A 3D printer, at its most elemental, is a machine that draws. It traces paths, continuous, uninterrupted lines of extruded material, that accumulate, layer by layer, into solid form. This is at once the technology's great strength and, until recently, one of its most underappreciated constraints. The moment a design demands that the printer stop mid-layer, retract, skip across dead space, and restart, the logic breaks. Seams appear. Structural continuity dissolves. The aspirational object and the fabricated one begin to diverge.

For "vase mode" printing, technically known as Thin-Walled 3D Printing, or TW3DP, this constraint becomes architectural. The method is prized precisely because it produces objects in a single, spiraling, uninterrupted bead of material, eliminating the need for dense volumetric infill. But conventional slicers are only capable of sustaining this logic for geometrically well-behaved inputs: manifold, watertight solids whose topological cleanliness keeps the path-planning tractable. Introduce a non-manifold geometry, such as intersecting fins, shared edges, or bifurcating internal channels, and the slicer either fails, or resorts to stop-and-go motions that scar the result.

This is the gap that Jutang Gao and Wes McGee, in a paper presented at AAG 2025, have set out to close. Their contribution is an Eulerian path-planning algorithm that treats non-manifold layer slices not as topological pathologies to be corrected, but as graph problems to be solved.

The Euler Circuit as Fabrication Logic

The appeal to Euler is not incidental. An Eulerian circuit, a path that traverses every edge of a graph exactly once and returns to its starting point, is, for a 3D printer, a statement of perfect material efficiency: no retraction, no dead travel, no seam. The obstacle is that non-manifold layers frequently produce graphs with odd-degree nodes, the precise condition that makes a classical Euler circuit impossible.

Gao and McGee's solution is elegant in its graph-theoretic simplicity: duplicate the offending edges to restore even degree throughout, then apply a depth-first-search-aided Hierholzer algorithm to recover the circuit. What makes the approach physically viable, rather than just theoretically tidy, is the handling of the second pass. Naively, a duplicated edge would send the nozzle retracing its own hot path. Instead, the authors apply an "ascending dimension" strategy: the second traversal is vertically offset by half the slice height, producing a helical spiral that climbs continuously through the layer while maintaining single-bead wall thickness.

"By directly transforming 3D designs into authentic, continuously printed structures, this work broadens the functional potential of TW3DP and enables a higher level of design freedom for architectural prototyping."

Seam management receives equal care. In most slicers, the layer start-and-stop point is arbitrary, or, at best, aligned globally, and the resulting seam reads as an artifact, a scar in the surface logic of the part. Gao and McGee implement a progressive inheritance of start points: the terminal position of each completed layer is projected onto the nearest segment of the next layer's path graph, so that the step-up seam is integrated into the object's geometry rather than imposed upon it.

The Dual Challenges of Connectivity and Continuity

The framework presented by the authors decomposes the problem into two distinct sub-problems: the layer connectivity problem and the layer continuity problem. Connectivity involves the vertical sequencing of deposited material, a task that becomes increasingly complex when the form is not monolithic, such as in bifurcated or porous structures. Continuity, conversely, concerns the horizontal plan for a single layer. By isolating these variables, the algorithm provides a modular solution that can be adapted to various architectural typologies, from simple vertical tubes to complex, multi-branched lattices.

To ensure the robustness of these paths, the algorithm first performs a rigorous cleaning of the geometric input. Slicing non-manifold geometry often introduces trivial flaws: tiny gaps, misaligned curve ends, or vanishingly small segments that increase computational load without adding structural value. The algorithm removes segments below a dimension threshold and consolidates proximate endpoints into shared nodes. This creates a cleaned topological graph that serves as the foundation for the edge-duplication process, ensuring that the resulting toolpath is not only continuous but also physically executable by the robotic arm.

From Theory to Architecture

The difference between algorithm and architecture is proven through fabrication. The case studies escalate in ambition. Triangular tubes with internal reinforcement pockets, Voronoi networks, branching internal structures: each would be an impasse for a standard manifold slicer. A series of six small-scale case studies validated the algorithm's ability to handle complex topologies, including minimal-surface infills that mimic the structural efficiency of biological bone.

The capstone prototype, a 2.4-meter-tall vertical planting structure featuring helical water channels, interior retention dams, and exterior soil pockets, was assembled without a single retraction event. In a conventional pipeline, such a design would demand either geometric simplification or hundreds of stitched sub-paths. Here, it emerges as a single, unbroken thread. Architectural-scale testing also produced a self-standing noise buffer and a perforated façade panel, demonstrating that the algorithm can facilitate the production of functionally rich components that combine acoustic performance with structural stability.

"Looking ahead, the direct production of non-manifold objects will enable lighter, more integrated, and functionally rich components, reducing discretization and assembly steps."

The implications extend well beyond ornamental complexity. When acoustic buffering, water management, and structural stiffening can be encoded directly into the topological DNA of a printed component, rather than approximated through layer-by-layer infill, the relationship between digital intent and physical result changes in kind, not just in degree. Material usage is minimized while structural utility is maximized, a core tenet of sustainable architectural production.

With plans to open-source their Rhino and Grasshopper plugin, Gao and McGee are setting the table for an architectural practice that no longer has to negotiate with its fabrication tools. The non-manifold geometry, once the domain of theoretical modeling, is on its way to becoming a printable one. Future iterations of the work will couple this pipeline with performance simulations for daylight, thermal, and fluid dynamics, allowing designers to optimize for structural performance and fabrication efficiency in a single, unified workflow.