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# Topic: Compass and straightedge

 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. Stated this way, compass and straightedge constructions appear to be a parlor game, rather than a serious practical problem. 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/Compass_and_straightedge   (2584 words)

 Angle Trisection I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle. Mark on the straightedge the length between A and B. Take the straightedge and line it up so that one edge is fixed at the point B. Let D be the point of intersection between the line from A parallel to BC. Move the marked straightedge until the line BD satisfies the condition AB = ED, that is adjust the marked straightedge until point E and point D coincide with the marks made on the straightedge. www.geom.uiuc.edu /docs/forum/angtri   (1278 words)

 Math Forum: Tessellations: Straightedge/Compass Constructions In principle, this method is reminiscent of the rope stretching techniques of surveying, using peg and rope for a pair of compasses, in the planning of buildings in Ancient Egypt. To construct a triangle using a straightedge and a compass. Adjust the compass to the desired length of the sides of the triangle. mathforum.org /sum95/suzanne/stcom.html   (412 words)

 ipedia.com: Ruler-and-compass construction Article   (Site not responding. Last check: 2007-10-19) The compass can be opened arbitrarily wide, but (unlike most real compasses) it also has no markings it. 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. Therefore the degree for the minimal polynomial for cos 20° is of degree three, so cos 20° is not constructible and 60° cannot be trisected. www.ipedia.com /ruler_and_compass_construction.html   (1369 words)

 Welcome to Cinderella Using straightedge and collapsible compass, construct the midpoint of a given segment. Using straightedge and collapsible compass only, drop a perpendicular to a given line from a given point. Using a straightedge and collapsible compass only, draw a circle with a given center and having a radius equal to a given line segment. www.ux1.eiu.edu /~cfdmb/cinderella/guide.html   (784 words)

 Untitled Document It was they who established the compass and straightedge as the allowable tools of geometry and who thereby determined the sorts of constructions that were and were not possible. First, the compass and straightedge existed as easy-to-use mechanical devices that could be employed by the geometer to draw figures on papyrus or in the sand. Although the compass and straightedge tradition predates Euclid, it is in his Elements that one finds its most perfect manifestation. www.acs.appstate.edu /orgs/centroid/whatsnew.htm   (1425 words)

 What is a construction? The compass establishes equidistance, and the straightedge establishes collinearity. The compass is anchored at a center point, and keeps the pencil at a fixed distance from that point. In many commonly accepted constructions (e.g., congruent angles), the compass radius is set by the distance between two points, and then the compass is centered on some third point, elsewhere on the drawing. whistleralley.com /construction/whatis.htm   (662 words)

 The CTK Exchange Forums Note that the original query was not on the subject of compass and straightedge constructions, but rather that of compass and marked straightedge constructions. I'm not completely sure of this one, but general quintics are not solvable, and this equation doesn't seem to be solvable by radicals, in the general case (arbitrary b). I have the feeling the marked straightedge does serve to solve the general quintic, but the feeling is not enough. www.cut-the-knot.org /htdocs/dcforum/DCForumID6/495.shtml   (1112 words)

 Trisecting An Angle   (Site not responding. Last check: 2007-10-19) Allow me to offer the more liberal argument that trisection of an arbitrary angle with straightedge and compass is indeed possible; it just takes a (countably) infinite number of steps. It is possible to exactly bisect an arbitrary angle using only straightedge and compass. Bisection, for example, requires an infinitely accurate straightedge and compass -- which I think can only be constructed in an uncountably infinite number of steps -- and nobody gets bent out of shape over assuming the existence of these ideal instruments. home.earthlink.net /~serotonin/proof.html   (514 words)

 Trisecting the Angle It's tempting to try to use a straightedge and compass to lay out the hatchet right on the desired angle, but it also can't be done without trial and error. You can't tighten nuts with a saw or cut a board with a wrench, and expecting a straightedge and compass to do something beyond their capabilities is equally futile. Geometry with straightedge and compass creates a similar illusion; eventually we believe the points we can construct are all the points that exist. www.uwgb.edu /dutchs/PSEUDOSC/trisect.HTM   (5079 words)

 From the time of Euclid, geometric constructions were done solely with a straightedge and compass   (Site not responding. Last check: 2007-10-19) It is not possible to construct with a ruler and compass a line whose length is the numerical value of a root of a cubic equation with rational coefficients having no rational roots (Davis, 227). Since traditional constructions are performed with the use of only a straightedge and compass there is no unit of measurement involved. The constructions that are possible with a compass and a straightedge are: drawing a line through two points, finding the intersection of two lines, drawing a circle with a given radius with a given center, and finding the intersection of a circle with another circle or line. www.cerebral-palsy.net /trisectangle/trisectangle2.html   (1751 words)

 Unsolvable Problems   (Site not responding. Last check: 2007-10-19) Trisecting an arbitrary angle with compass and straightedge. Doubling the cube, that is constructing a cube with twice the volume of a given cube (using only compass and straightedge). Squaring the Circle, constructing a square with the same areas a given circle (using only compass and straightedge). www.math.tamu.edu /~Michael.Pilant/math646/unsolvable.html   (47 words)

 Lab 10 For a given angle, use straightedge and compass to construct the ray that bisects the angle. Use straightedge and compass to construct the line perpendicular to the given line and passing through the given point. Use straightedge and compass to construct the line parallel to the given line and passing through the given point. www.mtholyoke.edu /courses/jmorrow/lab_10.html   (577 words)

 Activity 6 It would be possible to construct a regular 60-sided polygon with a compass and a straightedge, since the only odd factors of 60 are 3 and 5, and each different odd factor is unique. On the other hand, it is impossible to construct a regular 100-sided polygon with a compass and a straightedge, since 100 has more than one factor of 5. A 21-sided regular polygon cannot be constructed with compass and straightedge because 21 has an odd factor of 7, which is not in the table above. homepage.mac.com /efithian/Geometry/Activity-06.html   (524 words)

 Lecture Notes 5 - Math 3210 Constructions using compass and straightedge have a long history in Euclidean geometry. It is not possible to construct, with straightedge and compass alone, regular polygons of sides n = 7, 9, 11, 13, 14, 18, 19,.... It can also be shown that any construction that can be made with straightedge and compass can be made with straightedge alone, as long as there is a single circle with its center given (Steiner, 18??). www-math.cudenver.edu /~wcherowi/courses/m3210/hg3lc5.html   (1144 words)

 Confession of a Weekend Trisector: Mathematicians spent twenty centuries trying in vain to trisect an angle with straightedge and compass until, in 1837, it was proved that it could not be done. Maybe there is a loophole in the proof of the impossibility of angular trisection because, with this construction, you would not have to create the desired trisection solely from the original angle; you draw upon a new, accurate trisection that you have created. A compass and straightedge let you add and subtract distances, allowing you to generate all integers; and to multiply and divide distances, as in Figure 1, allowing you to generate rational numbers; but do not give you enough operations to generate irrational numbers. dagwood.dgrc.crc.ca /thom/trisection.htm   (4278 words)

 Ruler & compass   (Site not responding. Last check: 2007-10-19) Geometric construction by ruler and compass is a fascinating mathematical problem since ancient times. 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. 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. wims.unice.fr /wims/en_tool~geometry~rulecomp.en.html   (242 words)

 Planar Curves - Numericana The parabola is a curve that's constructible with straightedge and compass. Equilateral triangles, squares, regular pentagons and hexagons are all constructible with straightedge and compass, but it turns out that regular heptagons, enneagons, hendecagons, or tridecagons cannot be so obtained. The construction, with straightedge and compass of the center of the circle circumscribed to a triangle is a very early piece of mathematics, credited to Euphorbe, a Phrygian mathematician predating Thales... home.att.net /~numericana/answer/curve.htm   (3746 words)

 construction by a finite sequence of straightedge and compass operations. The ancient Greeks considered a geometric object to be ``solid'' constructible if if could be obtained using the ordinary straightedge and compass operations plus the ability to draw conic sections. He also includes a discussion of the marked straightedge case, but a precise characterization of the constructible numbers in that case is still unknown. mathcircle.berkeley.edu /BMC4/Handouts/construction/construction.html   (1348 words)

 Math Forum - Ask Dr. Math The only regular n-gons that can be drawn using only a straightedge and compass are those with the number of sides equal to a power of 2 times a product of distinct Fermat primes. This is because to solve the construction problem, you have to be able to construct a line segment whose length is the cosine of 2*Pi/n. With straightedge and compass you can construct line segments whose lengths are gotten by rational operations (adding, subtracting, multiplying, or dividing by nonzero rational numbers) and taking square roots. mathforum.org /library/drmath/view/51650.html   (398 words)

 [generic] Compass and straightedge whining I've heard one good argument against compass and straightedge, that being "on Friday there were not enough to go around". There is something appealing to me about having two separate pieces of stuff in my hands to mechanic doing something to a door or lock with two separate pieces of stuff (pick and torquewrench). Points 3 and 4 are valid, with the small exception that if the GM team is on the stick, they can buy "good compasses", "crappy compasses", "long hard rulers" and "short limp rulers" and put those into game as varying grades of lockpicking tools. mailman.mit.edu /pipermail/generic/2003-November/000032.html   (713 words)

 Talk - Trisecting the Angle In this talk, we will discuss why the compass and straightedge construction is impossible and show two simple methods for trisecting an angle with additional equipment: a compass and ruler construction known to Archimedes, and a more recent technique using origami. The mathematics behind this talk comes from the discussion of compass and straightedge constructions in What is Mathematics by Courant and Robbins and from the discussion of origami constructions on Thomas Hull's Origami Mathematics Page. After sketching a proof of the impossibility of trisection by compass and straightedge, we perform the Archimedes compass-and-ruler trisection and discuss the mathematics that makes trisection by paper folding possible. www.amherst.edu /~sgoldstine/exposition/tri.html   (337 words)

 Bisecting a line with compass and straightedge - Math Open Reference Set the compass width to a approximately two thirds the line length. Again without changing the compass width, place the compass point on the the other end of the line. Using a straightedge, draw a line between the points where the arcs intersect. www.mathopenref.com /constbisectline.html   (360 words)

 Bisecting an angle with compass and straightedge - Math Open Reference Without changing the compass width, draw an arc across each leg of the angle. Place the compass on the point where one arc crosses a leg and draw an arc in the interior of the angle. Without changing the compass setting repeat for the other leg so that the two arcs cross. www.mathopenref.com /constbisectangle.html   (336 words)

 [No title]   (Site not responding. Last check: 2007-10-19) In constructing geometric figures with straightedge and compass, the allowed tools are a straightedge (you can use a ruler, but you can't use its markings) and a compass for drawing circles. It is useful to think of a compass as a tool for copying lengths; after all, (Definition:) a circle is the set of all points that are the same fixed distance (radius) from a given point (the center). Without changing the compass opening, pick a point on the circle as a new center and draw another circle with the same radius. www.math.csusb.edu /courses/m129/eqtri.html   (513 words)

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