COrigami: AI Pipeline & Curved-Crease Mechanics

• COrigami is an AI-driven co-design framework that generates flat-foldable, visually recognizable origami from natural language using neuro-symbolic methods and geometric solvers.
• Its staged pipeline—transforming semantic stick figures into crease patterns—addresses challenges like strict flat-foldability, data scarcity, and aesthetic evaluation via reinforcement learning.
• COrigami also denotes research on curved-crease origami, where emergent curvature arises from mechanical frustration and differential crease actuation in elastic sheets.

Searching arXiv for the named COrigami paper and closely related curved-crease origami work. Search query: COrigami origami AI pipeline flat-foldable visually recognizable origami arXiv COrigami denotes two related but distinct usages in origami research. In the contemporary computational-design literature, it refers specifically to “COrigami: An AI Pipeline for Co-Designing Flat-Foldable Visually Recognisable Origami”, a neuro-symbolic, end-to-end system that generates flat-foldable, visually recognizable origami from natural-language prompts (Ertzbischoff et al., 17 Jun 2026). In mechanics and geometry, “COrigami” is also used as shorthand for curved origami or curved-crease origami, where curvature emerges either from explicitly curved fold lines or from elastic incompatibility in straight crease networks [(Dias et al., 2012); (Jules et al., 2021); (DeSimone et al., 2024)]. The former usage concerns AI-assisted co-design under flat-foldability constraints; the latter concerns the mechanics, kinematics, and morphogenesis of folded curved structures. The term therefore spans both a recent AI pipeline and an established research area in origami mechanics, linked by a common emphasis on mathematically constrained shape generation.

1. Terminological scope and research contexts

The AI-system usage of COrigami is explicit and title-level: the 2026 paper presents “an end-to-end AI-driven pipeline that assists the design cycle by generating crease patterns from natural language” (Ertzbischoff et al., 17 Jun 2026). Its stated domain is computational origami, described as “a mathematically rigid environment that grounds artistic design within the equations of flat foldability” (Ertzbischoff et al., 17 Jun 2026). In this setting, COrigami is a co-design framework rather than a geometric subfield.

A separate usage appears in the mechanics literature, where the 2021 paper on accordion-like elastic folding states that one can generate “curved origami, or COrigami,” without explicitly drawing a curved crease (Jules et al., 2021). There, COrigami means a curved deployed shell generated by a straight crease network, with curvature emerging from mechanical frustration rather than from intrinsic geodesic curvature in the crease pattern (Jules et al., 2021). Closely related work on curved crease origami analyzes folding along curved lines as a geometric-mechanical problem governed by developability, crease stiffness, and geometric frustration [(Dias et al., 2012); (DeSimone et al., 2024)].

These usages are not identical. The AI pipeline is concerned with text-conditioned generation of flat-foldable and visually recognizable designs (Ertzbischoff et al., 17 Jun 2026). The mechanics usage concerns curved folds, distributed actuation, and shell morphogenesis [(Dias et al., 2012); (Jules et al., 2021); (DeSimone et al., 2024)]. A plausible implication is that the shared term reflects a broader view of origami as a domain where geometry, mechanics, and algorithmic synthesis are tightly coupled.

2. COrigami as an AI pipeline for flat-foldable recognizable origami

The 2026 COrigami system is described as a neuro-symbolic, end-to-end AI system for co-designing flat-foldable, visually recognizable origami from natural-language prompts (Ertzbischoff et al., 17 Jun 2026). Its motivating claim is that origami is an unusually difficult testbed for generative AI because it combines strict geometric feasibility with subjective aesthetic/semantic fidelity (Ertzbischoff et al., 17 Jun 2026). The paper argues that these goals are often in tension and therefore should not be handled by unconstrained end-to-end generation.

The pipeline is explicitly staged as follows: natural-language prompt → semantic stick figure → discrete base packing on a box-pleated grid → flat-foldable crease pattern → shaped 3D origami → autonomous VLM aesthetic evaluation → RL refinement of shaping decisions using that evaluation as reward (Ertzbischoff et al., 17 Jun 2026). The architecture divides labor between language- and vision-language-model components for semantic conceptualization and aesthetic critique, and deterministic geometric algorithms for packing, tiling, foldability checking, and folding simulation (Ertzbischoff et al., 17 Jun 2026).

The paper states that the system acts as “a highly effective collaborative assistant, generating structural starting points that human artists can further expand and shape” (Ertzbischoff et al., 17 Jun 2026). It also states that human experts remain essential for final physical realization because the simulated designs ignore paper thickness, bulking, and craft-level post-processing (Ertzbischoff et al., 17 Jun 2026). This places COrigami within co-creative design rather than autonomous fabrication.

The system is motivated in part by a negative result: direct fine-tuning of a LLM for raw SVG crease-pattern generation “saturates at only about 60% flat foldability” (Ertzbischoff et al., 17 Jun 2026). The staged pipeline is therefore presented not as a stylistic preference but as a response to brittleness, long-output consistency, scarce data, and the difficulty of automatically defining aesthetic quality (Ertzbischoff et al., 17 Jun 2026).

3. Internal pipeline: semantic abstraction, packing, solving, and shaping

The first stage converts text prompts into a semantic stick figure, defined as a tree-structured, acyclic skeleton encoding the intended object in terms of labeled nodes and edges (Ertzbischoff et al., 17 Jun 2026). Each edge is parameterized by a unique label, length, azimuth angle, and elevation angle (Ertzbischoff et al., 17 Jun 2026). Leaf nodes correspond to flaps, and internal edges correspond to rivers (Ertzbischoff et al., 17 Jun 2026). The system uses constrained prompting to enforce properties such as symmetry, no graph cycles, and anatomically plausible part structure, then uses Gemini as a VLM to inspect rendered views from top, side, front, and isometric angles (Ertzbischoff et al., 17 Jun 2026).

The second stage maps the stick figure to a square sheet through a discrete rectangle-packing and tiling problem on an integer grid (Ertzbischoff et al., 17 Jun 2026). COrigami adopts a box-pleated representation in which axis-parallel creases and hinges lie on an orthogonal integer grid and diagonal ridges are limited to $45^\circ$ (Ertzbischoff et al., 17 Jun 2026). Leaf nodes become rectangles, internal edges become proportional-width paths, and rivers partition the paper into pockets (Ertzbischoff et al., 17 Jun 2026). The packing solver performs iterative backtracking over river placement, flap placement, pocket filling, symmetry constraints, overlap checks, and area-feasibility checks, with the first river placed by exhaustive enumeration and later rivers using a wall-following strategy (Ertzbischoff et al., 17 Jun 2026). After a feasible layout is found, a tiling step expands adjacent flaps to eliminate gaps so that the result is a complete contiguous tiling (Ertzbischoff et al., 17 Jun 2026).

The third stage deterministically converts the packing into a flat-foldable crease pattern (Ertzbischoff et al., 17 Jun 2026). The paper distinguishes local flat foldability from global flat foldability. For local flat foldability it explicitly invokes Kawasaki’s theorem, requiring alternating sums of sector angles to equal $180^\circ$, and Maekawa’s theorem, requiring $|M-V|=2$ at an interior flat-foldable vertex (Ertzbischoff et al., 17 Jun 2026). It also uses a recursive crimping algorithm as a sufficient local test (Ertzbischoff et al., 17 Jun 2026). For global flat foldability, the paper states that the system uses the facewise formulation from FlatFolder, treating the problem as a finite constraint-satisfaction graph over overlapping convex faces, with propagation through precomputed tables and DFS backtracking on remaining connected components (Ertzbischoff et al., 17 Jun 2026).

The deterministic solving stage is described as a sequence of pleat construction, pleat interleaving assignment, ridge construction and assignment, hinge assignment, and reassignment of trapped pleats as needed (Ertzbischoff et al., 17 Jun 2026). Pleats are assigned alternating mountain/valley orientations by BFS over connected paths, while diagonal $45^\circ$ ridges are propagated from anchor points such as Y-shaped vertices and boundary conditions (Ertzbischoff et al., 17 Jun 2026). Hinge assignment is treated as a combinatorial search over standard interleaved $M\!-\!V\!-\!M\!-\!V$ and symmetric $M\!-\!V\!-\!V\!-\!M$ orientations (Ertzbischoff et al., 17 Jun 2026).

The shaping stage occurs after flat-foldable synthesis. First, tree shaping via simple folds reconstructs the 3D skeleton of the stick figure (Ertzbischoff et al., 17 Jun 2026). The paper defines a simple fold by a cut line through two points $p_1,p_2$, assigned mountain or valley orientation (Ertzbischoff et al., 17 Jun 2026). The shaping algorithm traverses the tree in BFS order, computes each child orientation relative to its parent, and applies a physically realizable fold axis lying in the parent paper plane (Ertzbischoff et al., 17 Jun 2026). Second, a clip pattern algorithm handles narrowing, propagating a local 2D reference frame across layers, projecting shaping templates onto folded layers by affine transformation, detecting $Z$-axis flips, and clipping all lines to the hull of each face (Ertzbischoff et al., 17 Jun 2026). The paper distinguishes symmetric narrowing from asymmetric narrowing and notes that short rivers may be too short to narrow with this method (Ertzbischoff et al., 17 Jun 2026).

4. Reinforcement learning and autonomous aesthetic evaluation

A distinctive feature of COrigami is its reinforcement-learning stage guided by a vision-language evaluator (Ertzbischoff et al., 17 Jun 2026). The paper states that the heuristic tree-shaping stage reproduces the skeleton faithfully but not necessarily the best-looking or most semantically faithful final object, so RL is used to explore a broader shaping space (Ertzbischoff et al., 17 Jun 2026).

The policy model is Gemini 2.5 Flash Lite, and it outputs tool-use parameters for all flaps in one step rather than sequentially (Ertzbischoff et al., 17 Jun 2026). The action space includes simple folds, narrowing, and additional shaping variations (Ertzbischoff et al., 17 Jun 2026). The reward combines a hard invalidity penalty, VLM-derived feedback, and a small intrinsic reward for action diversity (Ertzbischoff et al., 17 Jun 2026). The paper gives the intrinsic term as

$r_i = \min\left(\frac{n}{10}, 1\right) * 0.6$

where $n$ is the number of successful tool calls (Ertzbischoff et al., 17 Jun 2026). Invalid trajectories or failures receive $180^\circ$0 (Ertzbischoff et al., 17 Jun 2026). Training uses batch size $180^\circ$1, learning rate $180^\circ$2, a simple policy-gradient algorithm, KL regularization to the base policy, and a KL coefficient decayed from 1 to $180^\circ$3 over 500 steps (Ertzbischoff et al., 17 Jun 2026).

The evaluator is Gemini 3 Flash with temperature 0 and no majority voting in the default deployment (Ertzbischoff et al., 17 Jun 2026). In Single Model Evaluation, the evaluator receives the prompt and seven rendered views of one candidate, then scores appendage count, topology, proportionality, differentiation between body regions, and aesthetic refinement on a $180^\circ$4 to $180^\circ$5 scale later normalized to $180^\circ$6 (Ertzbischoff et al., 17 Jun 2026). In Comparison Judge Mode, it compares reference and candidate images and chooses the better representation, with swapped ordering to reduce proximity bias (Ertzbischoff et al., 17 Jun 2026). The best-performing deployment for curation is described as a “double tournament” style comparison (Ertzbischoff et al., 17 Jun 2026).

This coupling of deterministic validity checking with autonomous aesthetic critique is central to the paper’s claim that AI can support mathematically constrained artistic co-design (Ertzbischoff et al., 17 Jun 2026).

5. Experimental results, benchmarks, and limitations

The paper benchmarks direct end-to-end generation against the staged pipeline. For the direct SVG-space baseline, a Gemini model is trained on 400k synthetic crease patterns totaling approximately 3.2B tokens (Ertzbischoff et al., 17 Jun 2026). The reported outcome is that syntax validity and flat foldability improve during early training, but test-set flat foldability plateaus at about 60% (Ertzbischoff et al., 17 Jun 2026).

For the VLM evaluator benchmark, the paper uses 87 positive examples and 152 negative examples (Ertzbischoff et al., 17 Jun 2026). It reports that Gemini Flash outperforms Gemini Pro, that prompt engineering matters substantially, and that the Double tournament setup is best, achieving 0.811 accuracy, 0.651 AP, and 0.74 $180^\circ$7 (Ertzbischoff et al., 17 Jun 2026).

For the full pipeline, the paper reports that from 560,000 initial tree candidates, COrigami yields 113,276 valid semantic stick figures (Ertzbischoff et al., 17 Jun 2026). It then reports a packing pass rate of 55.3%, solving pass rate of 79.2%, shaping pass rate of 92.0%, 27,869 final curated baseline models, and an overall survival rate of 5.0% (Ertzbischoff et al., 17 Jun 2026). It further states that 17,789 designs were filtered out during final verification, including 7,490 due to low VLM reward and 10,299 due to failing a tree-similarity threshold (Ertzbischoff et al., 17 Jun 2026). Failure is reported to increase with more flaps, more rivers, and denser trees, with packing and solving identified as the main bottlenecks for complex topologies (Ertzbischoff et al., 17 Jun 2026).

The deterministic folding engine is compared with a GPU mass-spring simulator on 87 complex crease patterns (Ertzbischoff et al., 17 Jun 2026). The paper states that the deterministic method achieves significantly lower reconstruction error, “in some cases by up to five orders of magnitude,” with vertex errors as low as $180^\circ$8 compared to a baseline around $180^\circ$9 (Ertzbischoff et al., 17 Jun 2026).

The limitations are stated explicitly. The current shaping repertoire mainly includes simple folds and narrowing, while more advanced methods such as Pythagorean stretches and level shifters are not integrated (Ertzbischoff et al., 17 Jun 2026). The simulator is geometric rather than thickness-aware and does not model layer bulk, fiber compression, paper creep, or tearing limits (Ertzbischoff et al., 17 Jun 2026). Dataset scarcity is also emphasized, including a foundational dataset of about 100 visually recognizable traditional origami models created with collaborating designers (Ertzbischoff et al., 17 Jun 2026). The paper also notes that the final top-10 figure involved human selection from RL samples by visual inspection of high-reward outputs (Ertzbischoff et al., 17 Jun 2026). These statements delimit the present system as a structural blueprint generator rather than a complete replacement for expert folding practice.

6. COrigami as curved origami and curved-crease mechanics

In the mechanics literature, COrigami refers to the generation or analysis of curved folded structures. The 2021 paper on an accordion-like fold network states that its contribution is to show that one can generate curved origami, or COrigami, without explicitly drawing a curved crease (Jules et al., 2021). The studied system consists of a thin elastic sheet with one central longitudinal mountain crease, a sequence of $|M-V|=2$0 equally spaced transverse creases crossing it perpendicularly, alternating mountain/valley directions, and rectangular facets of dimensions $|M-V|=2$1 (Jules et al., 2021). Although the imprinted crease network contains no geodesic curvature, the deployed structure exhibits an effective curvature generated by the deformed central fold (Jules et al., 2021).

The paper defines a discrete curvature

$|M-V|=2$2

where the $|M-V|=2$3 are angular mismatches between successive central-crease segments (Jules et al., 2021). Curvature is attributed to competition between crease stiffness and the kinematics of the crease network, producing mechanical frustration (Jules et al., 2021). It reports the scaling laws

$|M-V|=2$4

for simulations and experiments respectively (Jules et al., 2021). It also identifies three deformation regimes organized primarily by the aspect ratio $|M-V|=2$5: Region I: single-facet deformation for $|M-V|=2$6, Region II: faceting for $|M-V|=2$7, and Region III: buckling for $|M-V|=2$8 (Jules et al., 2021). The corresponding morphogenesis progresses from localized ridge-like bending to triangular faceting and then Euler-like edge buckling (Jules et al., 2021).

A more classical curved-crease formulation appears in the theory of an annular elastic strip folded along a central circular crease (Dias et al., 2012). That paper treats curved crease origami as a mechanics problem for a thin annulus of thickness $|M-V|=2$9, width $45^\circ$0, and crease radius $45^\circ$1, in the regime

$45^\circ$2

(Dias et al., 2012). The crease becomes a space curve with curvature $45^\circ$3, torsion $45^\circ$4, and dihedral angle $45^\circ$5, under the assumption that the sheet is isometrically deformed everywhere except at the crease (Dias et al., 2012). A central geometric relation is

$45^\circ$6

with $45^\circ$7 scaled by $45^\circ$8 (Dias et al., 2012). For an actual fold with $45^\circ$9, one obtains $M\!-\!V\!-\!M\!-\!V$0, leading to the conclusion that a closed circular crease cannot remain planar and therefore buckles out of plane (Dias et al., 2012). The total energy is written as

$M\!-\!V\!-\!M\!-\!V$1

with panel bending energy $M\!-\!V\!-\!M\!-\!V$2 and crease energy $M\!-\!V\!-\!M\!-\!V$3, the latter penalizing deviations from a preferred dihedral angle $M\!-\!V\!-\!M\!-\!V$4 via the parameter $M\!-\!V\!-\!M\!-\!V$5 (Dias et al., 2012). The paper distinguishes stiff folds, for which narrow stiff creases exhibit approximately constant curvature and oscillatory torsion, from softer folds, for which both curvature and torsion oscillate (Dias et al., 2012).

The 2024 paper extends this mechanics viewpoint to 1DoF mechanisms, distributed actuation by spontaneous curvature, and cross-talk between multiple folds (DeSimone et al., 2024). For a curved fold line $M\!-\!V\!-\!M\!-\!V$6 mapped isometrically to a space curve $M\!-\!V\!-\!M\!-\!V$7, the paper uses crease and surface frames to derive compatibility conditions and the relation

$M\!-\!V\!-\!M\!-\!V$8

so that the local fold opening is encoded by a scalar field $M\!-\!V\!-\!M\!-\!V$9 (DeSimone et al., 2024). It states that the folded curvature satisfies a restriction of the form

$M\!-\!V\!-\!V\!-\!M$0

and identifies a 1-DoF mechanism when the rulings on the two sides are collinear in the flat state, implying $M\!-\!V\!-\!V\!-\!M$1 and an ODE for $M\!-\!V\!-\!V\!-\!M$2 (DeSimone et al., 2024). The same paper compares a geometric approach with two mechanics-based actuation models—an active shell model with prescribed target curvature tensor and a 3D active elasticity model with prescribed target strain—and reports that both predict tapering near the crease due to incompatibility (DeSimone et al., 2024). It also argues that synchronous folding of multiple curved folds is energetically favored over sequential folding (DeSimone et al., 2024).

Across these works, COrigami in the mechanics sense denotes a family of problems in which curvature is not merely decorative but arises from developability, crease geometry, fold stiffness, elastic frustration, and multi-fold coupling [(Dias et al., 2012); (Jules et al., 2021); (DeSimone et al., 2024)].

7. Conceptual synthesis and significance

The AI and mechanics meanings of COrigami occupy different technical strata, but they share a common research logic: origami design is constrained not only by intended shape but by rigorous geometric compatibility. In the AI pipeline, this appears as the refusal to let a LLM freely emit long crease-pattern descriptions, replacing unconstrained generation with deterministic solvers for foldability-critical stages (Ertzbischoff et al., 17 Jun 2026). In curved-crease mechanics, it appears as the requirement that curvature, torsion, ruling directions, and developability satisfy nontrivial compatibility conditions [(Dias et al., 2012); (DeSimone et al., 2024)], or that emergent curvature must be understood through mechanical frustration and phase-structured deformation regimes (Jules et al., 2021).

The 2026 COrigami paper situates its contribution in a broader claim about structured creativity: AI works best when semantic understanding is delegated to LLMs, hard constraints to symbolic or geometric solvers, and subjective quality to a multimodal evaluator coupled to reinforcement learning (Ertzbischoff et al., 17 Jun 2026). The curved-crease literature, by contrast, shows that even seemingly simple fold networks can exhibit intricate couplings between local kinematics and global morphology [(Dias et al., 2012); (Jules et al., 2021); (DeSimone et al., 2024)]. This suggests a possible convergence between the two literatures, although such an overview is not claimed in the cited papers: AI-assisted origami systems may eventually need to incorporate richer mechanics if they are to move beyond zero-thickness flat-foldability toward physically faithful curved-shell design.

At present, the most precise usage of COrigami depends on context. In contemporary computational design, it names a specific AI-assisted pipeline for producing mathematically valid, visually meaningful origami blueprints from text (Ertzbischoff et al., 17 Jun 2026). In mechanics, it denotes curved origami structures whose geometry and actuation are governed by developability, stiffness, and elastic incompatibility [(Dias et al., 2012); (Jules et al., 2021); (DeSimone et al., 2024)]. The coexistence of these meanings reflects the breadth of origami research itself: a field spanning symbolic computation, geometric algorithms, differential geometry, elasticity, and co-creative design.