Neusis construction
In geometry, the neusis is a geometric construction method that was used in antiquity by Greek mathematicians.
Geometric construction
The neusis construction consists of fitting a straight line element of given length in between two given lines, in such a way that the extension of the line element passes through a given point. That is, one end of the line element has to lie on and the other end on while the line element is "inclined" towards.Point is called the pole of the neusis, line the directrix, or guiding line, and line the catch line. Length is called the diastema.
A neusis construction might be performed by means of a marked ruler that is rotatable around the point . In the figure one end of the ruler is marked with a yellow eye; this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance from the origin. The yellow eye is moved along line, until the blue eye coincides with line.
If we require both lines and to be straight lines, then the construction is called line–line neusis. Line–circle neusis and circle–circle neusis are defined analogously. The line–line neusis gives us precisely the power to solve quadratic and cubic equations while line–circle neusis and circle–circle neusis are strictly more powerful than line-line neusis. Technically, any point generated by either the line–circle neusis or the circle–circle neusis lies in an extension field of the rationals that can be reached by a tower of fields in which each adjacent pair has index either 2, 3, 5, or 6 while the adjacent-pair indices over the tower of the extension field of line–line neusis are either 2 or 3.
Trisection of an angle by line–circle neusis
Starting with two lines and that intersect at angle , let be the point of intersection and let be a second point at. Draw a circle through centered at. Place the ruler at line and mark it at and. Keeping the ruler touching, slide and rotate the ruler so that the mark touches, until mark again touches the circle. Label this point on the circle and let be the point where the ruler touches. The angle equals one-third of .Use of the neusis
Neuseis have been important because they sometimes provide a means to solve geometric problems that are not solvable by means of compass and straightedge alone. Examples are the trisection of any angle in three equal parts, and the doubling of the cube. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis; Isaac Newton followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use.Regular polygons
In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower of fields over,, such that the degree of the extension at each step is no higher than 6. Of all prime-power polygons below the 128-gon, this is enough to show that the regular 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, 89-, 103-, 107-, 113-, 121-, and 127-gons cannot be constructed with neusis. The 3-, 4-, 5-, 6-, 8-, 10-, 12-, 15-, 16-, 17-, 20-, 24-, 30-, 32-, 34-, 40-, 48-, 51-, 60-, 64-, 68-, 80-, 85-, 96-, 102-, 120-, and 128-gons can be constructed with only a straightedge and compass, and the 7-, 9-, 13-, 14-, 18-, 19-, 21-, 26-, 27-, 28-, 35-, 36-, 37-, 38-, 39-, 42-, 52-, 54-, 56-, 57-, 63-, 65-, 70-, 72-, 73-, 74-, 76-, 78-, 81-, 84-, 91-, 95-, 97-, 104-, 105-, 108-, 109-, 111-, 112-, 114-, 117-, 119-, and 126-gons with angle trisection. However, it is not known in general if all quintics have neusis-constructible roots, which is relevant for the 11-, 25-, 31-, 41-, 61-, 101-, and 125-gons. Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible; the 25-, 31-, 41-, 61-, 101-, and 125-gons remain open problems. More generally, the constructibility of all powers of 5 greater than 5 itself by marked ruler and compass is an open problem, along with all primes greater than 11 of the form p = 2r3s5t + 1 where t > 0.Squaring the Circle
Neusis can not square the circle, as all ratios constructible by neusis are algebraic, and so can not construct transcendental ratios like.Waning popularity
, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. The principle to avoid neuseis whenever possible may have been spread by Hippocrates of Chios, who originated from the same island as Oenopides, and who was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him Euclid too shunned neuseis in his very influential textbook, The Elements.The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were:
- constructions with straight lines and circles only ;
- constructions that in addition to this use conic sections ;
- constructions that needed yet other means of construction, for example neuseis.