Lang R.J. Origami and Geometric Constructions

Robert J. Lang, 2003. - 55 pages.Origami is the art of folding uncut sheets of paper into interesting and beautiful shapes. Within this text, the author presents a variety of techniques for origami geometric constructions. The field is rich and varied, with surprising connections to other branches of mathematics.Compass-and-straightedge geometric constructions are familiar to most students from highschool
geometry. Nowadays, they are viewed by most as a quaint curiosity of no more than
academic interest. To the ancient Greeks and Egyptians, however, geometric constructions were useful tools, and for some, everyday tools, used for construction and surveying, among other activities.
The classical rules of compass-and-straightedge allow a single compass to strike arcs and
transfer distances, and a single unmarked straightedge to draw straight lines; the two may not be used in combination (for example, holding the compass against the straightedge to effectively mark the latter). However, there are many variations on the general theme of geometric constructions that include use of marked rules and tools other than compasses for the
construction of geometric figures.
One of the more interesting variations is the use of a folded sheet of paper for geometric
construction. Like compass-and-straightedge constructions, folded-paper constructions are both
academically interesting and practically useful — particularly within origami, the art of folding
uncut sheets of paper into interesting and beautiful shapes. Modern origami design has shown
that it is possible to fold shapes of unbelievable complexity, realism, and beauty from a single
uncut square. Origami figures posses an aesthetic beauty that appeals to both the mathematician and the layman. Part of their appeal is the simplicity of the concept: from the simplest of beginnings springs an object of depth, subtlety, and complexity that often can be constructed by a precisely defined sequence of folding steps. However, many origami designs—even quite simple ones — require that one create the initial folds at particular locations on the square: dividing it into thirds or twelfths, for example. While one could always measure and mark these points, there is an aesthetic appeal to creating these key points, known as reference points, purely by folding.