EulerKnickung

EulerKnickung is a theoretical concept in geometry and origami theory describing a specific class of crease patterns on a planar sheet. The term combines the surname Euler with the German knickung, meaning crease or bend, and is used to denote patterns that enable efficient, continuous folding sequences.

Definition and core idea. Let P be a simply connected polygonal region in the plane, partitioned by straight creases into polygonal faces. The crease graph C has vertices at crease intersections and edges along the creases. A pattern is called an EulerKnickung if the dual graph C* is Eulerian, meaning it is connected and every vertex has even degree. Under this condition there exists a folding sequence that traverses every crease exactly once in a single continuous path, assuming ideal hinge behavior and no material thickness issues.

Construction and methods. Common construction approaches start from a polygonal subdivision and adjust the crease network to balance vertex degrees in the dual graph. Techniques include adding or removing auxiliary creases and pairing adjacent creases to preserve parity. A simple illustrative case is a square subdivided by a vertical and a horizontal crease through the center, forming four faces. The dual graph of this pattern is connected and all vertices have even degree, yielding an EulerKnickung configuration in the idealized setting.

Properties and limitations. EulerKnickung patterns are associated with parity conditions that guarantee a traversable folding path in theory. They emphasize non-overlapping, non-crossing traversal properties and are sensitive to material thickness and boundary constraints in practical origami. Existence and realization in real media depend on physical factors such as crease sharpness, hinge flexibility, and allowable intersections of creases.

See also. Origami mathematics, Euler characteristic, graph theory, crease pattern.

References. Conceptual and theoretical in nature; used as a term in discussions of foldable graph parity and pattern design.