Mathematics - Recreations

A virtual manipulation system for ORIGAMI (paper folding) is described. A piece of paper defined in a computer can be deformed interactively by picking and moving a corner vertex of a paper face on a graphic screen using a mouse. Three kinds of folding operations and a curving operation transform the paper into a three dimensional figure made by flat or curved surfaces. The roundness of the curved surfaces is calculated in real time by minimizing an elastic energy function. The simulated paper has traditional Japanese decoration applied as a texture to give a more realistic appearance. It is rendered with shading in real time.

This paper examines the mathematical modelling of rigid origami, a type of origami where all the panels are rigid and can only rotate about crease lines. The rotating vector model is proposed, which establishes the loop-closure conditions among a group of characteristic vectors. By building up an explicit relationship between the single-vertex origami and the spherical linkage mechanism, the rotating vector model can conveniently and directly describe arbitrary three-dimensional con?gurations and can detect some self-intersection. Quaternion and dual quaternion are then employed to represent the origami model, based on which two numerical methods have been developed. Through examples, it has been shown that the ?rst method can effectively track the entire rigid-folding procedure of an initially ?at or a non-?at pattern with a single vertex or multiple vertices, and thereby provide judgment for its rigid foldability and ?at foldability. Furthermore, its ability to rule out some self-intersecting con?gurations during folding is illustrated in detail, leading to its ability of checking rigid foldability in a more or less suf?cient way. The second method is especially for analysing the multi-vertex origami. It can also effectively track the trajectories of multiple vertices during folding.

Origami Simulation" was developed by Shinya Miyazaki when he was a doctor course student at Nagoya University. He is now a professor at Chukyo University. Origami Simulation was written with C/C++ Language and uses OpenGL library. It enables virtual paper folding (Origami) in a standard PC environment. It was developed on Silicon Graphics (SGI) workstation and translated into Windows.

Origami is the ancient Japanese art of folding paper. Even if origami is mainly an artistic product, it has received a great deal of attention from mathematicians, because of its interesting algebraic and geometrical properties.

Origami mathematics is the subset of mathematics that describes the underlying laws of origami. As part of mathematics, it is part of a deep, consistent (albeit incomplete) logical structure, but its applicability to real-world origami has its limits. Origami mathematics is always at most an approximation of real-world folding, and what one can construct, fold, or compute, using the operations of origami, depends, critically, on what one assumes as the underlying axioms, rules, or operations (depending on your choice of terminology).