Compass and Straightedge Constructions - Compass and Straightedge Tools

Compass and Straightedge Tools

The "compass" and "straightedge" of compass and straightedge constructions are idealizations of rulers and compasses in the real world:

• The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn using two existing points which give the centre and a point on the circle. The compass collapses when not used for drawing, it cannot be used to copy a length to another place.
• The straightedge is infinitely long, but it has no markings on it and has only one edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing line.

The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient compass. However, by Proposition 2 of Book 1 of Euclid's Elements, no computational power is lost by using such a collapsing compass; there is no need to transfer a distance from one location to another. Although the proposition is correct, its proofs have a long and checkered history.

Each construction must be exact. "Eyeballing" it (essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution.

Each construction must terminate. That is, it must have a finite number of steps, and not be the limit of ever closer approximations.

Stated this way, compass and straightedge constructions appear to be a parlour game, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proven to be exactly correct, and is thus important to both drafting (design by both CAD software and traditional drafting with pencil, paper, straight-edge and compass) and the science of weights and measures, in which exact synthesis from reference bodies or materials is extremely important. One of the chief purposes of Greek mathematics was to find exact constructions for various lengths; for example, the side of a pentagon inscribed in a given circle. The Greeks could not find constructions for three problems:

• Squaring the circle: Drawing a square the same area as a given circle.
• Doubling the cube: Drawing a cube with twice the volume of a given cube.
• Trisecting the angle: Dividing a given angle into three smaller angles all of the same size.

For 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be impossible generally (angles with certain values can be trisected, but not all possible angles).