,770 to 1,117 rectangles and the effective module vocabulary from 349 to 203 against the 100 mm grain, but 150 mm is the coarsest grid that secures that gain while keeping the 95th-percentile room within 50 mm of its drawn size; a 300 mm grid would double that displacement to 100 mm and would foreclose the substantive room variation the figures above record. The 150 mm meso grid therefore organises the 100 mm-grained stock rather than contradicting it.

2a. The meso-grid calibration: composability, distortion, and standardisation

The 150 mm meso-grid selection of Section 8.50 rests on the battery set out in full here, every figure descriptive over the 745-plan geometry census (7,287 dimensioned rooms; 3,655 interior adjacencies with both endpoints dimensioned); the corpus is a complete enumeration, so no sampling variability attaches. The selection turns on two axes that move in opposite directions as the grid coarsens. Fit (the share of room dimensions that are exact multiples of the grid) always favours the finer grain and is a degenerate criterion (Section 8.21). Composability, how small a module vocabulary the grid produces and how readily modules share a commensurate edge, improves as the grid coarsens, paid for in distortion (how far a room must move to reach the grid). Treating the grid as a lossy quantiser, the rate (bits to name a module, log-2 of the effective module vocabulary) falls and the distortion rises monotonically with coarseness:

Because composability has no interior optimum (it improves all the way to 600 mm), the meso grid is fixed by a distortion tolerance, the largest displacement a room may suffer while still reading as the same room. The coarsest grid whose 95th-percentile room stays within a tolerance T is unique:

The 150 mm meso grid is exactly the choice of a 50 mm tolerance: the ISO half-module M/2, the coarsest sub-module the standard recognises and a displacement below the threshold at which a room reads as a different size. Tightening the tolerance to 25 mm would select a 75 mm grid; loosening it to a perceptible 100 mm would admit 300 mm. The selection is therefore closed under a stated, audited tolerance rather than asserted.

The result holds across the census and across room types, with one informative exception. Both strata return a 50 mm 95th-percentile displacement at 150 mm (effective modules 222 in the August stratum, 186 in the October delta), and the selection is stable to subsampling: withholding any one-fifth of the dimensioned corpus leaves the 95th-percentile displacement at 150 mm unchanged at 50 mm and the tolerance-bounded selection at 150 mm across all five disjoint folds, mirroring the base-module robustness reported at Section 8.22. By module type at 150 mm, the habitable modules all hold within the 50 mm tolerance (bedroom (n = 2,587; effective modules 55; p95 50 mm), living (n = 1,696; 272; 50 mm), kitchen (n = 454; 116; 50 mm), outdoor (n = 1,561; 187; 50 mm), and dwelling-other (n = 127; 86; 50 mm)) while the small service modules reach 60 mm: sanitary (n = 373; 77; 60 mm), circulation (n = 49; 42; 60 mm), and entry (n = 82; 52; 60 mm). These are the tightly packed wet and circulation spaces, whose internal components are dimensioned at the finer 50 mm micro grain rather than at the room meso grid; the exception is the micro-meso separation showing through, not a failure of the meso grid. The 150 mm value is a design prescription, the grain at which the generator dimensions its room modules, not the 100 mm grain on which the existing stock is drawn.

3. Fixture packing density

Median floor area per contained fixture, for the categories that pack fixtures: toilet 1.57 m², bathroom 1.86, ensuite 2.08, laundry 2.68, kitchen 3.00, pantry 3.35. Wet rooms are the most tightly packed spaces: density peaks exactly where rooms are smallest and most obligatorily coupled to a host.

4. Kit reuse and the served-to-serving balance

A median plan draws its 17 spaces from about seven distinct geometric primitives, a reuse ratio of about 2.4: each primitive instantiated two to three times; a whole dwelling is drawn from roughly seven kinds of rectangle. A median plan has 7 serving spaces against 10 served, with serving spaces about 39 per cent of all spaces under the narrow definition used here (circulation, service and storage, sanitary, and entry); the share is sensitive to that definition, rising toward one-in-two if outdoor and parking infrastructure are counted as serving.

All figures descriptive over the complete 745-plan census; independently re-derived from the canonical datasets. CANDIDATE until operator MR-13.

Why this note exists

The geometry census yields a small set of design rules in the spirit of a pattern language, but with a property Alexander’s patterns lack: each is stated over a measured scope and carries an explicit falsifier: the observation that, had it held, would have refuted the rule.⁸⁷ The rules are descriptive regularities over a complete enumeration, not validated hypotheses about an unobserved population; the falsifier is what the census could have shown and did not. The set is deliberately small and honestly bounded: where an apparent regularity proved to be a base-rate coincidence, it is reported as such rather than presented as a pattern. Size-based rules describe the dimensioned subset (the habitable rooms); count-based rules use the complete inventory.

1. The rules

DR-1: Scale by count, not by dimension. A larger Australian project home is the same standard rooms repeated more often, not the same rooms enlarged. Scope: two- to five-plus-bedroom dwellings (bedroom count, independent of dimensioning). Evidence: the median space count rises about 69 per cent (13 → 22 spaces). Median room area moves only 12 per cent on the median, 27 per cent on the mean; living area is flat near 22 m². Falsifier: had per-room area scaled with the count, the rule would fail. Caveat: the pooled bedroom median falls as bedroom count rises, but this is a composition effect: a rising share of smaller secondary bedrooms, not a shrinking of any one room. The main bedroom in fact grows, 12.3 → 16.8 m².

DR-2: Rooms lengthen faster than they widen. As rooms enlarge they grow in both dimensions, but the long side grows faster than the short, so the aspect ratio rises. A strong modal short side sits at 3,000 mm. Scope: dimensioned rooms across all categories. Evidence: across long-side quintiles the median long side rises from 3.0 to 6.7 m (×2.2) while the median short side rises only from 2.6 to 4.6 m (×1.8). The median aspect ratio climbs from 1.11 to 1.55, and the single most common exact dimension is 3,000 × 3,000 mm. Falsifier: had the short side grown as fast as the long side, the rule would fail.

DR-3: Containment makes a space small and tight; openness makes it large and variable. A space’s size and variability are governed more by its containment than by its nominal type. Scope: all spaces, read through the componential primitives (enclosure, location, edge). Evidence: enclosed spaces sit at a median 12.2 m² (QCD 0.32), open spaces at 16.7 m² (0.48). Internal sit at 12.3 m² (0.32), external at 16.4 m² (0.51). Walled sit at 12.2 m² (0.32), unbounded at 17.7 m². Falsifier: had the enclosure and location primitives not separated size and spread, the rule would fail.

DR-4: Obligation predicts the opening. The opening between two spaces encodes the strength of their coupling: the doorless open-threshold share rises monotonically with how obligatory the adjacency is. Scope: realised interior openings, classified by coupling class. Evidence: the open-threshold share is 22.2 per cent on required (HARD) adjacencies, 8.6 per cent on preference (SOFT), and 4.0 per cent on insufficient; a plain door dominates throughout. Falsifier: had the open-threshold share been flat or higher on weak couplings, the rule would fail. Caveat: this characterises the opening type over the geometry graph (which realises only about 272 of roughly 680 category pairs); the canonical 49-HARD / 50-SOFT coupling split rests on the topology graph (the 15,957-zone access basis), not on the geometry read.

DR-5. A single genuine higher-order bundle: the premium tier. Beyond the obligatory core, only one room co-occurrence rises materially above its base rate: where a dedicated media room is drawn, an alfresco is present 86 per cent of the time against a 61 per cent base rate (co-occurrence ratio about 1.4): a discretionary premium-tier marker. Scope: the discretionary feature set across all 745 plans. Evidence: by contrast the candidate “master suite” (walk-in robe → ensuite, ratio 1.06) and the open kitchen-living-dining co-presence (ratio 1.01) sit at their base rates: coincidences, not patterns. The 745 plans realise about 723 distinct category-presence patterns. Falsifier: had media → alfresco sat at its base rate, there would be no premium-tier rule. The rule deliberately does not claim a rich bundle grammar. Australian plans are near-uniquely composed from a shared obligatory core (secondary bedroom, kitchen, bathroom, living, laundry, each in ≥ 90 per cent of plans), not assembled from recurring higher-order modules.

DR-6: Form follows fixed content. The rooms sized to a fixed physical object are the most dimensionally standardised; the rooms that absorb residual area are the most variable. Scope: all dimensioned categories, ranked by area QCD. Evidence: most standardised: garage (QCD 0.087), main bedroom (0.145), media (0.154); most variable: store (0.637), porch (0.568), alfresco (0.408). Falsifier: had fixed-content rooms shown spread comparable to leftover rooms, the rule would fail. Caveat: it is fixed content, not fixture count, that drives standardisation.

2. Positioning

This rule set extends Alexander’s pattern-language method (qualitative, author-asserted) by reading each rule’s context and parameters off a complete enumeration and attaching a numerical falsifier; it relates to the shape-grammar tradition by offering a measured rather than author-specified generative vocabulary;⁸⁸ and it reads the plan graph through a HARD/SOFT coupling contract rather than space- syntax integration measures.⁸⁹ All descriptive over a census; CANDIDATE until operator MR-13.

Why this note exists

The lexicon defines standard room names as formulae over distinctive feature bundles, so a category means the same thing for every reader and outside this corpus. Two names denote the same category exactly when they share a bundle; names with no real difference are collapsed. It answers a question the co-occurrence analysis (design rule DR-5) shows cannot be answered by co-presence: because plans are near-uniquely composed, a room’s identity is carried by its features, not by what it is drawn next to. As a classification of “what is,” the lexicon is a descriptive (analysis-and-description) construct.⁹⁰

1. The minimal load-bearing feature set is four

Across the 47 realised categories, four features fully differentiate every genuinely distinct category, function · host · fixture signature · edge, yielding 36 distinct bundles (the remaining eleven names are genuine collapses). Enclosure, location and substrate are readable descriptors that ride along; storey is excluded as under-marked.

2. Access is a topology, not “privacy”

The distinctive fact about how a space is reached is its access topology, read from the plan graph, not a vague notion of privacy: off-circulation (entered from a hall), nested (entered through a single host room: the walk-in / ensuite structure), open-plan (a doorless merge), attached (an external space attached to a host), and detached (freestanding). The host, the room one passes through, generalises “private from the bedroom”: a walk-in robe and a walk-in pantry are the same access structure (nested), differing only in host (bedroom versus kitchen).

3. Definitions that fall out of the features

• Ensuite = a bathroom nested in a bedroom: identical to a bathroom except in access topology; that, not its size or fixtures, is its definition.
• Walk-in family = a storage space nested in a host, parameterised by host and contents: walk-in robe (bedroom + robe), walk-in pantry (kitchen + pantry shelving), butler’s pantry (kitchen). The reach-in counterpart is the shallow, non-enterable cupboard (linen).
• Alfresco = the roofed outdoor room (covered + external + attached, host = living); porch = the open outdoor arrival threshold; balcony/deck = the open living-attached platform (balcony its upper-storey instance); garage versus carport = enclosure (walled versus open-sided).

4. Substantive vocabulary findings (collapses, operator-confirmed)

family / lounge → living; main + secondary bedroom → bedroom (main = a size-and-master-suite variant); studio → study; several small-n outdoor types → one detached outdoor amenity; activity / breakout → rumpus. Kept distinct: store ≠ walk-in robe (general versus clothes storage; a store is general-access, a walk-in robe is nested in a bedroom). Data-quality flag (not a merge): in the census a large share of spaces labelled “store” actually hold a robe and are bedroom-hosted: mislabelled built-in robes, flagged for an MR-13 reclassification; the label is wrong, the distinction is not.

5. How the lexicon predicts size (the bridge to DR-3)

Because each distinctive feature carries a size signature, a category’s size is the intersection of its features. A bathroom (median 5.5 m²) is enclosed (compact) + internal (compact) + off-circulation (a small leaf) + wet-function (smallest class) + a three-piece fixture set (packs about three fixtures into a few square metres): every feature pulls it small and tight. The alfresco is the mirror image (covered + external + attached → large and variable). The lexicon thus does explanatory work, not just naming: it says why a category sits where it does in the size distribution. Descriptive throughout; CANDIDATE until operator MR-13.

Why this note exists

The Chapter 9 generator’s vocabulary is not invented; it is measured off the census. This note records that vocabulary and the descriptive contract under which the generator uses it.

1. The kit and its conditional rules

Ten data-derived geometric primitives (size band × aspect band × rectangular/complex) cover 90.7 per cent of all 12,849 spaces (top three 65.2 per cent). The single largest class is the undimensioned-rectangle placeholder (41.7 per cent): not a true geometric primitive but the spaces without printed dimensions; restricting to the 7,245 spaces with a resolved dimensioned shape, the ten most common dimensioned primitives cover 88.7 per cent of them. Either way the generative kit is small: a whole dwelling is drawn from about seven distinct primitives (kit-reuse ≈ 2.4). Each primitive carries a size signature (for example M-square-rect ≈ 3.6 × 3.2 m, bedroom-dominated; L-oblong-rect ≈ 5.5 × 3.7 m, living; XL-square-rect ≈ 6.0 × 5.8 m, garage). The generator selects and sizes a primitive for a context using the conditional design rules of Appendix H.5 (the empirical distribution of size for a given category × host × graph-role × bedroom-count cell).

2. The generator is descriptive: a lookup-and-redraw sampler, not a fitted model

The conditional rules are empirical conditional distributions tabulated over the complete census. The generator’s primary output for a context is the observed band (median and IQR) a designer dials within: a deterministic lookup. Where a single value is wanted, it is redrawn from the literal observed multiset of that cell, so the generator can emit nothing the corpus did not exhibit in that context; it never synthesises an intermediate value through a fitted curve, and it makes no inferential claim, because over a complete enumeration the band is the population’s own value, not an estimate.⁹¹ This is categorically unlike procedures that manufacture pseudo-samples to estimate sampling variability (none is used) and unlike learned generators that fit weights to a training sample.

3. Two hard constraints on the generator

1. Out-of-corpus guard (required, mechanised). For any context cell the census never observed, the generator flags and refuses to synthesise rather than interpolating a size; interpolation across cells would be inferential. This must be a hard check in the generator contract, not a discipline.
2. Coupling stays on the topology graph. The adjacency vocabulary uses the opening grammar (design rule DR-4) for opening type, but the obligatory-versus-preferred coupling split (49 HARD / 50 SOFT) is read from the canonical topology graph (the 15,957-zone access basis), because the geometry graph realises only about 272 of roughly 680 category pairs and cannot carry the coupling contract.

Thin context cells are reported with their exact observed count and spread, with the wider parent-type band carrying the recommendation: a transparent dial-stability convention, not a sample-size correction.

4. Positioning

The kit extends the shape-grammar tradition by offering a measured generative vocabulary that is sampled rather than an author-specified rule set that is enumerated;⁹² and it differs from learned floor-plan generators (for example the RPLAN/Graph2Plan and House-GAN families) by exposing explicit, inspectable conditional distributions instead of design knowledge locked inside network weights.⁹³ Descriptive throughout; CANDIDATE until operator MR-13.