Constructions Using a Compass and Twice-Notched Straightedge

Abstract

1. INTRODUCTION. It is impossible to trisect an arbitrary angle. So mathematicians have claimed, with confidence, for more than 160 years. The statement is provocative. To a mathematician, the statement embodies the beauty of algebra and its applications to geometry, hints at Galois theory, and is a rare example of a statement of the nonexistence of a solution. To recreational mathematicians, it is often thought of as a challenge. Every year, mathematicians around the world receive letters from the general population making claims to the contrary. Their solutions fall into two main categories: they either are false or do not adhere to the rules of constructions. In our provocative statement, we often omit the qualifying phrase using only a straightedge and compass. This is a restriction whose popularity is most probably due to the writings of Plato (ca. 427‐347 BC )[ 6]. But according to Pappus (late third century, AD), the ancient Greeks (ancient already to him) classified problems in geometric construction into three types. A problem is called plane if it can be solved using only a straightedge and compass; it is called solid if it can be solved using one or more conic section(s); and it is called linear if the solution requires a more complicated curve. In particular, the ancient Greeks had already found solid solutions to the trisection problem, as well as to the problem of doubling the cube. They suspected that neither problem was plane, a fact that was finally established by Pierre L. Wantzel (1814‐1848) in 1837 (though some have argued that Gauss must have known how to do this soon after writing Disquisitiones in 1798 [4], [5]). I find this classification very intriguing, for it reflects a point of view that is appealing to modern algebraic geometers. The ancient Greeks somehow observed that the simplest problems are those that can be solved using quadratics, and that the next simplest class is the class of problems solvable with quadratics and cubics. Even their terminology (with the exception of “linear”) is very appropriate. The terms “plane” and “solid” are meant to suggest the two-dimensional and three-dimensional natures of the solutions. Thus, a plane construction should involve equations of degree two, and a solid construction should involve equations of degree three. The modern mathematician might be tempted to partition the linear problems (a poor choice of terminology) further into algebraic and transcendental problems, but we would probably do little more. I also find it intriguing that this classification scheme does not obviously include the following trisection algorithm, due to Archimedes (287‐212 BC).