Rigid Miura Tessellations Approximating Curved Surfaces

Definitions and mathematical statement

For a planar polygonal tessellation $T$ of a sheet domain $\Omega\subseteq{\mathbb{R}}^2$, a rigid folded realization, in the usual idealized interpretation, is a continuous map $f:\Omega\to{\mathbb{R}}^3$ whose restriction to each face is a Euclidean isometry onto a planar polygon. Thus faces remain rigid, and adjacent faces remain attached along their common edges. A rigid folding is a continuous family of such realizations. One degree of freedom ordinarily means that the configuration space, after identifying configurations differing by an ambient rigid motion, is locally one-dimensional along the relevant folding branch. These conventions explain the terminology; the supplied record does not specify the admissible sheets, tessellations, folding branches, or collision conditions.

Source question. Given an arbitrary surface with intrinsic curvature, does there exist a Miura-like tessellation of the plane whose folded realization approximates that surface? Can such a tessellation additionally be made rigidly foldable with one degree of freedom?

A possible quantitative structure for the first question is \[ \text{for every }S\in\mathcal S\text{ and every }\varepsilon>0,\quad \text{there exist }T\in\mathcal M\text{ and a folded realization }f \text{ with }\operatorname{err}(f,S)<\varepsilon. \] Here $\mathcal S$, $\mathcal M$, and $\operatorname{err}$ are placeholders for, respectively, the intended class of curved surfaces, the intended class of Miura-like tessellations, and the intended approximation criterion. They are not defined by the source record. The additional mechanism question asks whether the approximating realization can belong to a rigid folding mechanism with one degree of freedom. It is not specified whether the mechanism must start at a planar configuration or whether one-degree-of-freedom behavior must hold globally, locally near the approximating configuration, or away from singular configurations.

Unresolved qualifications. The record does not specify surface regularity, topology, compactness, boundary conditions, an ambient realization, or whether approximation is intrinsic or extrinsic. It does not define "Miura-like," specify whether the sheet is finite or the entire plane, or state whether cuts, holes, self-intersections, or self-contact are permitted. Consequently, the displayed quantifier pattern records a possible precision framework and is not a fully determined conjecture. The supplied dated assessment mentions rigidity obstructions for positive Gaussian curvature but gives no hypotheses or precise obstruction statement; no such theorem is asserted here.

Short English statement

The question asks whether every surface with intrinsic curvature can be approximated by a folded Miura-like planar tessellation, and whether such approximations can also have a rigid folding mechanism with one degree of freedom. The surface class, tessellation class, approximation criterion, and admissible folding motions remain unspecified. The supplied partially solved label and curvature-obstruction discussion are dated curation requiring independent review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30002815$, title "Rigid Miura Tessellations Approximating Curved Surfaces," exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2015; no individual proposer is supplied.
• Supplied report attribution: Discrete Differential Geometry (2015), Oberwolfach Reports, DOI 10.4171/owr/2015/13, pp. 661--729. https://doi.org/10.4171/owr/2015/13. This is also the supplied literature reference labelled "OWR Report 13/2015." The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it does not identify a problem-specific page or speaker.
• Supplied literature reference: Curved display based on origami tessellations. https://doi.org/10.1038/s41378-021-00319-1.
• Literature assessment dated exactly 2026-08-21: the supplied status is partially_solved. The supplied assessment says that origami constructions approximate broad classes of curved surfaces, that universal Miura-like approximation with a rigid one-degree-of-freedom folding mechanism is not established, and that positive Gaussian curvature has known rigidity obstructions. This is supplied dated curation derived from the OWR record, not independently verified here; the status and cited claims require independent verification.
• The registry marks the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This is supplied editorial metadata and does not establish mathematical correctness or resolve the formulation issues above.