Tetsuo Ida
University of Tsukuba, Japan

Title: Analysis of paper fold methods by geometric algebra

Abstract:

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.