TRISECTION

“Trisecting an angle” is one of the problems inherited from Greek antiquity. It asks for a series of constructions by straight edge and compass that trisects a given angle. This problem remained unsolved for 2,000 years and was finally shown to be algebraically impossible in the nineteenth century by Wantzel (1836). We saw earlier, in Figure 3.6 (p. 33), that it has been known for over a century that a linkage can trisect an angle. More recently it was established that it is possible to trisect an angle via origami folds; Figure 19.1 illustrates the elegant construction of Abe.

The reader may sense that it is not clear that this construction actually works, nor what are the exact rules of the game. We will attempt to elucidate both these issues.

HUZITA'S AXIOMS AND HATORI'S ADDITION

What is “constructible” by origami folds was greatly clarified by Humiaki Huzita in 1985, who presented a set of six axioms of origami construction (Hull 1996; Huzita and Scimemi 1989; Murakami 1987).

These axioms intend to capture what can be constructed from origami “points” and “lines” via a single fold. A line is a crease in a (finite) piece of paper or the boundary of the paper. A point is an intersection of two lines. Initially, the (traditionally square) paper has lines determined by the boundary edges. Crease the paper in half and you construct two points where the medial crease hits the paper boundary (see Figure 19.2).