Neusis construction

Neusis construction

In geometry, the neusis (νεῦσις; from Ancient Greek νεύειν (neuein) 'incline towards'; plural: νεύσεις, neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians.

Geometric construction

The neusis construction consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P.

Point P is called the pole of the neusis, line l the directrix, or guiding line, and line m the catch line. Length a is called the diastema (Greek: διάστημα, lit. 'distance').

A neusis construction might be performed by means of a marked ruler that is rotatable around the point P (this may be done by putting a pin into the point P and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue eye) indicates the distance a from the origin. The yellow eye is moved along line l, until the blue eye coincides with line m. The position of the line element thus found is shown in the figure as a dark blue bar.

Trisection of an angle

Neusis trisection of an angle θ > 135° to find φ = θ/3, using only the length of the ruler. The radius of the arc is equal to the length of the ruler. For angles θ < 135° the same construction applies, but with P extended beyond AB.

The neusis can be used to trisect angles, in the following ways (referring to the image):

• For an angle smaller than 45°: Mark the straightedge with the radius of the circle, then move its point A on the AO line while keeping B on its edge, until the end of the straightedge touches the circle—this will be at a point “higher” than B on the circle (it will look like the image, except for B and P being inverted). Then angle POA will be equal to 60° minus the third of angle BOA. As an example, if BOA = 27°, then POA will be equal to 60° − (27 ÷ 3)° = 51°.
• For an angle larger than 45° but smaller than 90°: The principle is the same, except this time, A will be on a vertical line extending upwards from O.
• For an angle larger than 90° but smaller than 135°: A is still on a vertical line extending upwards from O. However, let point C be anywhere on the horizontal line left of O, and let point D be anywhere on the horizontal line right of O. Angle POC will now be equal to a third of BOD.
• For an angle larger than 135° but smaller than 180°: This time, the drawing represents the situation accurately, with point D anywhere on the horizontal line right of O, so that θ = ∠DOB. AB is equal to OP = OB.

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 (287–212 BC) and Pappus of Alexandria (290–350 AD) freely used neuseis; Sir Isaac Newton (1642–1726) 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 $\mathbb {Q}$, $\mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K$, 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. (If a regular p-gon is constructible, then $\zeta _{p}=e^{\frac {2\pi i}{p}}$ is constructible, and in these cases p − 1 has a prime factor higher than 5.) 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 (fifth-order polynomials) 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 = 2^r3^s5^t + 1 where t > 0 (all prime numbers that are greater than 11 and equal to one more than a regular number that is divisible by 10).

Waning popularity

T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (ca. 440 BC) 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 (ca. 430 BC), 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:

1. constructions with straight lines and circles only (compass and straightedge);
2. constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas);
3. constructions that needed yet other means of construction, for example neuseis.

In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution. Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria (ca. 325 AD) as "a not inconsiderable error".

See also

• Angle trisection
• Constructible polygon
• Pierpont prime
• Quadratrix
• Steel square
• Tomahawk (geometry)
• Trisectrix

References

• R. Boeker, 'Neusis', in: Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa red. (1894–), Supplement 9 (1962) 415–461.–In German. The most comprehensive survey; however, the author sometimes has rather curious opinions.
• T. L. Heath, A history of Greek Mathematics (2 volumes; Oxford 1921).
• H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum [= The Theory of Conic Sections in Antiquity] (Copenhagen 1886; reprinted Hildesheim 1966).

External links

• MathWorld page
• Angle Trisection by Paper Folding