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COrigami: An AI Pipeline for Co-Designing Flat-Foldable Visually Recognisable Origami

Tom Zahavy, Shaobo Hou, Thomas Tumiel, James Doran, Francesco Faccio, Xidong Feng, Alex Havrilla, Igor Khytryi, Chenglei Li, Lisa Schut, Vivek Veeriah, Arijan Abrashi, Michał Kosmulski, Robert J. Lang, Nick Robinson, Brandon Wong, Marcus Chiam, Gloria Fang, Satinder Singh

arXiv:2606.26299v1cs.AI

TL;DR

Designing physical art requires satisfying both strict geometric constraints and subjective aesthetics. COrigami addresses this challenge with an end-to-end neuro-symbolic origami pipeline whose double-tournament evaluator achieves 0.811 classification accuracy, 0.651 average precision, and an F1 score of 0.74.

  • Problem

    Generating creative physical art that satisfies both strict physical viability and subjective human aesthetics remains an open challenge.

  • Method

    COrigami combines AI-driven semantic generation and shaping with algorithmic box-pleating, packing, flat-foldability solving, and autonomous aesthetic evaluation.

  • Results

    The double-tournament evaluator achieves 0.811 classification accuracy, 0.651 average precision, and an F1 score of 0.74.

  • Takeaways & Limitations

    The system generates mathematically sound structural starting points that expert origami artists can further expand and shape.

  • Takeaways & Limitations

    The pipeline focuses on pure origami within established box-pleating paradigms and relies on a limited set of shaping mechanisms.

Abstract

from arXiv · show

While generative AI has achieved remarkable success in solving problems with verifiable solutions, generating physical art that satisfies both strict geometric constraints and subjective visual aesthetics remains a challenge. This paper presents an approach to tackle these difficulties in the domain of computational origami, a mathematically rigid environment that grounds artistic design within the equations of flat foldability. We present COrigami, an end-to-end AI-driven pipeline that assists the design cycle by generating crease patterns from natural language. Our pipeline involves generating a semantic stick figure, computing a base packing, solving for a flat-foldable crease pattern, shaping the flat-folded crease pattern, and refining the generated model using reinforcement learning driven by an autonomous aesthetic evaluation loop. Our system acts as a highly effective collaborative assistant, generating structural starting points that human artists can further expand and shape. By integrating algorithmic optimisation with autonomous aesthetic critique, this work demonstrates how AI systems can satisfy multi-objective physical constraints to enable reliable, mathematically grounded co-creativity.

1. Introduction

COrigami addresses the challenge of generating origami that is both physically valid and aesthetically recognisable. Its neuro-symbolic pipeline combines neural semantic generation and shaping with an algorithmic foldability core on a discrete box-pleated grid.

  • Generating creative physical art requires satisfying strict physical viability alongside subjective human aesthetics.
  • Automated origami design is computationally hard because flat-foldability and valid mountain-valley assignments are provably intractable.
  • 60%: Directly fine-tuning frontier models for end-to-end crease-pattern generation leaves strict flat-foldability plateauing near 60%.
  • The project’s foundational dataset contains approximately 100 visually recognisable traditional origami models, reflecting a severe scarcity of fully fleshed-out crease patterns.
  • COrigami generates crease patterns on a discrete box-pleated grid, using Gemini and reinforcement learning for semantic generation and shaping while custom algorithms provide the structural core.

2. Background

This section defines crease patterns as geometric blueprints for origami and explains the local and global conditions required for flat foldability. It also traces origami design from traditional bases through computational methods and discrete box pleating.

  • Crease pattern: A crease pattern is a geometric blueprint specifying fold positions and orientations, representable computationally as SVG code.It contains outward-pointing mountain folds and inward-pointing valley folds.
  • Crease pattern: Within the framework, crease patterns comprise creases with endpoints, mountain-or-valley assignments, fold percentages in [0, 1], and vertices recording incident creases and sector angles.Vertices include crease endpoints and intersections.
  • Flat foldability: Flat foldability requires pressing paper into a plane without tearing or self-intersection, verified through local theorems and global intersection checks.Local foldability uses Kawasaki’s and Maekawa’s theorems, while global foldability concerns continuous self-intersections.
  • Origami design: Traditional origami uses one uncut square sheet, with design historically guided by trial-and-error sequences and later formalized through traditional bases.Traditional bases are pre-configured crease layouts that distribute paper area into a fixed number of termini.
  • Origami design: Computational origami emerged as detailed anatomies exceeded classical bases, but continuous circle-packing layouts produced irrational folding angles difficult to execute by hand.Constraining designs to finite angle sets alone was insufficient, according to the passage.
  • Origami design: Discrete box pleating restricts folds to an orthogonal integer grid and 45° diagonals, ensuring rational reproducible angles and a finite set of constructed creases.The grid framework is presented as a response to barriers in continuous computational layouts.

3. Methods

COrigami is an end-to-end neuro-symbolic pipeline that converts natural-language prompts into semantic stick figures, packed and tiled crease patterns, and flat-foldable designs through automated geometric algorithms. Its methods combine constrained AI generation, backtracking search, deterministic crease assignment, and geometric folding simulation.

  • Pipeline automation: The pipeline introduces the first fully automated implementation and evaluation of contiguous box-pleated base packing and flat-foldability solving.Previously, expert human designers manually performed these packing and solving steps.
  • Semantic stick figure: A Gemini-based workflow converts a user prompt into a semantic stick figure that specifies design topology and guides subsequent shaping.Each stick carries a label and spatial parameters, while leaf nodes represent flaps and internal edges represent rivers.
  • Packing and tiling: The packing stage maps the stick figure onto a square integer grid by solving rectangle packing and tiling under graph-adjacency constraints.An iterative backtracking search places rivers and flaps into pockets, while flap expansion eliminates residual gaps to produce a perfect tiling.
  • Crease-pattern solving: The solver constructs uniformly oriented pleats within ridge-partitioned regions and assigns mountain/valley orientations through interleaving and deterministic ridge anchoring.The resulting packing layout defines hinge geometry and minimizes the grid size required for the stick figure.
  • Combinatorial hinge assignment: A priority-driven greedy search assigns remaining hinges as interleaved or symmetric folds, ranking states by local flat-foldability scores and allowing pleat reassignment.Localized partitioning, heuristic guidance, and pruning support convergence to a globally valid crease pattern.
  • Folding simulation: To avoid strain errors associated with physics-based simulation in automated pipelines, the method develops a purely geometric folding simulator.Existing tools generally emphasize either 2D foldability analysis or dynamic physics-based simulation.

4. Experiments

Experiments show that unconstrained raw crease-pattern generation plateaus near 60% strict flat-foldability, while the staged pipeline filters candidates through packing, solving, shaping, and VLM-based selection. The VLM performs best with temperature-scaled best-of-N sampling and structured rubric prompts, and RL produces topologically consistent but morphologically different shapes.

  • Unconstrained generation: Directly fine-tuning a language model to generate raw SVG crease patterns reaches a hard ceiling near 60% strict mathematical flat-foldability.Structural syntax validity improves during training, but flat-foldability plateaus.
  • VLM evaluation: Temperature T=1.0 with best-of-N sampling at N=4 achieves 0.766 classification accuracy and 0.689 F1 score for VLM origami evaluation.Gemini Flash outperforms the Pro model on this spatial reasoning and structural evaluation task.
  • VLM evaluation: The structured Rubrics prompt vastly outperforms Rubrics, V0, and simpler zero-shot scoring or binary classification prompts.The prompt explicitly verifies appendage counts, topology, proportionality, and differentiation before scoring, and was refined through manual inspection with origami designers.
  • Candidate generation: The pipeline retains candidates through sequential stages with 20.2% valid semantic stick-figure generation, 55.3% packing feasibility, 79.2% flat-foldability solving, and 92.0% algorithmic 3D shaping pass rates.These rates are measured from 560,000 initial tree candidates.
  • Reinforcement learning: RL operates on the top-1000 structurally validated base crease patterns and produces shaped models that differ from algorithmic sources while maintaining the same topology.The RL phase expands shaping with structural narrowing and more flexible simple folding, followed by another tournament.

5. Discussion

COrigami frames computational creativity as a problem requiring AI to balance subjective aesthetics with rigid physical constraints through an end-to-end neuro-symbolic pipeline. The discussion identifies limited creative scope, unresolved component-level improvements, and scalability barriers as key challenges.

  • Contributions: COrigami uses an end-to-end neuro-symbolic pipeline to address the joint demands of subjective human aesthetics and rigid physical constraints.The framework is presented as a response to standard generative AI’s limitations in multi-step spatial reasoning.
  • Limitations: The current system is limited to pure origami within established box-pleating paradigms and a limited set of shaping methods.Future progress requires inventing novel folding methods rather than only optimizing existing ones.
  • Limitations: Substantial room for improvement remains across the pipeline’s individual components despite the establishment of a fully automated, end-to-end system.The authors describe the current pipeline as a foundation for future research.
  • Computational Complexity: Flat-foldability testing and valid mountain-valley assignment for general crease patterns are strictly NP-complete, including on restricted box-pleating grids.The stated intractability arises from cascading parity constraints and exponential branching factors.
  • Computational Complexity: Greedy algorithms fail to scale to complex topologies because semantic-tree structural density creates efficiency bottlenecks and higher failure rates in densely constrained designs.The discussion calls for advanced machine learning and robust exploration strategies to handle the most computationally demanding designs.

A. Related Work · Axiomatic Foundations and Local Flat-Foldability · Global Flat-Foldability and Computational Complexity

The related work traces origami computation from local geometric foldability axioms to global flat-foldability analysis and advanced neuro-symbolic AI systems for end-to-end design. It emphasizes the transition from continuous geometric verification toward discrete generative optimization and multimodal spatial reasoning.

  • A. Related Work: Origami computation has evolved over four decades from formal local geometric axioms toward neuro-symbolic AI systems for end-to-end design.The reviewed progression also includes a shift from continuous to discrete generative optimization.
  • A. Related Work: The related work reviews foundational foldability theorems, discrete generative optimization, and multimodal spatial reasoning.These themes define the section’s scope across mathematical foundations and contemporary AI methods.
  • Axiomatic Foundations and Local Flat-Foldability: Kawasaki’s Theorem states that an isolated vertex folds flat if and only if its alternating incident-sector-angle sum equals 180 degrees.This theorem addresses continuous angular constraints at a vertex.
  • Axiomatic Foundations and Local Flat-Foldability: Maekawa’s Theorem governs Mountain and Valley fold assignments at a flat-foldable vertex by requiring M−V= ±2.It addresses the combinatorial assignment of fold orientations rather than continuous sector angles.
  • Global Flat-Foldability and Computational Complexity: Global flat-foldability requires ensuring that a crease pattern folds physically without continuous self-intersection, a problem characterized as strictly NP-hard.This differs from local flat-foldability, which the passage describes as trivially verified.
  • Global Flat-Foldability and Computational Complexity: Historically, global verification used a pointwise definition requiring geometric checks over infinite point sets across a continuous manifold.The passage identifies this pointwise approach as intractable and notes a later shift toward a facewise definition.

Generative Design: From Continuous to Discrete Frameworks

Computational origami generation evolved from continuous optimization methods such as TreeMaker and Origamizer toward discrete box-pleating frameworks constrained by orthogonal integer grids and 45-degree diagonals. COrigami adopts an integrated orthogonal-grid backtracking packer because prior automated and hybrid workflows struggled with complex topologies, Pythagorean stretches, and contiguous tiling.

  • Continuous frameworks: TreeMaker formalized crease-pattern generation as circle-packing optimization, while Origamizer mapped arbitrary 3D polyhedral meshes to flat-foldable crease patterns.These approaches used continuous spatial optimization to minimize square paper dimensions.
  • Discrete frameworks: Box pleating restricts structural creases to orthogonal integer grids with 45-degree diagonal ridges, but algorithmic design remains NP-hard because of cascading parity constraints.Box Pleating Studio introduced routing mechanisms such as Generalized Offset Pythagorean Stretches to improve packing efficiency on discrete grids.
  • COrigami motivation: Automating TreeMaker and combining it with BP Studio failed on complex topologies, while resulting solutions frequently contained Pythagorean stretches and could not enforce contiguous tiling.Contiguous tiling, meaning elimination of empty space on the sheet, is required for a valid base crease pattern.
  • COrigami approach: COrigami therefore uses an integrated backtracking packer on a strictly orthogonal box-pleating grid, prioritizing success rate and visual recognizability over packing efficiency.The resulting designs are intended to be easier to understand, fold, and verify by hand than efficiency-optimized alternatives.
  • Evaluation caveat: TreeMaker’s fully automated benchmark may underrepresent its efficacy because the system normally relies on human intervention to resolve complex topologies.Critical manual interventions were bypassed to enable programmatic evaluation.
  • Alternative grids: Hex pleating offers an alternative grid with 30° crease increments and rational folding angles, but no computational model has yet been developed for it.The passage attributes this absence to a lack of pr…

Spatial Intelligence and Neuro-Symbolic Pipelines

MLLMs have enabled computational design assistance, but current models struggle with invariant geometric properties, multi-step spatial reasoning, and structural validity in origami tasks.

  • Model limitations: MLLMs introduced potential for computational design assistance, while deep neural networks struggle with invariant geometric properties.This limitation is especially relevant to mathematically constrained computational design.
  • Benchmark evidence: OrigamiSpace and OrigamiBench empirically exposed severe deficits in modern models’ multi-step spatial reasoning.The reported deficits include models such as GPT-4o and Gemini 2.5.
  • Structural validity: The benchmark findings also revealed structural violations, including paper self-intersection.These violations demonstrate failures to preserve valid origami structure.

Origami Software and Editors

Origami software has progressed from tools for digitizing and editing crease patterns to GPU-accelerated interactive simulation. However, current folding engines still idealize paper as zero-thickness, limiting their ability to model physical layer accumulation, while newer AI systems shift artists toward high-level direction.

  • Static pattern editing: ORIPA digitizes crease patterns while enforcing line typologies and exporting designs in .opx or ASCII .cp formats.Created by Jun Mitani in 2005, ORIPA is a Java-based Origami Pattern Editor.
  • Dynamic simulation: The Origami Simulator enables fast, interactive folding evaluation through a highly parallelised, GPU-accelerated WebGL application.It evaluates the entire crease pattern simultaneously by iteratively solving for small geometric displacements rather than calculating sequential rigid folding steps.
  • Physical limitation: Current computational folding engines, including ORIPA and Origami Simulator, assume zero-thickness paper despite physical sheets having finite thickness t>0.Dense box-pleated layouts can accumulate dozens of overlapping layers in a single appendage, causing severe bulking that requires substantial handling.
  • Creative generation: Generative AI systems increasingly address creative generation in structured domains by combining natural-language interfaces with geometric or algorithmic constraints.Examples include LLM- and VLM-based CAD generation and evolutionary systems for organic digital art.
  • Artist collaboration: The framework offloads low-level geometric and topological curation to specialized algorithmic and tokenized learning blocks, allowing artists to focus on high-level collaborative direction.This positions the system as a collaborative creative tool rather than replacing the artist’s directional role.

B. Generating crease patterns in SVG space · C. Stick figure generation

The paper evaluates direct language-model generation of crease patterns, finds persistent flat-foldability limitations, and motivates a discrete box-pleating approach; it then introduces implementation details for stick-figure generation.

  • B. Generating crease patterns in SVG space: 400k synthetic crease patterns, totaling approximately 3.2B tokens, were used to fine-tune Gemini for direct crease-pattern generation.The patterns came from a scalable TreeMaker-based pipeline and were diverse and flat-foldable, but not visually recognizable.
  • B. Generating crease patterns in SVG space: The TreeMaker-based data-generation pipeline produced diverse flat-foldable patterns but could not generate visually recognizable origami designs.This established a baseline for unconstrained generative architectures.
  • B. Generating crease patterns in SVG space: 60%: flat foldability plateaued near this level on the test set, showing that Gemini did not reliably produce fully flat-foldable crease patterns.The passage attributes this ceiling to the model’s difficulty with invariant geometric properties and multi-step spatial reasoning.
  • B. Generating crease patterns in SVG space: The negative results identify strict physical viability, long generation length, and scarce mathematically rigorous training data as core barriers to unconstrained end-to-end generation.Real-world crease patterns traditionally function as abstract structural guidelines rather than exhaustive 3D blueprints.
  • B. Generating crease patterns in SVG space: These limitations motivated a transition to discrete box pleating, restricting axis-parallel creases and hinges to an orthogonal integer grid.The restriction discretises the design space and maps continuous geometric packing to tractable combinations.
  • C. Stick figure generation: The stick-figure-generation subsection provides further implementation details for that stage of the pipeline.The supplied passage does not specify the implementation methods or results.

C.1. Category and Example Generation

The synthetic-data process hierarchically generates broad object categories and then objects within each category. It constrains both stages to real, structurally defined objects that can be represented by simple origami and acyclic, stick-figure-like skeletons.

  • Category Generation: Category generation first produces broad categories for objects representable by acyclic, stick-figure-like skeletons and simple origami designs.The prompt excludes simple solid shapes and entities without defined structure.
  • Object Generation: Object generation then suggests many physical objects per category, requiring tree-like skeletons and simple origami representations.These requirements prevent unstructured entities such as water and clouds and solid shapes such as bricks and cups.
  • Object Generation: The prompts favor real-world objects understandable by children and reject imaginary objects or designs lacking sufficient detail.Good examples have distinct structure, while bad examples include solid shapes, undefined forms, or objects that cannot reach the required detail level.

C.2. Tree similarity score · D. Flat Foldability · E. Packing

The pipeline evaluates folded shapes against simplified stick-figure structures using an invariant tree-similarity score, while testing local and global flat foldability through geometric conditions and finite constraints. Its packing interface relies on a merged structural representation that emphasizes leaf extremities and branching.

  • C.2. Tree similarity score: The tree-similarity score is invariant to scale, rotation, and translation when comparing simulated folded models with target stick-figure configurations.It evaluates how closely the simulated 3D folded origami matches the target structure.
  • C.2. Tree similarity score: Linear chains are merged by collapsing straight degree-two segments into single sticks before similarity evaluation.This simplification maps the stick figure to the packing solver.
  • C.2. Tree similarity score: The merged stick-figure representation focuses 3D-shape evaluation on leaf extremities and branching.It serves as a structural approximation for the target configuration.
  • E. Packing: The merged structural approximation facilitates mapping the similarity evaluation to the packing solver.Packing uses the simplified stick representation produced by merging linear chains.
  • D. Flat Foldability: Kawasaki’s theorem requires alternating sums of consecutive sector angles at each vertex to equal 180°, while Maekawa’s theorem requires |M−V| = 2.These are classical necessary conditions for local foldability.
  • D. Flat Foldability: Because Kawasaki’s and Maekawa’s conditions are necessary but insufficient with mountain–valley assignments, the pipeline uses crimping for a sufficient local test.The crimping algorithm generalises the Big-Little-Big lemma.
  • D. Flat Foldability: Global flat foldability is made computationally viable by replacing continuous pointwise verification with a finite constraint-satisfaction problem over overlapping convex-face pairs.The facewise formulation was introduced by Akitaya et al. (2024) and used in the Python reimplementation.
  • D. Flat Foldability: The facewise formulation determines whether a full crease pattern admits a valid folded state without self-intersection.It operationalizes global flat-foldability verification through overlapping face pairs.

E.1. Grid Size Initialization Heuristic … G.2. Narrowing Templates for the Clip Pattern Algorithm

The pipeline initializes packing with a circle-packing-based grid heuristic, constructs and assigns foldable crease patterns through deterministic and greedy procedures, then shapes models using simple folds and narrowing templates tailored to segment geometry and symmetry.

  • E.1. Grid Size Initialization Heuristic: The grid-size heuristic estimates required area from stick-specific contributions based on flap or river type, connectivity, and the stick-figure tree’s diameter.The heuristic is applied before backtracking packing search; the four longest flaps are modeled as quarter circles.
  • F. Solving: The solving stage provides packing creases, assigns pleats and ridges deterministically, and uses a greedy algorithm to assign hinges until global flat foldability is achieved.The hinge-assignment step is combinatorial, but the paper reports the greedy decision process as remarkably efficient.
  • F.1.1. Pleat construction: Pleats are generated by filtering a dense orthogonal grid into candidate segments retained by perpendicularity or X-shape structural conditions.The grid is mapped across the sheet, split at hinge and ridge intersections, and filtered before border-intersecting segments are handled.
  • F.2. Interleaving Assignment: Initial pleat orientations use connected paths sorted by distance from the origin, with BFS assigning alternating Mountain and Valley orientations to adjacent paths.Alternation applies to adjacent paths separated by a single grid unit.
  • F.3. Ridge Construction and Assignment: Ridge orientations are anchored at constrained Y-shape vertices or borders, then propagated using consistency rules for X-shape, parallel, and perpendicular intersections.X-shape and perpendicular ridge intersections share orientations, while adjacent parallel ridges use opposite orientations.
  • G.1. Orchestrating simple fold: The shaping algorithm converts a generated stick figure into simple folds and applies them in breadth-first-search order from the stick figure’s root.The folds are applied to the base crease pattern to shape the model into 3D in a way resembling the stick figure.
  • G.2. Narrowing Templates for the Clip Pattern Algorithm: Narrowing templates combine a base adapter with a narrowing pleat so narrowing is diverted from the rest of the model while preserving flat foldability.The starting edge contains no interior vertices, enabling the narrowing effect without impacting the remaining model.
  • G.2. Narrowing Templates for the Clip Pattern Algorithm: Symmetric templates pinch both edges toward the center, whereas asymmetric templates fold one edge inward; flaps use one adapter, while rivers require adapters at both ends.The configurations support different mass distributions and boundary conditions for terminal appendages versus internal structural bridges.

H. Folding · I. VLM feedback · I.1. VLM prompt baselines

The paper computes folded spatial configurations through graph-based affine transformations and evaluates them with deterministic geometric accuracy and VLM-based visual judgment. The VLM framework supports both standalone scoring and candidate–reference comparison, alongside simpler prompt baselines.

  • H. Folding: H. Folding: A face-adjacency graph is traversed breadth-first to compute each face’s global 4 × 4 affine transformation.Each transformation translates the shared edge to the origin, rotates around the edge axis by the fold angle, reverses the translation, and composes with the parent transformation.
  • H. Folding: H. Folding: Parametric symmetric flap-narrowing templates reduce structural-segment width by dynamically adjusting internal pleat networks.An asymmetric template shifts mass to one side and extends outward until clipped by the layer hull; mirrored adapter blocks reintegrate narrowed segments with the surrounding orthogonal grid.
  • H. Folding: H. Folding: Across 87 complex crease patterns with up to several thousand creases, the deterministic solver yields lower geometric errors than the mass-spring baseline.The comparison uses sorted vertex reconstruction errors on a logarithmic scale against the Origami Simulator model of Ghassaei et al. (2018).
  • I. VLM feedback: I. VLM feedback: The evaluation pipeline has single-model evaluation and comparison-judge modes, using prompts for Gemini 3 Flash.The prompts assess monochrome folded models by form, structure, geometry, and aesthetics while ignoring color, shading, and surface patterns.
  • I. VLM feedback: I. VLM feedback: Comparison judging determines whether a candidate is better or worse than a reference across recognizability, structural accuracy, proportional fidelity, geometric quality, and aesthetics.The reference serves as a comparison baseline rather than necessarily being high quality, and the output uses a 0–10 comparative rubric.
  • I.1. VLM prompt baselines: I.1. VLM prompt baselines: Simpler prompts evaluate recognizability with either a binary <true>/<false> response or a 0–10 quality score.The more detailed baseline additionally requests feature identification, comparison with the canonical target shape, flaw detection, and view-aware handling of flat-foldable samples.
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