University of Tsukuba, Japan
Title: Analysis of paper fold methods by geometric algebra
We discuss an application of the geometric algebra (GA) to the computational origami. GA is called for, to extend computational origami system Eos to deal with construction and motion of 3D origamis more flexibly and systematically. The algebra commonly used to link it to geometry is the affine variety defined over the field of reals. The affine variety is derived via the Cartesian coordinate system. Whether it is a polynomial of standard form or of matrix form, the geometric meanings that originally pertain to our geometric statements in natural language are often lost during the translation. Here, we strongly need adequate abstraction layers between the language of our daily use in logic and geometry and the language for the computational purposes.
We first recall Huzita’s origami basic operation set HO, which plays a fundamental role in origami geometry. HO is written in the natural language using the terminology of daily life. Even geometric notions are hidden in the statements. For abstract reasoning about and for computation of origami objects, they are first transcribed to the language of the fragment of the first-order logic and then to the language of the algebra, i.e. the affine variety.
We observe the importance of the geometric algebra as the layer of abstraction between the language of the logic for the origami geometry and the affine variety. Especially, we do recognize the importance when we study 3D objects symbolically and constructively.
Since we intend to use GA to be integrated into the computational system tightly, we start with the formalization of GA in Isabelle/HOL with the help of Mathematica, and then for most computational, i.e. simplification, purposes, we use Mathematica. In this talk We focuss on HO. We show how HO expressed in GA and reason HO algebraically.