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Topic: Ruler-and-compass construction

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MATH 302: Topics in Geometry A Ruler and Compass construction is a drawing (construction) that you create using only a straight edge and a compass. Use a ruler and compass to construct the bisector of a given angle. Use a ruler and compass to construct the perpendicular bisector of a line segment. www.math.uiuc.edu /~stolman/m302/labs/labs/lab2.html   (1715 words)

Ruler-and-compass construction - Wikipedia, the free encyclopedia Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) operations of paper folding, or origami. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed. The straightedge and compass give you the ability to produce ratios which are solutions to quadratic equations, but doubling the cube and trisecting the angle require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. en.wikipedia.org /wiki/Ruler-and-compass_construction   (1715 words)

GRACE - Graphical Ruler and Compass Editor GRACE is an interactive ruler and compass construction editor for use in teaching the fundamental concepts of geometry to high school students. GRACE allows dynamic creation and modification of ruler and compass constructions, allowing students to easily visualize the steps of a construction and how it varies for different inputs. Constructions may be built from one of five geometric primitives (Line, Line Segment, Ray, Circle, Perpendicular Bisector, and Intersection), and from other constructions; thus constructions may be built by composing more basic constructions. www.cs.rice.edu /~jwarren/grace   (319 words)

PlanetMath: Euclidean field Then it is not hard to see that the set of all numbers that can be ``constructed'' by the ruler and compass construction forms a Euclidean field over Conversely, we can define the ruler and compass construction as follows: can be ``constructed'' by the ruler and compass contruction. planetmath.org /encyclopedia/EuclideanField.html   (157 words)

Ruler-and-compass construction - Wikipedia, the free encyclopedia Stated this way, ruler and compass constructions appear to be a parlor game, rather than a serious practical problem. Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books. en.wikipedia.org /wiki/Ruler-and-compass_construction   (1580 words)

First Lecture (21.10.96) Two thousand and more years ago the Greeks asked the question whether one can construct with ruler and compass a square with the same volume as the unit circle, or whether one construct cube twice is voluminous as a given cube. The way the Greeks posed the question was algorithmical: find a straight line ruler and compass algorithm such that given a specified input the out put satisfies other specified properties. The way the answer was proven is the leading theme of this course: The algorithmic description of the problem via ruler and compass was first replaced by an equivalent algebraic definition (Galois, Abel). www.cs.technion.ac.il /~janos/COURSES/CD/W96/lect-1.html   (1111 words)

Geometric construction with the compass alone The assertion that every ruler-and-compass construction could be accomplished with a compass is due to Lorenzo Mascheroni (1750-1800) and appeared in his 1797 tractate The Geometry of Compasses. Everything you can do with a ruler and a compass you can do with the compass alone. The question of geometric construction with the compass alone is not concerned with such kinds of geometries. www.cut-the-knot.org /do_you_know/compass.shtml   (753 words)

The quadratrix Once we admit that we can use the limit point L, only ruler and compass are needed for the further construction of the square with side length sqrt(pi). It is possible to construct an angle of 60 degrees using only ruler and compass, but it has been proved that it's impossible to construct an angle of 20 degrees using only the same tools. During the 19th century the French mathematician Pierre Wantzel proved that under these circumstances the first two of those constructions are impossible and for the squaring of the circle it lasted until 1882 before a proof had been given by Ferdinand von Lindemann! cage.rug.ac.be /~hs/quadratrix/quadratrix.html   (713 words)

Ruler and Compass Constructions Take your compass and spread it apart to be the length of the segment. Determining how to perform constructions was a major component of the study of geometry for the ancient Greeks, and it continues to be a component of geometry today. If you use a previous construction, say so and be specific about how you use it, but do not write down its steps. sierra.nmsu.edu /morandi/CourseMaterials/RulerAndCompass.html   (655 words)

The construction of arctan(1/2)/Pi Keywords : Binary expansion, A004715 of the On-Line Encyclopedia of Integer Sequences, constant, ruler and compass construction, Pi. In this context it means that we can't construct an arc length of 1 radian with the ruler and compass. It is known that rational numbers of the form 1/q can be computed with a ruler and compass in small bases. www.lacim.uqam.ca /~plouffe/compass.html   (1035 words)

Ruler & compass This application allows you to simulate ruler and compass constructions on the net, either freehand, or for predified goals going from basic operations to sophisticated ones such as the regular polygon of 17 sides. If on the theoretical side, today one knows exactly what are the constructions which are possible by ruler and compass, on the practical side, it is always interesting to know how to construct which or which configuration concretely. Geometric construction by ruler and compass is a fascinating mathematical problem since ancient times. wims.unice.fr /wims/en_tool~geometry~rulecomp.en.html   (242 words)

Ruler and Compass Construction The number of steps for using ruler and compass must be finite.  It is not allowed to repeat certain process infinitely many times. The compass is used for drawing circles with given radius. Ruler is only used to draw straight line. www.scienceoxygen.com /mathnote/geo110.html   (429 words)

Construction of a Regular Pentagon. Such a construction is provided by a free to download program "Ruler and Compass". It is a very well known fact that a regular pentagon can be constructed with a ruler and a compass (unlike a regular polygon with 7 sides). In this picture point A is denoted by P and point B by Q. This picture was drawn by a program "Ruler and Compass" mentioned above. www.geocities.com /literka/mathcountry/pentagon.htm   (712 words)

Construction of a Regular Polygon with 257 Sides. We use the procedures of the page of Literka Construction of roots of quadratic equation with ruler and compass. It is a command of a new version of program "Ruler and Compass", which cannot be downloaded free yet. If you have a free version of a program "Ruler and Compass", these commands must be erased for a program to run. www.literka.addr.com /mathcountry/regular257.htm   (966 words)

preface to Geom. Const. As specified by Plato, the game is played with a ruler and a compass, where the ruler can be used only to draw the line through two given points and the compass can be used only to draw the circle with a given center and through a given point. For example, we will prove that angle trisection is generally impossible with only the ruler and compass (Chapter 2), and we will see how to trisect any given angle with a marked ruler. The most famous of the other construction tools is the marked ruler, which is simply a ruler with two marks on its edge (Chapter 9). math.albany.edu:8000 /math/pers/preface.html   (829 words)

Zef Damen Non ruler and compass constructions (1) Zef Damen Non ruler and compass constructions (1) On the next page, I'll show that this can be seen as a way of constructing a regular 9-sided (and 18-sided) polygon, and how this can be generalised to any odd-numbered regular polygon. I first do the construction a second time, but with an arbitrary circle 3. home.wanadoo.nl /zefdamen/Constructions/NonRaCConstructions1_en.htm   (689 words)

Math Forum - Ask Dr. Math Archives: High School Constructions Construct the triangle using only ruler and compass if you know A, h, m. Given a circle with two points inside it, construct another circle that passes through the given points and is tangent to the given circle. How can I construct a triangle ABC given AM, BN, and CP, the respective medians from the vertices A, B, and C? Constructing Tangents to Circles [05/08/2002] www.mathforum.org /library/drmath/sets/high_constructions.html   (689 words)

Compass and Straightedge Constructions in Geometry -- Step-By-Step Instructions unmarked ruler) and compass construction in Euclidean geometry. These constructible numbers are the key to understanding what is possible and what is not when it comes to compass and straightedge constructions. Compass and Straightedge Constructions in Geometry - What is possible and impossible and why. mtl.math.uiuc.edu /modules/module13/what_to_do.htm   (689 words)

Ruler-and-compass construction - Wikipedia, the free encyclopedia Stated this way, ruler and compass constructions appear to be a parlor game, rather than a serious practical problem. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed. The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of Galois theory. en.wikipedia.org /wiki/Ruler-and-compass_construction   (689 words)

Compass and straightedge - Wikipedia, the free encyclopedia Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books. The set of ratios constructible using compass and straightedge from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. en.wikipedia.org /wiki/Ruler-and-compass_construction   (2535 words)

Ruler-and-compass construction - One Language Without the constraint of requiring solution by ruler and compass alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity. The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of Galois theory. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books. www.onelang.com /encyclopedia/index.php/Angle_trisection   (1476 words)

Encyclopedia: Compass-(drafting) A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. This is about drafting, the art and science of technical drawing. www.nationmaster.com /encyclopedia/Compass_(drafting)   (268 words)

Ruler An interactive ruler and compass construction editor for use in teaching thefundamental concepts of geometry to high school students. Madame X — the cruel, uncrowned ruler of the China seas — promises "gold, love,and adventure" to all women who'll leave their humdrum lives behind. In mathematics, the term "Golomb Ruler" refers to a set of non-negative...An Optimal Golomb Ruler (OGR) is the shortest Golomb Ruler possible for a given... www.classicgame.org /ruler.html   (943 words)

Ruler-and-compass construction Information The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of is possible using geometric constructions, but not possible using ruler and compass alone. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed. Without the constraint of requiring solution by ruler and compass alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity. www.articleshead.com /show_article/ruler_a%257e_and_a%257e_compass-construction   (943 words)

Trisecting an angle The construction of regular polygons using ruler and compass was certainly one of the major aims of Greek mathematics and it was not until the discoveries of Gauss that further polygons were constructed with ruler and compass which the ancient Greeks had failed to find. The problem is therefore to trisect an arbitrary angle and the aim is to make the construction using ruler and compass (which is impossible) but failing that to devise some method to trisect an arbitrary angle. Gauss had stated that the problems of doubling a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Trisecting_an_angle.html   (2043 words)

Cabri - Cabri World 1999 - Anais Mohr in 1672 and Mascheroni in 1797 proved the fundamental theorem according to which every ruler and compass construction can be made using only the compass. The proof of the equivalence of the ruler and compass model and the compass model is reduced then to proving that the problems A and B can be solved using the compass alone. The Danish geometer G. Mohr in 1672 discovered an important result about the compass model; but his book "Euclides Danicus" was lost and was rediscovered only three centuries later. www.cabri.com.br /pesquisas/c99_anais/pa/pa_boieri.htm   (808 words)

construct As an introduction, the author describes a simple construction of the regular 7-gon (heptagon) which besides ruler and compass only requires the trisection of an angle and which is related to an older construction of J. Plemelj [Monatsh. The main goal of the paper is the answer to the question: which regular polygons can be constructed with the aid of ruler, compass and angle trisector? An application of this result is the construction of the regular 13-gon (triskaidecagon), which requires one angle trisection. www.math.niu.edu /~rusin/known-math/97/construct   (542 words)

construct As an introduction, the author describes a simple construction of the regular 7-gon (heptagon) which besides ruler and compass only requires the trisection of an angle and which is related to an older construction of J. Plemelj [Monatsh. The main goal of the paper is the answer to the question: which regular polygons can be constructed with the aid of ruler, compass and angle trisector? An application of this result is the construction of the regular 13-gon (triskaidecagon), which requires one angle trisection. www.math.niu.edu /~rusin/known-math/97/construct   (542 words)

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