The art of paper folding, known as ''origami'', provides the methodology of constructing a geometrical object out of a sheet of paper solely by means of folding by hands. Computational origami studies the mathematical and computational aspects of origami, including geometrical theorem proving and visualization. By the assistance of software tools for modeling, reasoning and verifying properties of origami, we expect to be able to formalize origami with rigor and capability beyond the methods performed by hands.

In this talk I will show the importance of symbolic and algebraic meth- ods in computational origami, that are employed by our computational origami system called ''Eos'' (E-Origami System). I discuss (1) Huzita's axiomatization of origami,(2) application of Gobner bases method and the cylindrical and algebraic decomposition, and (3) the algebraic graph rewriting of abstract origami. Issue (1) is discussed with relation to the algorithmic treatment of origami foldability, issue (2) for origami geometrical theorem proving, and issue (3) for modeling origami fold. On the whole, I would like to emphasize the importance of symbolic and algebraic computations on discrete geometrical objects, and of the separation of the domains of concern between symbolic and numeric computations. It leads to clearer and more abstract formulation of origami theories.