Mathematical Teardown Diagram Generator

// ═══ PART A: THE CONSERVATION BIJECTION ═══ // A teardown is not "the object broken up." It is a mapping with an inverse. Let A = {a₁ … aₙ} the parts of the ASSEMBLED object (left panel) Let K = {k₁ … kₘ} the slots of the KNOLLED layout (right panel) Define φ : A → K, φ(aᵢ) = the slot occupied by part aᵢ TEARDOWN AXIOM: φ is a BIJECTION (i) injective - no two parts share a slot (ii) surjective - no slot is empty, no part is unplaced ⇒ n = m exactly. COROLLARY 1 (NO ORPHANS): ∀a ∈ A, ∃k ∈ K : φ(a) = k COROLLARY 2 (NO PHANTOMS): ∀k ∈ K, ∃a ∈ A : φ(a) = k COROLLARY 3 (COMPLETE DISASSEMBLY): ∀ aᵢ, aⱼ ∈ A : φ(aᵢ) ≠ φ(aⱼ) ∧ ¬subassembly_intact(aᵢ, aⱼ) // if two parts still touch as a unit on the right panel, the teardown is incomplete VERIFICATION: checksum_left = Σ mass(aᵢ) over A checksum_right = Σ mass(kⱼ) over K assert checksum_left == checksum_right // mass is conserved too, not just count // ═══ PART B: THE TAXONOMIC LATTICE SOLVER ═══ // Given A, find the coordinate assignment c : A → ℝ² that satisfies every // hard constraint while optimizing the layout objective. HARD CONSTRAINTS (non-negotiable): C1 ∀i≠j : rect(c(aᵢ), dims(aᵢ)) ∩ rect(c(aⱼ), dims(aⱼ)) = ∅ // no overlap C2 ∀i : clearance(c(aᵢ), nearest_neighbor) ≥ g_min // min gutter C3 ∀i : rotation(aᵢ) ∈ {0°, 90°} // 90° alignment C4 tier(aᵢ) < tier(aⱼ) ⇒ y(c(aᵢ)) < y(c(aⱼ)) // tier order top→bottom C5 same_class(aᵢ,aⱼ) ⇒ aligned_to_shared_column(aᵢ,aⱼ) // identical parts in ranks C6 aspect_ratio(bounding_box(K)) ∈ [0.9, 1.6] // fits the right panel SOFT OBJECTIVE (maximize): J(c) = 0.34 · row_regularity(c) // rows share a common baseline rail + 0.24 · tier_readability(c) // tier bands visually distinct, gutters clear + 0.18 · size_gradient(c) // within tier: large → small, left → right + 0.12 · symmetry_balance(c) // visual mass balanced about the vertical axis + 0.12 · compactness(c) // minimal wasted ground; generous but not sparse SOLVER: 1. classify every aᵢ into T1…T5 (satisfies C4 by construction) 2. within each tier, sort by (-area, name) (satisfies C5, boosts size_gradient) 3. pack rows left→right at pitch p, wrap at panel width 4. insert tier gutters = 2p between bands (boosts tier_readability) 5. center each row on the vertical axis (boosts symmetry_balance) 6. while J(c) not converged: nudge parts along row rail to even the spacing (boosts row_regularity) re-check C1, C2 after every nudge (never trade a hard constraint for J) // ═══ PART C: WHY THIS BEATS "ORGANIZED CHAOS" ═══ // "Organized chaos" is unfalsifiable - any scatter can claim to satisfy it. // A solved lattice is falsifiable: C1–C6 either hold or they don't, // and J(c) tells you how good the arrangement is. // The result reads as ENGINEERED, because it was. RENDER: {{SUBJECT}}. Solve φ, then solve c. Two bijections, one teardown plate.