Frustration Propagation in Tubular Foldable Mechanisms

Shell mechanisms are patterned surface-like structures with compliant deformation modes that allow them to change shape drastically. Examples include many origami and kirigami tessellations as well as other periodic truss mechanisms. The deployment paths of a shell mechanism are greatly constrained by the inextensibility of the constitutive material locally, and by the compatibility requirements of surface geometry globally. With notable exceptions (e.g., Miura-ori), the deployment of a shell mechanism often couples in-plane stretching and out-of-plane bending. Here, we investigate the repercussions of this kinematic coupling in the presence of geometric confinement, specifically in tubular states. We demonstrate that the confinement in the hoop direction leads to a frustration that propagates axially as if by buckling. We fully characterize this phenomenon in terms of amplitude, wavelength, and mode shape, in the asymptotic regime where the size of the unit cell of the mechanism~$r$ is small compared to the typical radius of curvature~$\rho$. In particular, we conclude that the amplitude and wavelength of the frustration are of order $\sqrt{r/\rho}$ and that the mode shape is an elastica solution. Derivations are carried out for a particular pyramidal truss mechanism. Findings are supported by numerical solutions of the exact kinematics.Comment: 7 figures, added figures and references, corrected typo

Instrumentalization of origami in construction of folded plate structures: Design, research and education

The paper deals with the origami used as an abstract tool to describe and represent the form and the structure of physical objects. In that respect, the potentials of this interdisciplinary technique as a medium of exploration of structural forms was introduced in the semester project done within the course Structural Systems at the Belgrade University, Faculty of Architecture. The technique was used as an interface to gain cognitive experience on spatial transformation and computational design. Throughout the intensive project period divided into three successive stages, the objective was to test method which enabled students to analyze geometrical principles of folding in order to apply these principles in the development of new designs. The generative algorithm inspired by the technique of paper folding assisted form-finding. Resulting shapes were verified by a production of small scale prototype models. The applied method, as a guiding design principle, facilitated formal exploration and augmentation of the design process. At the end of the course, students got cognitive experience on structural forms, while this simple technique delivered richness in terms of design solutions

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

Tiling Euclidean Polygons Mapped From Their Hyperbolic Equivalent

University of Minnesota M.S. thesis.June 2017. Major: Computer Science. Advisor: Douglas Dunham. 1 computer file (PDF); vi, 76 pages.The concept of repeating artistic patterns, for instance spirals, waves, snail shells, tilings etc., have been in existence for centuries now. It was during 1900's that a noted Dutch graphic artist M.C. Escher worked extensively in this world of art which was inspired by mathematics. Escher painstakingly hand-drew such perceptive repeating patterns (which were mostly Euclidean in nature) and his famous hyperbolic patterns: Circle Limit I, II, III and IV which were based on regular tessellations. This research work concentrates on leveraging hyperbolic and Euclidean geometry in art, drawing inspiration from Escher's work. Various Euclidean, non-Euclidean and spherical repeating patterns are special forms of tessellations. At the core of these patterns lies an idea, proposed by Dr. Dunham, that a small congruent sub-pattern, called a motif, which when reflected and rotated will generate the entire pattern. This work focuses on transforming the central polygon in a hyperbolic pattern to obtain its Euclidean counterpart. This counterpart will further tile a planar region to generate a Euclidean tiling. There are various interesting applications that allow the user to draw such repeating patterns programmatically, however none of them show the reverse mapping from a hyperbolic to a Euclidean pattern. We enhance an existing Java application by creating a bridge that connects hyperbolic patterns to their Euclidean equivalents and facilitates the user to work with tilings. The results are expected to show a transformation from hyperbolic to Euclidean patterns followed by tiling of the Euclidean pattern on a planar region