Straightedge and compass construction
The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it.
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.)
"Eyeballing" distances (looking at the construction and guessing at its accuracy) or using markings on a ruler, are not permitted.
(If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of infinite sequences converging to a limit.)
But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or regular polygons with other numbers of sides.[2]: p.
29 Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass.[2]: p.
30 In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle;[2]: p. 37 but these methods also cannot be followed with just straightedge and compass.
No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of n sides to be constructible.[2]: pp.
is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.[2]: p.
Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results.
Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry.
The most-used straightedge-and-compass constructions include: One can associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors.
The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of the points on the unit circle viewed as complex numbers).
Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π).
The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of addition, subtraction, multiplication, division, complex conjugate, and square root, which is easily seen to be a countable dense subset of the plane.
For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ/(2π)) is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio.
The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable.
(The problems themselves, however, are solvable, and the Greeks knew how to solve them without the constraint of working only with straightedge and compass.)
Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity.
[8] The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction).
Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes.
However, there are only 5 known Fermat primes, giving only 31 known constructible regular n-gons with an odd number of sides.
The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution.
A complex number that can be expressed using only the field operations and square roots (as described above) has a planar construction.
Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful.
[16] The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions.
Archimedes gave a neusis construction of the regular heptagon, which was interpreted by medieval Arabic commentators, Bartel Leendert van der Waerden, and others as being based on a solid construction, but this has been disputed, as other interpretations are possible.
Therefore, regular n-gon admits a solid, but not planar, construction if and only if n is in the sequence The set of n for which a regular n-gon has no solid construction is the sequence Like the question with Fermat primes, it is an open question as to whether there are an infinite number of Pierpont primes.
In 1998 Simon Plouffe gave a ruler-and-compass algorithm that can be used to compute binary digits of certain numbers.
[23] The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits.
