Construction, An Introduction

Introduction

Given an initial set of points in the plane, an image can be "constructed" if it can be produced by a finite number of successive drawing operations using a straightedge and compass. The straightedge allows one to draw a line through any two designated points. The compass allows one to draw a circle of a given radius around a designated point. The radius can be set to some random value, or to the distance between two designated points. Additional points are determined by the intersections of lines and circles.

The ancient greeks developed this concept, and established procedures for constructing many common shapes. For example, given two points one unit distance apart (setting the scale), a unit equilateral triangle can be constructed. Let the segment determined by the two initial points be the base of the triangle, draw a circle of radius 1 about each endpoint, and let the intersection of these two circles be the apex of the triangle.

Actually the circles intersect in two points, one above the base and one below. So you can draw an equilateral triangle pointing up and another one pointing down, both sharing a common base. Draw a line through the two apexes and you have done two things: cut the base in half, and drawn a line perpendicular to the base. These are both useful operations. For instance, you can draw two perpendicular walls one unit apart, then a ceiling, to make a unit square.

But other shapes, such as the heptagon, eluded the best mathematical minds for 2,000 years. And for good reason; most of these shapes cannot be constructed. We need field theory to describe exactly what can be drawn and what cannot.