Print&Fold: Printing and Folding Shape-accurate 3D Models

Abstract: This paper introduces Print&Fold, a tool to allow FDM 3D printing of complex models with less time and material while preserving shape accuracy. Key to this work is a folding algorithm that planarizes foldable faces internal to the 3D model. While folding techniques typically discretize a target model's surface, thereby fabricating low fidelity counterparts, our method preserves the surface features in the physical print. Our design tool allows users to unfold 3D models to be FDM-printed flat before manually folding these into their target shapes. We showcase a variety of applications and evaluate the material and time savings across a range of 3D models.

• The paper introduces Print&Fold, a computational pipeline that wraps mesh panels around a maximally inscribed rectangular hollow core, preserving the original surface geometry while enabling adhesive-free assembly with printed hinges and connectors.
• Evaluation on nine PLA models shows complex objects achieved average savings of 41% in mass and 19% in printing time, with savings increasing as model size grows; fold-angle error remained below 1° and hinges survived 100 fatigue cycles.
• The method is best suited to larger, box-like complex models with planar faces, while thin or highly concave shapes and small convex polyhedra provide limited savings or increased print times because of connector and chamfer overhead.

Print&Fold is a computational design tool that converts 3D models into flat, FDM-printable nets which are then manually folded into shape-accurate 3D objects (2608.13279). Its central contribution is an inscribed-box folding algorithm: rather than unfolding or discretizing a model's surface, the system identifies the maximum-volume rectangular prism inscribable within the mesh, carves it out as a hollow core, and unfolds the remaining material into six panels around this core. Because the folded panels are rigid sub-regions of the original mesh—rotated, not resampled—the printed surface is geometrically identical to the input at printer resolution, distinguishing the approach from prior folding methods that approximate surfaces with low-polygon discretizations.

Motivation and positioning

The work targets the tension in rapid prototyping between speed, material cost, and shape fidelity. Prior approaches—wireframe printing (WirePrint), laser-cut substitution (Platener), Lego-brick hybridization (faBrickation), and self-folding shape-memory techniques such as Thermorph and Inkjet 4D Print—achieve savings largely by degrading fidelity. Among prior systems, only Scrappy preserves full resolution with FDM alone, but it requires manual intervention during printing to insert scrap material. Print&Fold requires no mid-print intervention, uses only PLA on commodity single-nozzle printers, and additionally produces reversibly foldable objects suitable for flat-packed transport or insertion of electronics into the hollow interior.

Technical pipeline

For complex meshes, the pipeline proceeds in four stages. First, the tool detects the largest planar face of the input (a hard requirement for overlap-free folding, since unfolded faces must become co-planar on the buildplate) and aligns the model to it. It then computes the maximum-volume axis-aligned rectangular prism $\mathcal{B}$ contained in the mesh, maximizing the volume efficiency ratio $\eta = V_{\mathcal{B}}/V_{\mathcal{M}}$. Because exact computation relates to the open Convex Skull Problem, the authors adopt a two-phase heuristic: sampling $k{=}64$ horizontal cross-sections, inscribing the largest rectangle in each, and refining the best candidate's six face positions via coordinate-wise search. Second, triangles are partitioned into six panels using the box's outward face planes augmented with eight diagonal bisector planes, yielding contiguous, non-overlapping regions with clean seams. Third, each panel is planarized by rotating about its hinge edge—the intersection of its face plane with the mesh—with straddling triangles clipped by linear interpolation. Fourth, export thickens panels to a default shell thickness of 0.8 mm, generates living hinges (0.2 mm wide, 0.4 mm thick) printed monolithically with the shell, chamfers edges for correct dihedral angles, and places peg-and-hole connectors (5×2.5 mm pegs with 45° chamfers; holes toleranced +0.2 mm) so models assemble without adhesives.

For convex polyhedra, where spanning-tree unfolding already yields volumetrically optimal hollows, the tool integrates a modified spanning-tree approach: Quickhull checks convexity, a BFS spanning tree rooted at the largest face orders unfold rotations composed in the cumulative world frame, and panels are thickened to 3 mm with hinges spanning 95% of edge lengths. Leaf-face connectors bound accumulated fold-angle error.

Evaluation

The authors evaluated nine models printed in PLA on Bambu A1/X1C printers at 0.2 mm layer height, holding wall, top, and bottom loop counts constant across conditions for fair comparison.

Model	η	Mass gain	Time gain
Stanford Bunny	0.48	−43%	−24%
Stormtrooper	0.51	−45%	−27%
Glasses case	0.79	−35%	−14%
Lightbulb	0.46	−41%	−9%
Tetrahedron	—	0%	+37%
Cube	—	−17%	+28%
Dodecahedron	—	−28%	+31%
Cuboctahedron	—	−23%	+26%
Icosahedron	—	−22%	+38%

For complex models, average savings were 41% mass and 19% time, exceeding Scrappy's reported averages of 29.4% and 26.4% on different objects. On the Stanford bunny benchmark, Print&Fold achieves 43%/24% savings versus Pop-up print's reported 5%/25%. Savings derive primarily from infill elimination within the hollow core and secondarily from reduced support material due to flat printing. Scaling experiments confirm that savings grow with model size—as expected, since the hollow scales with length cubed while shells scale with length squared: the bunny's mass savings rise from 38% to 43% across 60–120 mm sizes.

A notable negative result concerns convex polyhedra: despite averaging 18% mass savings, printing time increased by 32% on average, because chamfered edges and connectors print up to 25× slower than infill under Bambu defaults. The tetrahedron showed no mass benefit at all. This indicates the method is best suited to larger, box-like complex models rather than small polyhedra.

Shape accuracy is architecturally exempt from discretization error: where Thermorph approximates the bunny with 26 faces and Inkjet 4D Print with 635, Print&Fold modifies no surface geometry. Residual error arises only from fold-angle error at hinges, bounded below 1° by leaf-face connectors. Repeatability tests showed 20 fold/unfold cycles without connector degradation and 100 hinge fatigue cycles to 180° without tearing.

Limitations and open questions

Savings scale directly with $\eta$: thin-shelled or strongly concave models yield modest gains, illustrated by the Benchy ($\eta = 0.08$). The requirement of at least one planar face constrains admissible inputs, though many practical models satisfy it. Replacing the rectangular prism with a maximum inscribed convex polyhedron would raise $\eta$ but depends on progress on the open Convex Skull Problem. For polyhedra, manual fold count grows with edge number, and the time penalty from slow-printed chamfers remains unresolved. Finally, flattening onto a single printbed limits achievable object size relative to conventional printing; multi-sheet assemblies with automatic seam placement could address this, though partitioning forfeits the kinematic constraint benefits of a single sheet. Mechanical rigidity is demonstrated qualitatively but not established for load-bearing use.

Conclusion

Print&Fold demonstrates that FDM printing of fully shape-accurate objects with substantial material and time savings is achievable on inexpensive desktop hardware by unfolding around a maximally inscribed hollow core rather than approximating surfaces. The strong results on complex models contrast with the unfavorable time economics on small convex polyhedra, delineating clearly where the method applies.