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Straightedge and compass construction
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{{Short description|Method of drawing geometric objects}} {{redirect-distinguish|Constructive geometry|Constructive solid geometry}} [[File:Regular Hexagon Inscribed in a Circle 240px.gif|thumb|right|Creating a regular [[hexagon]] with a straightedge and compass]] {{general geometry|Concepts Features}} In [[geometry]], '''straightedge-and-compass construction''' – also known as '''ruler-and-compass construction''', '''Euclidean construction''', or '''classical construction''' – is the construction of lengths, [[angle]]s, and other geometric figures using only an [[Idealization (science philosophy)|idealized]] [[ruler]] and a [[Compass (drawing tool)|compass]]. The idealized ruler, known as a [[straightedge]], is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (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]]. 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 [[Straightedge and compass construction#Markable rulers|Markable rulers]] below.) More formally, the only permissible constructions are those granted by the first three [[Euclidean geometry#Axioms|postulates]] of [[Euclid's Elements|Euclid's ''Elements'']]. It turns out to be the case that every point constructible using straightedge and compass [[Mohr–Mascheroni theorem|may also be constructed using compass alone]], or by [[Poncelet–Steiner theorem|straightedge alone if given a single circle and its center.]] [[Greek mathematics|Ancient Greek mathematicians]] first conceived straightedge-and-compass constructions, and a number of ancient problems in [[Euclidean plane geometry|plane geometry]] impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. [[Gauss]] showed that some [[polygon]]s are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by [[Pierre Wantzel]] in 1837 using [[field theory (mathematics)|field theory]], namely [[Angle trisection|trisecting an arbitrary angle]] and [[doubling the cube|doubling the volume of a cube]] (see [[Straightedge and compass construction#Impossible constructions|§ impossible constructions]]). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems. 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. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of [[square root]]s but of no higher-order roots. ==Straightedge and compass tools== [[File:Régua e compasso.jpg|thumb|300px|Straightedge and compass]] [[File:Cyrkiel RB1.jpg|thumb|110px|right|A compass]] The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world [[Ruler|rulers]] and [[Compass (drawing tool)|compasses]]. *The '''straightedge''' is an infinitely long edge with no markings on it. It can only be used to draw a line segment between two points or to extend an existing line segment. *The '''compass''' can have an arbitrarily large radius with no markings on it (unlike certain real-world compasses). [[Circle|Circles]] and [[Circular arc|circular arcs]] can be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse (i.e. fold after being taken off the page, erasing its 'stored' radius). *Lines and circles constructed have infinite precision and zero width. Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be a less powerful instrument. 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. Although the proposition is correct, its proofs have a long and checkered history.<ref>Godfried Toussaint, "A new look at Euclid’s second proposition," ''The Mathematical Intelligencer'', Vol. 15, No. 3, (1993), pp. 12-24.</ref> In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass. Each construction must be mathematically ''exact''. "Eyeballing" distances (looking at the construction and guessing at its accuracy) or using markings on a ruler are not permitted. Each construction must also ''terminate''. That is, it must have a finite number of steps and not be the limit of ever closer approximations. (If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of [[Sequence|infinite sequences]] converging to a [[Limit (mathematics)|limit]].) 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 ''proven'' to be ''exactly'' correct. ==History== The [[Greek mathematics|ancient Greek mathematicians]] first attempted straightedge-and-compass constructions, and they discovered how to construct [[summation|sums]], [[difference (mathematics)|differences]], [[product (mathematics)|products]], [[ratio]]s, and [[square root]]s of given lengths.<ref name=Bold/>{{rp|p. 1}} They could also construct [[Bisection#Angle bisector|half of a given angle]], a square whose area is twice that of another square, a square having the same area as a given polygon, and [[Regular polygon|regular polygons]] of 3, 4, or 5 sides<ref name=Bold/>{{rp|p. xi}} (or one with twice the number of sides of a given polygon<ref name=Bold/>{{rp|pp. 49–50}}). But they could not construct [[Angle trisection|one third of a given angle]] except in particular cases, or a square with [[Squaring the circle|the same area as a given circle]], or regular polygons with other numbers of sides.<ref name=Bold>Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).</ref>{{rp|p. xi}} Nor could they construct the side of a cube whose volume is [[Doubling the cube|twice the volume of a cube]] with a given side.<ref name=Bold/>{{rp|p. 29}} [[Hippocrates of Chios|Hippocrates]] and [[Menaechmus]] showed that the volume of the cube could be doubled by finding the intersections of [[hyperbola]]s and [[parabola]]s, but these cannot be constructed by straightedge and compass.<ref name=Bold/>{{rp|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 (mathematician)|Nicomedes]] in the second century BCE showed how to use a [[conchoid (mathematics)|conchoid]] to trisect an arbitrary angle;<ref name=Bold/>{{rp|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.<ref name=Bold/>{{rp|pp. 51 ff.}} In 1837 [[Pierre Wantzel]] published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube,<ref name=Wantzel>{{cite journal|last=Wantzel|first=Pierre-Laurent|title=Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas.|journal=[[Journal de Mathématiques Pures et Appliquées]]|date=1837|volume=2|series=1|pages=366–372|url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1837_1_2_A31_0.pdf|access-date=3 March 2014}}</ref> based on the impossibility of constructing [[cube root]]s of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary.<ref name=Kazarinoff /> Then in 1882 [[Ferdinand von Lindemann|Lindemann]] showed that <math>\pi</math> 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.<ref name=Bold/>{{rp|p. 47}} ==The basic constructions== [[File:Basic constructions animation with text.gif|thumb|right|800px|The basic constructions]] All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are: *Creating the [[Line (geometry)|line]] through two points *Creating the [[circle]] that contains one point and has a center at another point *Creating the point at the [[intersection]] of two (non-parallel) lines *Creating the [[Tangent|one point]] or [[Secant line|two points]] in the intersection of a line and a circle (if they intersect) *Creating the one point or two points in the intersection of two circles (if they intersect). For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic [[Abstract Algebra|algebra]], replacing its elements by symbols. Probably [[Carl Friedrich Gauss|Gauss]] first realized this, and used it to prove the impossibility of some constructions; only much later did [[David Hilbert|Hilbert]] find a complete set of [[Hilbert's axioms|axioms for geometry]]. {{clear}} ==Common straightedge-and-compass constructions== The most-used straightedge-and-compass constructions include: * Constructing the [[Bisection#Line segment bisector|perpendicular bisector]] from a segment * Finding the [[midpoint#Construction|midpoint]] of a segment. * Drawing a [[Perpendicular#Construction of the perpendicular|perpendicular line]] from a point to a line. * [[Bisection#Angle bisector|Bisecting an angle]] * [[Reflection (mathematics)#Construction|Mirroring a point in a line]] * [[Tangent lines to circles#Compass and straightedge constructions|Constructing a line through a point tangent to a circle]] * [[circle#Construction through three noncollinear points|Constructing a circle through 3 noncollinear points]] * Drawing a line through a given point parallel to a given line. ==Constructible points== {{main|Constructible number}} {| style="float:right;" !- COLSPAN=3 | Straightedge-and-compass constructions corresponding to algebraic operations |- | [[File:Number construction multiplication.svg|thumb|x150px|''x'' = ''a''·''b'' ([[intercept theorem]])]] | [[File:Number construction division.svg|thumb|x150px|''x'' = ''a''/''b'' ([[intercept theorem]])]] | [[File:SqrtGeom.gif|thumb|x150px|''x''={{sqrt|''a''}} ([[geometric mean theorem#Constructing_a_square_root|geometric mean theorem]])]] |} One can associate an algebra to our geometry using a [[Cartesian coordinate system]] made of two lines, and represent points of our plane by [[Ordered pair|vector]]s. Finally we can write these vectors as complex numbers. Using the equations for lines and circles, one can show that the points at which they intersect lie in a [[Kummer theory|quadratic extension]] of the smallest field ''F'' containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form <math>x + y = \sqrt{k}</math>, where ''x'', ''y'', and ''k'' are in ''F''. Since the field of constructible points is closed under ''square roots'', it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an [[algebraic number]], though not every algebraic number is constructible; for example, {{math|{{radic|2|3}}}} is algebraic but not constructible.<ref name=Wantzel /> ===Constructible angles=== There is a [[bijection]] between the angles that are constructible and the points that are constructible on any constructible circle. 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). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular [[heptadecagon]] (the seventeen-sided [[regular polygon]]) is constructible because :<math>\begin{align} \cos{\left(\frac{2\pi}{17}\right)} &= \,-\frac{1}{16} \,+\, \frac{1}{16} \sqrt{17} \,+\, \frac{1}{16} \sqrt{34 - 2 \sqrt{17}} \\[5mu] &\qquad +\, \frac{1}{8} \sqrt{ 17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}} } \end{align}</math> as discovered by [[Carl Friedrich Gauss|Gauss]].<ref>{{MathWorld | urlname=TrigonometryAnglesPi17 | title=Trigonometry Angles--Pi/17}}</ref> The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in the complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct [[Fermat primes]]. In addition there is a dense set of constructible angles of infinite order. ===Relation to complex arithmetic=== Given a set of points in the [[Euclidean plane]], selecting any one of them to be called '''0''' and another to be called '''1''', together with an arbitrary choice of [[orientation (vector space)|orientation]] allows us to consider the points as a set of [[complex number]]s. 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 (mathematics)|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 (mathematics)|division]], [[complex conjugate]], and [[square root]], which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations. Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and compass from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. 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 :<math>\mathrm{Re}(z)=\frac{z+\bar z}{2}\;</math> :<math>\mathrm{Im}(z)=\frac{z-\bar z}{2i}\;</math> :<math>\left | z \right | = \sqrt{z \bar z}.\;</math> ''Doubling the cube'' and ''trisection of an angle'' (except for special angles such as any ''φ'' such that ''φ''/(2{{pi}})) is a [[rational number]] with [[denominator]] not divisible by 3) require ratios which are the solution to [[cubic equation]]s, while ''squaring the circle'' requires a [[transcendental number|transcendental]] ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists. ==Impossible constructions== The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable.<ref>{{cite book|last1=Stewart|first1=Ian|title=Galois Theory|page=75}}</ref> With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and the Greeks knew how to solve them without the constraint of working only with straightedge and compass.) ===Squaring the circle=== {{Main article|Squaring the circle}} The most famous of these problems, [[squaring the circle]], otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proved impossible, as it involves generating a [[transcendental number]], that is, {{math|{{sqrt|{{pi}}}}}}. Only certain [[algebraic number]]s can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. 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.<ref>{{cite web|url=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html|title=Squaring the circle|publisher=J J O'Connor and E F Robertson|website=www-gap.dcs.st-and.ac.uk/|archive-date=25 June 2017|archive-url=http://web.archive.org/web/20170625130837/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html|url-status=dead}}</ref> A method which comes very close to approximating the "quadrature of the circle" can be achieved using a [[Kepler triangle]]. ===Doubling the cube=== {{Main article|Doubling the cube}} Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its [[minimal polynomial (field theory)|minimal polynomial]] over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass. ===Angle trisection=== {{Main article|Angle trisection}} Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2{{pi}}/5 [[radian]]s (72° = 360°/5) can be trisected, but the angle of {{pi}}/3 [[radian]]s (60[[Degree (angle)|°]]) cannot be trisected.<ref>Instructions for trisecting a [https://commons.wikimedia.org/wiki/File:Trisection_of_a_72_degree_angle_revised.svg 72˚ angle.]</ref> The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a [[neusis]] construction). ===Distance to an ellipse=== The line segment from any point in the plane to the nearest point on a [[circle]] can be constructed, but the segment from any point in the plane to the nearest point on an [[ellipse]] of positive [[eccentricity (mathematics)|eccentricity]] cannot in general be constructed. See <ref>Azad, H., and Laradji, A., "Some impossible constructions in elementary geometry", ''Mathematical Gazette'' 88, November 2004, 548–551.</ref> Note that results proven here are mostly a consequence of the non-constructivity of conics. If the initial conic is considered as a given, then the proof must be reviewed to check if other distinct conic needs to be generated. As an example, constructions for normals of a parabola are known, but they need to use an intersection between circle and the parabola itself. So they are not constructible in the sense that the parabola is not constructible. ===Alhazen's problem=== In 1997, the [[University of Oxford|Oxford]] mathematician [[Peter M. Neumann]] proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient [[Alhazen's problem]] (billiard problem or reflection from a spherical mirror).<ref>{{Citation|last=Neumann|first=Peter M.|author-link=Peter M. Neumann|title=Reflections on Reflection in a Spherical Mirror|journal=[[American Mathematical Monthly]]|volume=105|issue=6|pages=523–528|year=1998|jstor=2589403|mr=1626185|doi=10.1080/00029890.1998.12004920}}</ref><ref>{{Citation|last=Highfield |first=Roger |date=1 April 1997 |title=Don solves the last puzzle left by ancient Greeks |journal=[[Electronic Telegraph]] |volume=676 |url=https://www.telegraph.co.uk/htmlContent.jhtml?html=/archive/1997/04/01/ngre01.html |access-date=2008-09-24 |url-status=dead |archive-url=https://web.archive.org/web/20041123051228/http://www.telegraph.co.uk/htmlContent.jhtml?html=%2Farchive%2F1997%2F04%2F01%2Fngre01.html |archive-date=November 23, 2004 }}</ref> ==Constructing regular polygons== {{Main article|Constructible polygon}} [[File:Pentagon construction.gif|thumb=Pentagon construction small.gif|right|Construction of a regular [[pentagon]]]] Some [[regular polygon]]s (e.g. a [[pentagon]]) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass? [[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 factor]]s of ''n'' are distinct [[Fermat prime]]s. Gauss [[conjecture]]d that this condition was also [[necessary condition|necessary]]; the conjecture was proven by [[Pierre Wantzel]] in 1837.<ref name=Kazarinoff>{{cite book|last=Kazarinoff|first=Nicholas D.|title=Ruler and the Round|year=2003|orig-year=1970|publisher=Dover|location=Mineola, N.Y.|isbn=978-0-486-42515-3|pages=29–30}}</ref> The first few constructible regular polygons have the following numbers of sides: :[[equilateral triangle|3]], [[square|4]], [[pentagon|5]], [[hexagon|6]], [[octagon|8]], [[decagon|10]], [[dodecagon|12]], [[pentadecagon|15]], [[hexadecagon|16]], [[heptadecagon|17]], [[icosagon|20]], [[icositetragon|24]], [[triacontagon|30]], [[triacontadigon|32]], [[Triacontatetragon|34]], [[tetracontagon|40]], [[tetracontaoctagon|48]], 51, [[hexacontagon|60]], [[hexacontatetragon|64]], 68, [[octacontagon|80]], 85, [[enneacontahexagon|96]], 102, [[120-gon|120]], 128, 136, 160, 170, 192, 204, 240, 255, 256, [[257-gon|257]], 272... {{OEIS|A003401}} There are known to be an infinitude of constructible regular polygons with an even number of sides (because if a regular ''n''-gon is constructible, then so is a regular 2''n''-gon and hence a regular 4''n''-gon, 8''n''-gon, etc.). However, there are only 5 known Fermat primes, giving only 31 known constructible regular ''n''-gons with an odd number of sides. ==Constructing a triangle from three given characteristic points or lengths== Sixteen key points of a [[triangle]] are its [[vertex (geometry)|vertices]], the [[Bisection#Bisectors of the sides of a polygon|midpoints of its sides]], the feet of its [[altitude (geometry)|altitudes]], the feet of its [[bisection#Angle bisector|internal angle bisectors]], and its [[circumcenter]], [[centroid]], [[orthocenter]], and [[incenter]]. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points.<ref>Pascal Schreck, Pascal Mathis, Vesna Marinkoviċ, and Predrag Janičiċ. "Wernick's list: A final update", ''Forum Geometricorum'' 16, 2016, pp. 69–80. http://forumgeom.fau.edu/FG2016volume16/FG201610.pdf {{Webarchive|url=https://web.archive.org/web/20160408121045/http://forumgeom.fau.edu/FG2016volume16/FG201610.pdf |date=2016-04-08 }}</ref> Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39 the required triangle exists but is not constructible. Twelve key lengths of a triangle are the three side lengths, the three [[altitude (geometry)|altitudes]], the three [[median (geometry)|medians]], and the three [[bisection#Angle bisector|angle bisectors]]. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined.<ref>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|pp. 201–203}} == Restricted constructions == Various attempts have been made to restrict the allowable tools for constructions under various rules, in order to determine what is still constructible and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can. ===Constructing with only ruler or only compass=== It is possible (according to the [[Mohr–Mascheroni theorem]]) to construct anything with just a compass if it can be constructed with a ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of [[Archimedean property | Archimedes' axiom]],<ref>{{Cite journal | doi=10.1007/BF01222890|title = On strict strong constructibility with a compass alone| journal=Journal of Geometry| volume=38| issue=1–2| pages=12–15|year = 1990|last1 = Avron|first1 = Arnon| s2cid=1537763 |author1-link=Arnon Avron}}</ref> which is not first-order in nature. Examples of compass-only constructions include [[Napoleon's problem]]. It is impossible to take a square root with just a ruler, so some things that cannot be constructed with a ruler can be constructed with a compass; but (by the [[Poncelet–Steiner theorem]]) given a single circle and its center, they can be constructed. ==Extended constructions== The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories.<ref>T.L. Heath, "A History of Greek Mathematics, Volume I"</ref> This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described [[#Constructible points and lengths|above]]) has a planar construction. A complex number that includes also the extraction of cube roots has a solid construction. In the language of fields, a complex number that is planar has degree a power of two, and lies in a [[algebraic extension|field extension]] that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3. ===Solid constructions=== A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x<sup>2</sup> together with the points (0,0) and (1,0), one can construct any complex number that has a solid 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.<ref>P. Hummel, "Solid constructions using ellipses", ''The Pi Mu Epsilon Journal'', '''11'''(8), 429 -- 435 (2003)</ref> 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.<ref>{{citation | last = Knorr | first = Wilbur R. | author-link = Wilbur Knorr | doi = 10.1111/j.1600-0498.1989.tb00848.x | issue = 4 | journal = Centaurus | mr = 1078083 | pages = 257–271 | title = On Archimedes' construction of the regular heptagon | volume = 32 | year = 1989}}</ref> The quadrature of the circle does not have a solid construction. A regular ''n''-gon has a solid construction if and only if ''n''=2<sup>''a''</sup>3<sup>''b''</sup>''m'' where ''a'' and ''b'' are some non-negative integers and ''m'' is a product of zero or more distinct [[Pierpont prime]]s (primes of the form 2<sup>''r''</sup>3<sup>''s''</sup>+1). Therefore, regular ''n''-gon admits a solid, but not planar, construction if and only if ''n'' is in the sequence :[[heptagon|7]], [[enneagon|9]], [[tridecagon|13]], [[tetradecagon|14]], [[octadecagon|18]], [[enneadecagon|19]], [[icosihenagon|21]], 26, 27, 28, 35, 36, 37, 38, 39, [[tetracontadigon|42]], 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... {{OEIS|A051913}} The set of ''n'' for which a regular ''n''-gon has no solid construction is the sequence :[[hendecagon|11]], [[icosidigon|22]], [[icositrigon|23]], 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... {{OEIS|A048136}} Like the question with Fermat primes, it is an open question as to whether there are an infinite number of Pierpont primes. ===Angle trisection=== What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with such a tool.<ref>[[Andrew Gleason|Gleason, Andrew]]: "Angle trisection, the heptagon, and the triskaidecagon", ''Amer. Math. Monthly'' '''95''' (1988), no. 3, 185-194.</ref> On the other hand, every regular n-gon that has a solid construction can be constructed using such a tool. ===Origami=== {{main article|Huzita–Hatori axioms}} The [[Mathematics of paper folding|mathematical theory of origami]] is more powerful than straightedge-and-compass construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, [[origami]] can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems.<ref>{{cite book|last=Row|first=T. Sundara|title=Geometric Exercises in Paper Folding|title-link= Geometric Exercises in Paper Folding |year=1966|publisher=Dover|location=New York}}</ref> ===Markable rulers=== {{main article|Neusis construction}} [[Archimedes]], [[Nicomedes (mathematician)|Nicomedes]] and [[Apollonius of Perga|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 the two given lines, 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 [http://www.cut-the-knot.org/pythagoras/archi.shtml Archimedes' trisection]) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a [[cubic equation|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.<ref>Conway, John H. and Richard Guy: ''[[The Book of Numbers (math book)|The Book of Numbers]]''</ref> 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 [[quintic]]s that are [[Abel–Ruffini theorem|not solvable using radicals]].<ref>A. Baragar, "Constructions using a Twice-Notched Straightedge", ''The American Mathematical Monthly'', '''109''' (2), 151 -- 164 (2002).</ref> 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.<ref>{{cite journal | last1=Benjamin | first1=Elliot | last2=Snyder | first2=C. | title=On the construction of the regular hendecagon by marked ruler and compass | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] | volume=156 | issue=3 | pages=409—424 | date=2014 | doi=10.1017/S0305004113000753}}</ref> It is still open as to whether a regular 25-gon or 31-gon is constructible using this tool. ===Trisect a straight segment=== [[File:Trisectionofstraightedge.gif|thumb|right|Trisection of a straight edge procedure.]] Given a straight line segment called AB, could this be divided in three new equal segments and in many parts required by the use of [[intercept theorem]]. ==Computation of binary digits== In 1998 [[Simon Plouffe]] gave a ruler-and-compass [[algorithm]] that can be used to compute [[binary digit]]s of certain numbers.<ref name="Plouffe1998">{{cite journal|title=The Computation of Certain Numbers Using a Ruler and Compass|author=Simon Plouffe |journal=Journal of Integer Sequences|year=1998|issn=1530-7638|volume=1|page=13 |bibcode=1998JIntS...1...13P |url=http://www.cs.uwaterloo.ca/journals/JIS/compass.html}}</ref> The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits. ==See also== *[[Carlyle circle]] *[[Geometric cryptography]] *[[Geometrography]] *[[List of interactive geometry software]], most of them show straightedge-and-compass constructions *[[Mathematics of paper folding]] *[[Underwood Dudley]], a mathematician who has made a sideline of collecting false straightedge-and-compass proofs. ==References== {{Reflist}} ==External links== *[http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html Regular polygon constructions] by Dr. Math at ''The Math Forum @ Drexel'' *[http://www.cut-the-knot.org/do_you_know/compass.shtml Construction with the Compass Only] at ''[[cut-the-knot]]'' * [http://www.cut-the-knot.org/Curriculum/Geometry/Hippocrates.shtml Angle Trisection by Hippocrates] at ''cut-the-knot'' * {{MathWorld | urlname=AngleTrisection | title=Angle Trisection}} {{Ancient Greek mathematics}} {{Authority control}} [[Category:Straightedge and compass constructions| ]] [[Category:Greek mathematics]]
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