1. Angle Trisection Angle Trisection. Most people are familiar from high school geometry with compass and straightedge constructions. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

2. Trisection Of An Angle I read your article on angle trisection. You seem intelligent enough, We havea proof that angle trisection (using certain simple tools) cannot be done. http://www.jimloy.com/geometry/trisect.htm

Return to my Mathematics pages Go to my home page Trisection of an Angle Under construction (just kidding, sort of). This page is divided into seven parts: Part I - Possible vs. Impossible Part II - The Rules Part III - Close, But No Banana Part IV - "Cheating" (violating the rules) ... Part VIII - Comments Part I - Possible vs. Impossible In Plane Geometry, constructions are done with compasses (for drawing circles and arcs, and duplicating lengths, sometime called "a compass") and straightedge (without marks on it, for drawing straight line segments through two points). 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. Another way to say this is that a general arc cannot be trisected. The public and the newspapers seem to think that this means that mathematicians don't know how to trisect an angle; well they don't, not with these tools. But they can estimate a trisection to any accuracy that you want. What can be done with these tools? Given a length

3. An Angle Trisection An angle "trisection" http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. A Real Trisection It interactively demonstrates the actual trisection of an angle, apparentlyinvented by CA Laisant in 1875. Move the bright red dot in order to change the http://www.jimloy.com/cindy/trisect1.htm

Return to my Cinderella pages Return to my Mathematics pages Go to my home page A Real Trisection It interactively demonstrates the actual trisection of an angle, apparently invented by C. A. Laisant in 1875. Move the bright red dot in order to change the angle. That dot is a hinge, as are the other red dots that far from the vertex. The two red dots farthest from the vertex are hinges for the short bars, but slide along the angle trisectors. The whole device is just a couple of parallelograms and a couple diagonals. Please enable Java for an interactive construction (with Cinderella). We cannot use a pair of compasses and a straightedge to trisect a general angle. That has been proven. But we can make other tools (such as the device simulated here) to trisect such an angle. In fact, we can use a pair of compasses and a straightedge to make this device. All that this device does is triple the smallest angle. The above Java interactive demonstration was created with Cinderella (a geometry program).

5. Trisection Of An Angle trisection of an angle Copyright 1997 and 2003, Jim Loy. Under construction (just kidding, sort of). This page is divided into seven parts http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Archimedes And Latitude Archimedes Trisection of the Angle http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Euclid Challenge - Trisection Of Any Angle By Straightedge And Trisection of Any Angle by Straightedge and Compass Turn the "chart page" 90 to the left, from "portrait" to " "landscape". http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. The Trisection Of An Angle It turns out that trisecting the angle is equivalent to solving a cubic equation . Figure 7.1 Trisection of the Angle with a marked ruler http://db.uwaterloo.ca/~alopez-o/math-faq/node57.html

Next: Which are the Up: Famous Problems in Mathematics Previous: The Four Colour Theorem The Trisection of an Angle This problem, together with Doubling the Cube Constructing the regular Heptagon and Squaring the Circle were posed by the Greeks in antiquity, and remained open until modern times. The solution to all of them is rather inelegant from a geometric perspective. No geometric proof has been offered [check?], however, a very clever solution was found using fairly basic results from extension fields and modern algebra. It turns out that trisecting the angle is equivalent to solving a cubic equation. Constructions with ruler and compass may only compute the solution of a limited set of such equations, even when restricted to integer coefficients. In particular, the equation for degrees cannot be solved by ruler and compass and thus the trisection of the angle is not possible. It is possible to trisect an angle using a compass and a ruler marked in 2 places. Suppose X is a point on the unit circle such that is the angle we would like to ``trisect''. Draw a line

9. Trisection Of An Angle. The Columbia Encyclopedia, Sixth Edition . trisection of an angle. The Columbia Encyclopedia, Sixth Edition. 2001 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. The Trisection Of An Angle The trisection of the angle by an unmarked ruler and compass alone is in generalnot possible. Figure 7.1 Trisection of the Angle with a marked ruler http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node28.html

Next: Which are the 23 Up: Famous Problems in Mathematics Previous: The Four Colour Theorem The Trisection of an Angle Theorem 4. The trisection of the angle by an unmarked ruler and compass alone is in general not possible. This problem, together with Doubling the Cube Constructing the regular Heptagon and Squaring the Circle were posed by the Greeks in antiquity, and remained open until modern times. The solution to all of them is rather inelegant from a geometric perspective. No geometric proof has been offered [check?], however, a very clever solution was found using fairly basic results from extension fields and modern algebra. It turns out that trisecting the angle is equivalent to solving a cubic equation. Constructions with ruler and compass may only compute the solution of a limited set of such equations, even when restricted to integer coefficients. In particular, the equation for theta = 60 degrees cannot be solved by ruler and compass and thus the trisection of the angle is not possible. It is possible to trisect an angle using a compass and a ruler marked in 2 places.

11. Sci.math Message sci.math Search. All Discussions sci.math Archive Topic Message previous next Message Re Trisection of angle http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Sci.math FAQ: The Trisection Of An Angle 1998 Version 7.5 The trisection of an angle Theorem 4. The trisection of theangle by an unmarked ruler and compass alone is in general not possible. http://www.faqs.org/faqs/sci-math-faq/trisection/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: The Trisection of an Angle There are reader questions on this topic! Help others by sharing your knowledge Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca alopez-o@unb.ca http://daisy.uwaterloo.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Rate this FAQ N/A Worst Weak OK Good Great Related questions and answers Usenet FAQs Search Web FAQs ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 15 2004 @ 00:31 AM

13. Sci.math Message All Discussions sci.math Archive Topic Message previous next Message Re tools for trisection of angle ? http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. Questions And Answers Related To Current FAQs Q As sci.math FAQ The trisection of an angle. Home Toprated FAQs. Back tosci.math FAQ The trisection of an angle http://www.faqs.org/qa/fqa3235.html

Home Top-rated FAQs Back to sci.math FAQ: The Trisection of an Angle Are you an expert in this area? Share your knowledge and earn expert points by giving answers or rating people's questions and answers! This section of FAQS.ORG is not sanctioned in any way by FAQ authors or maintainers. Questions somewhat related to this FAQ: what is impossibilities on the problem of constructing a cube having twice the volume of a... by Rana (1/19/2004) Other questions awaiting answers: 23690 questions related to other FAQs 3985 general questions 12723 answered questions

15. Demonstration Of The Archimedes' Solution To The Trisection Problem Angle Trisection by Archimedes of Syracuse (circa 287 212 B.C.) http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Angle Trisection Origami trisection of an angle. How can you trisect an angle? http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

17. About "Trisection Of An Angle" trisection of an angle. _ Library Home Full Table of Contents Suggest a Link Library Help http://mathforum.org/library/view/8045.html

Trisection Of An Angle Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.jimloy.com/geometry/trisect.htm Author: Jim Loy Description: A discussion of what can and can't be constructed using compasses (for drawing circles and arcs, and duplicating lengths) and straight-edge (without marks on it, for drawing straight line segments). Levels: Middle School (6-8) High School (9-12) Languages: English Resource Types: Articles Math Topics: Constructions Home The Math Library Quick Reference ... Help http://mathforum.org/

18. Sci.math FAQ: The Trisection Of An Angle Subject sci.math FAQ The trisection of an angle; Fromalopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz); Date Fri, 27 Feb 1998 193859 GMT http://www.uni-giessen.de/faq/archiv/sci-math-faq.trisection/msg00000.html

Index sci.math FAQ: The Trisection of an Angle Subject : sci.math FAQ: The Trisection of an Angle From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : Fri, 27 Feb 1998 19:38:59 GMT Newsgroups sci.math news.answers sci.answers Sender news@undergrad.math.uwaterloo.ca (news spool owner) Summary : Part 18 of 31, New version http://daisy.uwaterloo.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.trisection

19. Trisection Of An Angle A proof of how to trisect an angle using a straight edge, a compass, and successiveapproximations. http://www.trevorstone.org/trisection.html

Essays Poetry Pictures Schoolwork ... More Trisection by Successive Approximation Note I wrote this a couple years ago while I was taking geometry. I know it is not precise, but it is nice to see. At last! I have proved the trisection of an angle using successive approximations! This is set up for a graphical browser. If you don't have a graphical browser, you MAY be able to follow it, but it would be a good idea to download the picture. If you wish to be more accurate, split the arc into 6 segments, then 9, etc. Then take the endpoint of the segment one third the distance from the midpoint (with 3 segments it would be the segment next to the midpoint, with 6 the second segment from the midpoint, etc.) and connect it to the vertex. Congratulations! You have just trisected an angle using successive approximations. The exactness depends on how many segments you split the arc into. Back to my homepage Created by Trevor Stone Last modified: March 30 2002 17:18:59 Bo knows rehab. me

20. Trisection Of An Angle - Columbia Encyclopedia® Article About Trisection Of An Columbia Encyclopedia® article about trisection of an angle. trisection of an angle.Information about trisection of an angle in the Columbia Encyclopedia®. http://columbia.thefreedictionary.com/trisection of an angle

Domain='thefreedictionary.com' word='trisection of an angle' Your help is needed: American Red Cross The Salvation Army join mailing list webmaster tools Word (phrase): Word Starts with Ends with Definition subscription: Dictionary/ thesaurus Computing dictionary Medical dictionary Legal dictionary Financial dictionary Acronyms Columbia encyclopedia Wikipedia encyclopedia Hutchinson encyclopedia trisection of an angle Also found in: Wikipedia 0.01 sec. Page tools Printer friendly Cite / link Email Feedback trisection of an angle: see geometric problems of antiquity geometric problems of antiquity, three famous problems involving elementary geometric constructions with straight edge and compass, conjectured by the ancient Greeks to be impossible but not proved to be so until modern times. The three problems are: (1) the duplication of the cube, also known as the Delian problem because it is said to have originated with the task of constructing a cubical altar at Delos having twice the volume of the original cubical altar; (2) the trisection of an arbitrary angle; (3) the squaring, or quadrature, of the circle, i. Click the link for more information.

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