Neusis construction The neusis (from Greek νεῦσις from νεύειν neuein "incline towards"; plural: νεύσεις neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians.
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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".
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Rotating a ruler around it, one discovers the distances to the section, from which the minimum and maximum can be discerned. The technique is not applied to the situation, so it is not neusis. The authors use neusis- like, seeing an archetypal similarity to the ancient method.
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The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction).
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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 (διάστημα; Greek for "distance"). A neusis construction might be performed by means of a 'neusis ruler': 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.
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Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube. The construction of the fold guaranteed by Axiom 6 requires "sliding" the paper, or neusis, which is not allowed in classical compass and straightedge constructions. Use of neusis together with a compass and straightedge does allow trisection of an arbitrary angle.
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This is because PAQ, A'AQ and A'AR are three congruent triangles. Aligning the two points on the two lines is another neusis construction as in the solution to doubling the cube.
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These are not code words for future concepts, but refer to ancient concepts then in use. The authors cite Euclid, Elements, Book III, which concerns itself with circles, and maximum and minimum distances from interior points to the circumference. Without admitting to any specific generality they use terms such as “like” or “the analog of.” They are known for innovating the term “neusis-like.” A neusis construction was a method of fitting a given segment between two given curves.
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Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.
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Thus the regular myriagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.
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The concept of constructibility as discussed in this article applies specifically to compass and straightedge construction. More constructions become possible if other tools are allowed. The so-called neusis constructions, for example, make use of a marked ruler. The constructions are a mathematical idealization and are assumed to be done exactly.
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As 22 = 2 × 11, the icosidigon can be constructed by truncating a regular hendecagon. However, the icosidigon is not constructible with a compass and straightedge, since 11 is not a Fermat prime. Consequently, the icosidigon cannot be constructed even with an angle trisector, because 11 is not a Pierpont prime. It can, however, be constructed with the neusis method.
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It can be also constructed using neusis, or by allowing the use of an angle trisector. Tomahawk, at the end 10 s break trisection of the angle according to Archimedes,Retrieved on 14 July 2019. animation, at the end 10 s break Nonagon constructed with 36 meccano equal bars. The nonagon can be constructed with 36 meccano equal bars.
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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. Neusis trisection of an angle θ > 135° to find φ = θ/3, using only the length of the ruler.
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Angles may be trisected via a neusis construction using tools beyond an unmarked straightedge and a compass. The example shows trisection of any angle by a ruler with length equal to the radius of the circle, giving trisected angle . Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
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For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.
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It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23-gon or 29-gon using this tool. Benjamin and Snyder proved that it is possible to construct the regular 11-gon, but did not give a construction.E. Benjamin, C. Snyder, "On the construction of the regular hendecagon by marked ruler and compass", Mathematical Proceedings of the Cambridge Philosophical Society, 156 (3), 409 -- 424 (2014).
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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 (compass and straightedge); # constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas); # constructions that needed yet other means of construction, for example neuseis.
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Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Pandrosion, a female mathematician of ancient Greece, found a numerically-accurate approximate solution using planes in three dimensions, but was heavily criticized by Pappus of Alexandria for not providing a proper mathematical proof. Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
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Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects three lines, and such that the distance between the points of intersection equals the given segment. This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes).
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Thus the regular chiliagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three. Therefore, construction of a chiliagon requires other techniques such as the quadratrix of Hippias, Archimedean spiral, or other auxiliary curves. For example, a 9° angle can first be constructed with compass and straightedge, which can then be quintisected (divided into five equal parts) twice using an auxiliary curve to produce the 0.36° internal angle required.
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Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding. Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is usually treated alongside them. The Greeks knew how to construct regular -gons with (for any integer ) or the product of any two or three of these numbers, but other regular -gons eluded them.
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Constructible Polygon Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three. It is not known if the regular hectogon is neusis constructible. However, a hectogon is constructible using an auxiliary curve such as an Archimedean spiral. A 72° angle is constructible with compass and straightedge, so a possible approach to constructing one side of a hectogon is to construct a 72° angle using compass and straightedge, use an Archimedean spiral to construct a 14.4° angle, and bisect one of the 14.4° angles twice.
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Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them.Conway, John H. and Richard Guy: The Book of Numbers The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some quintics that are not solvable using radicals.A. Baragar, "Constructions using a Twice-Notched Straightedge", The American Mathematical Monthly, 109 (2), 151 -- 164 (2002).
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300px There is a simple neusis construction using a marked ruler for a length which is the cube root of 2 times another length. #Mark a ruler with the given length; this will eventually be GH. #Construct an equilateral triangle ABC with the given length as side. #Extend AB an equal amount again to D. #Extend the line BC forming the line CE. #Extend the line DC forming the line CF #Place the marked ruler so it goes through A and one end, G, of the marked length falls on ray CF and the other end of the marked length, H, falls on ray CE. Thus GH is the given length. Then AG is the given length times .
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Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a Neusis construction, i.e., that uses tools other than an un-marked straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees. This requires three facts from geometry (at right): # Any full set of angles on a straight line add to 180°, # The sum of angles of any triangle is 180°, and, # Any two equal sides of an isosceles triangle will meet the third in the same angle.
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Doubling the cube: PB/PA = cube root of 2 The classical problem of doubling the cube can be solved using origami. This construction is due to Peter Messer: A square of paper is first creased into three equal strips as shown in the diagram. Then the bottom edge is positioned so the corner point P is on the top edge and the crease mark on the edge meets the other crease mark Q. The length PB will then be the cube root of 2 times the length of AP. The edge with the crease mark is considered a marked straightedge, something which is not allowed in compass and straightedge constructions. Using a marked straightedge in this way is called a neusis construction in geometry.
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