Geometric construction

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Geometric construction

In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean constructions. Such constructions lay at the heart of the geometric problems of antiquity of circle squaring, cube duplication, and angle trisection. The Greeks were..

Neusis construction

A geometric construction, also called a verging construction, which allows the classical geometric construction rules to be bent in order to permit sliding of a marked ruler. Using a Neusis construction, cube duplication, angle trisection, and construction of the regular heptagon are soluble. The conchoid of Nicomedes can also be used to perform many Neusis constructions (Johnson 1975). Conway and Guy (1996) give Neusis constructions for the 7-, 9-, and 13-gons which are based on angle trisection.

Delian constant

The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).

Triangle squaring

Let be the altitude of a triangle and let be its midpoint. Thenand can be squared by rectangle squaring. The general polygon can be treated by drawing diagonals, squaring the constituent triangles, and then combining the squares together using the Pythagorean theorem.

Cycloid of ceva

The polar curve(1)that can be used for angle trisection. It was devised by Ceva in 1699, who termed it the cycloidum anomalarum (Loomis 1968, p. 29). It has Cartesian equation(2)It has area(3)and arc length(4)(5)(OEIS A138497), with , where , , and are complete elliptic integrals of the first, second, and third, respectively.The arc length function is a slightly complicated expression that can be expressed in closed form in terms of elliptic functions, and the curvature is given by(6)

Steiner construction

A construction done using only a straightedge. The Poncelet-Steiner theorem proves that all constructions possible using a compass and straightedge are possible using a straightedge alone, as long as a fixed circle and its center, two intersecting circles without their centers, or three nonintersecting circles are drawn beforehand. For example, the centers of two intersecting circles can be found using a straightedge alone (Steinhaus 1999, p. 142).

Mascheroni construction

A geometric construction done with a movable compass alone. All constructions possible with a compass and straightedge are possible with a movable compass alone, as was proved by Mascheroni (1797). Mascheroni's results are now known to have been anticipated largely by Mohr (1672).An example of a Mascheroni construction of the midpoint of a line segment specified by two points and illustrated above (Steinhaus 1999, Wells 1991). Without loss of generality, take . 1. Construct circles centered at and passing through and . These are unit circles centered at (0, 0) and (1, 0). 2. Locate , the indicated intersection of circles and , and draw a circle centered on passing through points and . This circle has center (1/2, ) and radius 1. 3. Locate , the indicated intersection of circles and , and draw a circle centered on passing through points and . This circle has center (3/2, ) and radius 1. 4. Locate , the indicated intersection of circles and , and draw a circle..

Simplicity

The number of operations needed to effect a geometric construction as determined in geometrography. If the number of operations of the five geometrographic types are denoted , , , , and , respectively, then the simplicity is and the symbol . It is apparently an unsolved problem to determine if a given geometric construction is of smallest possible simplicity.

Rectangle squaring

Given a rectangle , draw on an extension of . Bisect and call the midpoint . Now draw a semicircle centered at , and construct the extension of which passes through the semicircle at . Then has the same area as . This can be shown as follows:(1)(2)

Circle squaring

Construct a square equal in area to a circle using only a straightedge and compass. This was one of the three geometric problems of antiquity, and was perhaps first attempted by Anaxagoras. It was finally proved to be an impossible problem when pi was proven to be transcendental by Lindemann in 1882.However, approximations to circle squaring are given by constructing lengths close to . Ramanujan (1913-1914), Olds (1963), Gardner (1966, pp. 92-93), and (Bold 1982, p. 45) give geometric constructions for . Dixon (1991) gives constructions for and (Kochanski's approximation).While the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky Space (Gray 1989).

Apollonius' problem

Given three objects, each of which may be a point, line, or circle, draw a circle that is tangent to each. There are a total of ten cases. The two easiest involve three points or three lines, and the hardest involves three circles. Euclid solved the two easiest cases in his Elements, and the others (with the exception of the three circle problem), appeared in the Tangencies of Apollonius which was, however, lost. The general problem is, in principle, solvable by straightedge and compass alone.The three-circle problem was solved by Viète (Boyer 1968), and the solutions are called Apollonius circles. There are eight total solutions. The simplest solution is obtained by solving the three simultaneous quadratic equations(1)(2)(3)in the three unknowns , , for the eight triplets of signs (Courant and Robbins 1996). Expanding the equations gives(4)for , 2, 3. Since the first term is the same for each equation, taking and gives(5)(6)where(7)(8)(9)(10)and..

Geometrography

A quantitative measure of the simplicity of a geometric construction which reduces geometric constructions to five steps. It was devised by È. Lemoine. Place a straightedge's graph edge through a given point, Draw a straight line, Place a point of a compass on a given point, Place a point of a compass on an indeterminate point on a line, Draw a circle. Geometrography seeks to reduce the number of operations (called the "simplicity") needed to effect a construction. If the number of the above operations are denoted , , , , and , respectively, then the simplicity is and the symbol is . It is apparently an unsolved problem to determine if a given geometric construction is of the smallest possible simplicity.

Geometric problems of antiquity

The Greek problems of antiquity were a set of geometric problems whose solution was sought using only compass and straightedge: 1. circle squaring. 2. cube duplication. 3. angle trisection. Only in modern times, more than years after they were formulated, were all three ancient problems proved insoluble using only compass and straightedge.Another ancient geometric problem not proved impossible until 1997 is Alhazen's billiard problem. As Ogilvy (1990) points out, constructing the general regular polyhedron was really a "fourth" unsolved problem of antiquity.

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