21. Www.faqs.org/ftp/faqs/sci-math-faq/trisection ca!neumann.uwaterloo.ca!alopezo From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)Subject sci.math FAQ The trisection of an angle Summary Part 18 of 31 http://www.faqs.org/ftp/faqs/sci-math-faq/trisection

Newsgroups: sci.math,news.answers,sci.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!news.kodak.com!news-pen-14.sprintlink.net!207.41.200.16!news-pen-16.sprintlink.net!newsfeed.nysernet.net!news.nysernet.net!news.sprintlink.net!Sprint!128.122.253.90!newsfeed.nyu.edu!newsxfer3.itd.umich.edu!news-peer.gip.net!news-lond.gip.net!news.gsl.net!gip.net!newsfeed.icl.net!btnet-feed2!btnet!bmdhh222.bnr.ca!bcarh8ac.bnr.ca!bcarh189.bnr.ca!nott!kwon!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: The Trisection of an Angle Summary: Part 18 of 31, New version Originator: alopez-o@daisy.uwaterloo.ca Message-ID:

22. Trisection Of An Angle these up). Angle BDC is a perfect trisection of angle BAC. Proof Call lineBC. Again, this is not a real trisection of angle A. Here is http://www2.ittu.edu.tm/math/bosna.net/Geometry/papers/Trisecting an angle/Trise

Return to my Mathematics pages Go to my home page Trisection Of An Angle In Plane Geometry, constructions are done with compasses (for drawing circles and arcs, and duplicating lengths) and straight-edge (without marks on it, for drawing straight line segments). See Geometric Constructions . With these tools (see the diagram), an amazing number of things can be done. But, it is fairly well known that it is impossible to trisect (divide into three equal parts) a general angle, using these tools. Certain angles (90° for example) can be trisected. A general angle cannot. In fact, a 60° angle cannot. It is usually much more difficult to prove that something is impossible, than it is to prove that something is possible. In this case, mathematicians had to show just what kinds of lengths could be constructed. And then they could show that other lengths could not be constructed, because they were not the right kinds of lengths. What can be done with these tools? Given a length a , we can multiply this length by any integer, and divided it by any integer. Together, these allow us to multiply this length by any rational number. Given two lengths a and b (and sometimes a unit length), we can add them together, subtract them, multiply them, divide one by the other. Given a length

23. GEOMETRY CONSTRUCTION. Trisecting an angle, Conic Sections, Angle Trisection, Conic Sections,Trisection_page, Trisecting the Angle, trisection of an angle, GEOMETRIC ARTS. http://www2.ittu.edu.tm/math/bosna.net/Geometry/geometry.htm

GEOMETRY PAPERS Geometric Puzzles ,Constructions Squaring the circle Pythagorean Theorem Kissing spheres ... Geometry and Origami SUBJECTS TRISECTING AN ANGLE CONIC SECTIONS CONSTRUCTION Trisecting an angle Conic Sections Angle Trisection ... ART OF GEOMETRY GEOMETRY BOOKS Imagination and Geometry Geometry Problem Book Analytic Geometry Book GENERAL GEOMETRY PROGRAMS Cinderalla Geometry Software WCabri (Home Page) (Mathematics) ... alilafcina@hotmail.com

24. The Quadratrix 2/pi. The trisection of an angle using the quadratrix First we considera special case with historical importance. It is possible http://cage.rug.ac.be/~hs/quadratrix/quadratrix.html

THE QUADRATRIX Trisecting an angle - Squaring the circle Introduction Three famous geometrical construction problems, originating from ancient Greek mathematics occupied many mathematicians until modern times. These problems are the duplication of the cube: construct (the edge of) a cube whose volume is double the volume of a given cube, angle trisection: construct an angle that equals one third of a given angle, the squaring of a circle: given (the radius of) a circle, construct (the side of) a square whose area equals the area of the circle. In the ancient Greek tradition the only tools that are available for these constructions are a ruler and a compass . 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 If we extend the range of tools the problems can be solved. New tools can be material tools (ex. a "marked ruler", that's a ruler with two marks on it, a "double ruler", that's a ruler with two parallel sides,...), or

25. Untitled sci.math,news.answers,sci.answers From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz)Subject sci.math FAQ The trisection of an angle Message-ID Ep1yKz http://www.jmas.co.jp/FAQs/sci-math-faq/trisection

Newsgroups: sci.math, news.answers http://daisy.uwaterloo.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

26. Akolad News| Romain The trisection of an angle is one of the infamous three problemsof antiquity which have stumped mathematicians for centuries. http://www.akolad.com/news/romain.htm

Haitian Math Whiz May Have Unraveled Age-old Geometry Mystery HAITI PROGRES ( http://www.haiti-progres.com), October 9 - 15, 2002 Vol. 20, No. 30 by Kim Ives PHOTO: : Leon Romain has devised a theorem for trisecting any angle, one of geometry's great puzzles. If he is right, it could change your life. So far, nobody has proved him wrong Around 450 B.C., the Greek mathematician, Hippias of Ellis, began searching for a way to trisect an angle. Over 2000 years later, in 1837, a French mathematician named Pierre Wantzel proclaimed that it was impossible to trisect an angle using just a compass and a straightedge, the only tools allowed in geometric construction. But now, at the dawn of the twenty-first century, a Haitian computer program designer, Leon Romain, claims he has proven, with a "missing theorem," that it is possible to trisect an angle with those simple tools, disproving Wantzel's assertion and exploding centuries of mathematical gospel. "This discovery shows us that the notions that every mathematician has held for the past 200 years as absolute certainty are actually false," Romain told Haiti Progres. "The mathematical and even philosophical ramifications are huge."

27. Haitian Math Whiz May Have Unraveled Age ramifications are huge. The trisection of an angle is one of the infamous threeproblems of antiquity which have stumped mathematicians for centuries. http://www.radiolakay.com/romain.htm

please download macromedia Listen Live : Real Player or Windows Media Player Live Chat with Radio Lakay on AIM chat AIM:LAKAYINTER Links Global Outlook Library of Congress Bob Corbett's Home Page Haitian Math Whiz May Have Unraveled Age-old Geometry Mystery HAITI PROGRES ( http://www.haiti-progres.com ), October 9 - 15, 2002 Vol. 20, No. 30 by Kim Ives PHOTO: Leon Romain has devised a theorem for trisecting any angle, one of geometry's great puzzles. If he is right, it could change your life. So far, nobody has proved him wrong. Around 450 B.C., the Greek mathematician, Hippias of Ellis, began searching for a way to trisect an angle. Over 2000 years later, in 1837, a French mathematician named Pierre Wantzel proclaimed that it was impossible to trisect an angle using just a compass and a straightedge, the only tools allowed in geometric construction. But now, at the dawn of the twenty-first century, a Haitian computer program designer, Leon Romain, claims he has proven, with a "missing theorem," that it is possible to trisect an angle with those simple tools, disproving Wantzel's assertion and exploding centuries of mathematical gospel. "This discovery shows us that the notions that every mathematician has held for the past 200 years as absolute

28. Www.math.niu.edu/~rusin/known-math/93_back/trisect Newsgroups sci.math From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) SubjectRe NEW sci.math FAQ The trisection of an angle Date Thu, 10 Nov 1994 01 http://www.math.niu.edu/~rusin/known-math/93_back/trisect

Newsgroups: sci.math From: ramsay@unixg.ubc.ca (Keith Ramsay) Subject: Re: Trisect an angle? Date: Thu, 17 Sep 1992 17:26:11 GMT In article jurjus@kub.nl (H. Jurjus) writes: > In Article Greg Griffiths

29. Trisecting An Angle Response 3 of 3 Author jlu Here is an intuitive argument that may make itclearer why trisection of an angle is impossible with ruler and compass. http://newton.dep.anl.gov/newton/askasci/1995/math/MATH101.HTM

Ask A Scientist Mathematics Archive Trisecting an angle Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

30. Math Forum: Ask Dr. Math: FAQ trisection of an angle Given an angle, construct an angle one thirdas large. The problem must be solved for an arbitrary angle. http://www2.sunysuffolk.edu/wrightj/MA28/Greek/Impossible.htm

Ask Dr. Math: FAQ "I mpossible" G eometric C onstructions Three geometric construction problems from antiquity puzzled mathematicians for centuries: the trisection of an angle, squaring the circle, and duplicating the cube. Are these constructions impossible? Whether these problems are possible or impossible depends on the construction "rules" you follow. In the time of Euclid, the rules for constructing these and other geometric figures allowed the use of only an unmarked straightedge and a collapsible compass. No markings for measuring were permitted on the straightedge (ruler), and the compass could not hold a setting, so it had to be thought of as collapsing when it was not in the process of actually drawing a part of a circle. Following these rules, the first two problems were proved impossible by Wantzel in 1837, although their impossibility was already known to Gauss around 1800. The third problem was proved to be impossible by Lindemann in 1882. The impossibility proofs depend on the fact that the only quantities you can obtain by doing straightedge-and-compass constructions are those you can get from the given quantities by using addition, subtraction, multiplication, division, and by taking square roots. These numbers are called Euclidean numbers, and you can think of them as the numbers that can be obtained by repeatedly solving the quadratic equation. These three problems require either taking a cube root or constructing pi. A cube root is not a Euclidean number, and Lindemann showed that pi is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, making it too non-Euclidean.

31. Trisecting The Angle trisection of an angle. Given angle ABC, trisect angle ABC. Step1. This requires the use of a marked straight edge. This can be http://www.geocities.com/robinhuiscool/Trisectionofangle.html

TRISECTION OF AN A N G L E Given angle ABC, trisect angle ABC. Step 1. This requires the use of a marked straight edge. This can be something like a slip of paper or ruler. First draw a line parallel to line BC at point A. Step 2. Draw a perpendicular line from point A, intersecting BC at D. Step 3. Mark off on the straight edge points E,F and G where EF=FG=AB. Step 4. Position the straight edge so that it crosses point B, point E touches AD, and point G touches line A. Angle CBG is 1/3 of ABC.

32. Trisection Of An Angle Home. Encyclopeadia. T. Tra Tri. trisection of an angle. Index. Help.Encyclopedie. trisection of an angle. see geometric problems of antiquity. http://nl.slider.com/enc/53000/trisection_of_an_angle.htm

Home Encyclopeadia T Tra - Tri ... Rope Ladders Slider Zoeken: Het Web Encyclopedie Winkelen Index Help Encyclopedie trisection of an angle see geometric problems of antiquity Add URL Advertise Contact Us ... geometric problems of antiquity

33. Angle Two angles that add up to a straight angle are supplementary. One ofthe geometric problems of antiquity is the trisection of an angle. http://www.infoplease.com/ce6/sci/A0804037.html

Trace your family history with Ancestry.com All Infoplease All Almanacs General Entertainment Sports Biographies Dictionary Encyclopedia Infoplease Home Almanacs Atlas Dictionary ... Fact Monster Kids' reference Info:Daily Fun facts Homework Center Newsletter You've got info! Help Site Map Visit related sites from: Family Education Network Encyclopedia angle angle, /2 radians; the sides of a right angle are perpendicular to one another. An angle less than a right angle is acute, and an angle greater than a right angle is obtuse. Two angles that add up to a right angle are complementary. Two angles that add up to a straight angle are supplementary. One of the geometric problems of antiquity Angkor angler Search Infoplease Info search tips Search Biographies Bio search tips About Us Contact Us Link to Infoplease ... Privacy

34. > Learning > Reference And Documentation > Other Computations ; Pirelated programs ; How to Test Whether SQRT is Rounded Correctly; Beastly Numbers ; Approximate trisection of an angle ; Gregorian calendar http://www.mathtools.net/Learning/Reference_and_documentation/Other/

Mathtools.net Learning Reference and documentation Other ... Email page to friend ACM SIGPLAN - SIGPLAN is a Special Interest Group of ACM that focuses on Programming Languages. In particular, SIGPLAN explores programming language concepts and tools, focusing on design, implementation and efficient use. Its members are programming language users, developers, implementers, theoreticians, researchers and educators. ACM: Digital Library - Tap into the ACM Digital Library, a vast resource of bibliographic information, citations, and full-text articles. Access to full-text is by subscription only: ACM members who are Digital Library subscribers have access to all full-text articles. Members and nonmembers who subscribe to electronic publications (but not to the entire Library) have full-text access to their subscriptions only. Algorithms and Complexity Best Book Buys - Best Book Buys is an on-line shopping agent that searches bookstores on the internet. It helps users find books they are looking for at the best prices. People use Best Book Buys to search for a book and then it returns a list of stores that are selling that book and for what price. Thus making it easy to compare several store prices for the same book. Computer Science Portal - Offers free information pertaining to computer science. Read articles and site news. Learn the how-to's from our many lessons. You can obtain help by posting questions on our message board or asking the author. Have your code or file reviewed for rating. Receive free code listings and quick guides to get what you need fast.

35. Interactive Mathematics Miscellany And Puzzles, Geometry Preservation Property Java; angle trisection Java; angle Trisectorson Circumcircle Java; an Old Japanese Theorem; Apollonian Gasket http://www.cut-the-knot.com/geometry.shtml

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36. Angle Trisection angle trisection. When someone mentions angle trisection I immediatelythink of trying to trisect an angle via a compass and straightedge. http://freeabel.geom.umn.edu/docs/forum/angtri/

Up: Geometry Forum Articles Angle Trisection Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a circle among other constructions. A few weeks ago I explained my job to a group of professors visiting the Geometry Center . I mentioned that I wrote articles on a newsgroup about geometry and that sometimes people write to me with geometry questions. For instance one person wrote asking whether it was possible to divide a line segment into any ratio, and also whether it was possible to trisect an angle. In response to the first question I explained how to find two-thirds of a line segment. 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 . One professor I told this story to replied by saying, "Bob it is possible to trisect an angle." Before I was able to respond to this shocking statement he added, "You just needed to use a MARKED straightedge and a compass." The professor was referring to Archimedes' construction for trisecting an angle with a marked straightedge and compass. When someone mentions angle trisection I immediately think of trying to trisect an angle via a compass and straightedge. Because this is impossible I rule out any serious discussion of the manner. Maybe I'm the only one with this flaw in thinking, but I believe many mathematicians make this same serious mistake.

37. Trisection -- From MathWorld trisection, angle trisection is the division of an arbitrary angle into three equalangles. angle trisection. http//www.geom.umn.edu80/docs/forum/angtri/. http://mathworld.wolfram.com/Trisection.html

Geometry Geometric Construction Geometry Trigonometry ... Angles Trisection Angle trisection is the division of an arbitrary angle into three equal angles . It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836). Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary angle can be accomplished using a marked ruler (a Neusis construction ) as illustrated above (Courant and Robbins 1996). An approximate trisection is described by Steinhaus (Wazewski 1945, Steinhaus 1999, p. 7). Given an angle , draw the bisector , with , then divide BC such that . From the SAS theorem , the length s is given by the formula with s = b and L is then The angle can then be computed from the formula to obtain is then given by the formula for an SAS triangle The Maclaurin series is then to a very good approximation.

38. The Problem Of Angle Trisection In Antiquity The Problem of angle trisection in antiquity. As it turns out, the trisectionof an angle is not a `plane' problem, but a `solid' one Heath. http://www.math.rutgers.edu/courses/436/436-s00/Papers2000/jackter.html

The Problem of Angle Trisection in Antiquity A. Jackter History of Mathematics Rutgers, Spring 2000 The problem of trisecting an angle was posed by the Greeks in antiquity. For centuries mathematicians sought a Euclidean construction, using "ruler and compass" methods, as well as taking a number of other approaches: exact solutions by means of auxiliary curves, and approximate solutions by Euclidean methods. The most influential mathematicians to take up the problem were the Greeks Hippias, Archimedes, and Nicomedes. The early work on this problem exhibits every imaginable grade of skill, ranging from the most futile attempts, to excellent approximate solutions, as well as ingenious solutions by the use of "higher" curves [Hobson]. Mathematicians eventually came to the empirical conclusion that this problem could not be solved via purely Euclidean constructions, but this raised a deeper problem: the need for a proof of its impossibility under the stated restriction. The trisection of an angle, or, more generally, dividing an angle into any number of equal parts, is a natural extension of the problem of the bisection of an angle, which was solved in ancient times. Euclid's solution to the problem of angle bisection, as given in his Elements , is as follows: To bisect a given rectilineal angle: Let the angle BAC be the given rectilineal angle. Thus it is required to bisect it. Let a point D be taken at random on AB; let AE be cut off from AC equal to AD; let DE be joined, and on DE let the equilateral triangle DEF be constructed; let AF be joined. I say that the straight line AF has bisected the angle BAC. For, since AD is equal to AE, and AF is common, the two sides DA, AF are equal to the two sides EA, AF respectively. And the base DF is equal to the base EF; therefore the angle DAF is equal to the angle EAF. Therefore the given rectilineal angle BAC has been bisected by the straight line AF

39. An Angle Trisection an angle trisection . A highly accurate approximate constructionby Mark Stark. Drag the point B to change the angle AOB Drag the http://www.math.umbc.edu/~rouben/Geometry/trisect-stark.html

An angle "trisection" A highly accurate approximate construction by Mark Stark Drag the point B to change the angle AOB Drag the point E to change the initial guess The angle AOE' is a very close to being 1/3 of angle AOB Note how insensitive G and E' are with respect to displacements of E Type "r" to reset the diagram to its initial state The construction shown above, which trisects an arbitrary angle with great accuracy, was first proposed by Mark Stark in the geometry-puzzles discussion list. In a followup article Eric Bainville noted that the iteration of this trisection algorithm "will effectively converge to the trisection with a cubic convergence rate." Here is the outline of the construction, as restated by Mark in a later article Draw an arc with origin at O crossing both lines of the angle at points A and B. Draw line AB making an isosceles triangle. Using point A as the origin, draw an arc crossing line AB and the earlier arc somewhere between 1/4 and 1/2 way between points A and B. Label where this new arc crosses line AB point D. Label where this new arc crosses the first arc point E. Draw line DE and extend it well past O. If line DE passes exactly through point O (it wont) stop, your first guess was an exact trisection.

40. An Angle Trisection an angle trisection . A pretty simple approximate construction dueto C. R. Lindberg and Free Jamison. Drag the point B to change http://www.math.umbc.edu/~rouben/Geometry/trisect-jamison.html

An angle "trisection" A pretty simple approximate construction due to C. R. Lindberg and Free Jamison Drag the point B to change the angle AOB The angle E'OB is approximately 1/3 of angle AOB Type "r" to reset the diagram to its initial state The construction The construction shown above, which trisects an arbitrary angle with a pretty good accuracy, is described in: Free Jamison, Trisection Approximation , American Mathematical Monthly, vol. 61, no. 5, May 1954, pp. 334-336. The construction, the main idea of which, according to Jamison, comes from an unpublished work by C. R. Lindberg, is as follows: Draw a circle centered at the angle's vertex O. Let the circle intersects the angle's sides at A and B. Extend BO to intersect the circle at a point C. Draw the bisector of the angle AOB and let it intersect the circle at D. Draw the line CD and extend it to a point E such that DE equals the circle's diameter. Draw the line OE and Let it cut the circle at the point E'. Then the angle E'OB approximately 1/3 of angle AOB The function e(a) is monotonically increasing, therefore the worst error occurs at a=Pi. We have: e(Pi) = 0.0063 radians = 0.361 degrees. This corresponds to a relative error of approximately 0.2%.

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