The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. Stated this way, straightedge-and-compass 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 proved to be exactly correct. That is, it must have a finite number of steps, and not be the limit of ever closer approximations. "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.Įach construction must terminate. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass.Įach construction must be exact.
However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass.Īlthough the proposition is correct, its proofs have a long and checkered history. A 'collapsing compass' would appear to be a less powerful instrument. The compass may or may not collapse when it is not drawing a circle.Īctual compasses do not collapse and modern geometric constructions often use this feature. The arc that is drawn is infinitesimally thin point-width. Circles can only be drawn starting from two given points: the centre and a point on the circle, and aligned to those points with infinite precision. The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it.It can only be used to draw a line segment between two points, with infinite precision to those points, or to extend an existing segment. The line drawn is infinitesimally thin point-width. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers.The "straightedge" and "compass" of straightedge-and-compass constructions are idealizations of rulers and compasses in the real world: A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In spite of existing proofs of impossibility, some persist in trying to solve these problems.
Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. Gauss showed that some polygons are constructible but that most are not. The ancient Greeks developed many constructions, but in some cases were unable to do so. The ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane geometry impose this restriction. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center.
Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass see compass equivalence theorem. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. Straightedge-and-compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.