Abstract

Symmetry: Culture and Science
Volume 31, Number 4, pages 403-416 (2020)
https://doi.org/10.26830/symmetry_2020_4_403

THE FIFTH FOLD: COMPLEX SYMMETRIES IN KRESLING–ORIGAMI PATTERNS

Biruta Kresling^*

^* Experimental Design and Bionics, 170 Rue Saint-Charles, 75015 Paris, France
E-mail: kresling.b@outlook.fr

Abstract: The present study focuses on particular symmetry properties of a twist buckling pattern in thin-walled cylinders, named 'Kresling-pattern' (Stewart, 1999; Kresling, 2008 and 2012). The pattern is formed by wrapping a sheet of paper around two mandrels, coaxial with a gap between them. On twisting the mandrels in opposite directions about their common axis, a regular pattern of folds is generated and the mandrels are drawn closer together. The cylindrical wall collapses to a diaphragm. There is neither stretching nor warping, only twist by folding. The pattern is able to unfold. Five folds meet at vertices, the fifth fold lays on a generatrix of the initial cylinder and stabilizes the polygonal borders of the diaphragm. Parallel lines of transverse shear, of identical length, help to define the geometries of:

• foldable cylinders and polygonal prisms;
• nearly stress-free foldable bellows;
• a cylindrical Miura-ori;
• the foldable 'skeleton' of hyperboloids of revolution;
• their portions, individually foldable saddle-shaped tesselations.

All five solutions had been held impossible to be realized from developable surfaces, by means of straight folds. This presentation shows the contrary.

Keywords: origami engineering; vertex-based design; diaphragm; chirality and antichirality; twist buckling; cyclic, radial and translational symmetries.

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