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Symmetry issues in collapsible origami. (English) Zbl 1274.00027

Summary: ‘Collapsible Origami’ is the field which invents and investigates folded forms made from flat uncut sheets that can be squeezed into smaller 2D or 3D shapes and be re-expanded, repeatably and reliably, in a single fluid movement. In this paper we consider the effects of a fold-pattern’s symmetry and asymmetry on its collapsibility. Symmetry tends to distribute compressive/tensile forces more evenly in a sheet: for many purposes that is a good thing, but with excess equilibrium there is no clear channel of release for these forces. So, introducing a single small asymmetry often aids in compaction. That is said, folding and unfolding are rarely the SOLE purpose of collapsible origami. Often the sheet also has to unfold into a pretty shape, or compress into a specific shape, or do something interesting as it unfolds, or work using rigid hinged parts like steel (unlike paper which can flex), or compress only partway and resist compression thereafter, or fold along curved lines which have a distinct logic of their own, etc. For each of these varied requirements, pattern and symmetry – but also the breaking of symmetry – are relevant. We illustrate this theme by considering a few novel patterns of our own design, as well as some famous cases.

MSC:

00A66 Mathematics and visual arts
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
51M20 Polyhedra and polytopes; regular figures, division of spaces
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