41. Angle Trisection How can you trisect an angle? It can be shown it s impossible to do this with However, in origami, you can get accurate trisection of an acute angle. http://www.math.lsu.edu/~verrill/origami/trisect/

http://hverrill.net/pages~helena/origami/trisect/ Origami Trisection of an angle How can you trisect an angle? It can be shown it's impossible to do this with ruler and compass alone, (using Galois theory) - so don't try it!!! But you may be able to find some good approximations. However, in origami, you can get accurate trisection of an acute angle. You can read about this in several places, but since it's so neat, I thought I'd put instructions up here too - more people should be able to do this for a party trick! Jim Loy has informed me that this construction is due to to Hisashi Abe in 1980, (see "Geometric Constructions" by George E. Martin). See Jim Loy's page at http://www.jimloy.com/geometry/trisect.htm for a description of many other ways to trisec an angle. Since we're working with origami, the angle is in a piece of paper: So what we want is to find how to fold along these dotted lines: Note, if you don't start with a square, you can always make a square, here's the idea. We're going to trisect this angle by folding. I'm going to try and describe this in a way so that you'll remember what to do. Suppose we could put three congruent triangles in the picture as shown: These triangles trisect the angle. So we need to know how to get them there.

42. Ruler-and-compass Construction: Information From Answers.com Trisecting the angle Dividing a given angle into three smaller angles all of angle trisection using only ruler and compass, construct an angle that is http://www.answers.com/topic/ruler-and-compass-construction

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping ruler-and-compass construction Wikipedia ruler-and-compass construction A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass , or more properly a straightedge and compass The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of Galois theory . In spite of these impossibility proofs, some mathematical novices persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially solvable provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compass alone. 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.

43. Trisection Of An Angle. The Columbia Encyclopedia, Sixth Edition. 2001-05 an angle trisectionan angle trisection . A highly accurate approximate construction by Mark Stark Line OE is a good trisection. However this is only the start. http://www.bartleby.com/65/x-/X-trisecti.html

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44. An Angle Trisection an angle trisection . A pretty simple approximate construction due to CR Lindbergand Free Jamison. Drag the point B to change the angle AOB 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%.

45. Explorations In Math This exploration looks at various ways to trisect an angle. However, attemptsto use this simple trisection on an angle quickly proved useless as you http://jwilson.coe.uga.edu/emt669/Student.Folders/Godfrey.Paul/work/proj2/tri.ht

Tri as I Might by Paul Godfrey This exploration looks at various ways to trisect an angle. First we look at using an unmarked straight-edge and compass. We will also look at trisections that can be performed with marked straight-edge and compass. Then we explore using trisectrices to perform the job. Geometer's Sketchpad [1] was used for most of these explorations. For those having GSP, the GSP files can be obtained by clicking on the figure number. Reading the College Mathematics Journal [2] I noticed an article about trisecting an angle . It talked about something called a trisectrix. A trisectrix is a curve that can help us easily trisect an angle. One example given was a curve with equation called a trisectrix of Maclaurin. A graph of this curve looks like this Fig-1 next At first, this sounded like a complicated way to trisect an angle. After all, trisecting a line was a simple matter as Fig-2 shows. Fig-2 next Further, we know that given triangle DBC with rays BJ and BK as shown, any line segment parallel to DC with endpoints on rays BD and BC will be trisected by the rays BD, BJ, BK, BC due to the proportionality principle of similar triangles. Fig-3 next So, we reason that the angle we wish to trisect could be trisected using this method. Since the arc on a circle defined by the legs of the angle is the same measure as the angle, we merely need to trisect the arc. However, attempts to use this simple trisection on an angle quickly proved useless as you can see in

46. Dividing One Angle Into Three Equal Angles Seems A Trivial Problem But, since a 90 degree(pi/2) angle can be trisected with the use of an With this curve, the problem of trisecting an angle could be reduced to the http://www.perseus.tufts.edu/GreekScience/Students/Tim/Trisection.page.html

Dividing one angle into three equal angles seems a trivial problem. That is probably why it irked the Greeks so. Instead of being a simple problem, it is a complex, non-planar problem, as the Greeks soon discovered. The trisection problem can probably credit its origin to the construction of regular polygons. The discover of the construction of a perfect pentagon(see The Golden Section One of the earliest ways discovered was that of Hippias of Elis(circa 425 BC). Hippias used a curve he had invented, called the quadratrix . With this curve, the problem of trisecting an angle could be reduced to the trisection of a line segment. The following picture is one construction of such segment trisect. The great benefit of this method was that it could be generalized to divide any angle into any number of parts. I don't really like this next solution, but maybe you will. This second method, perhaps the most well known of all, can be credited to Nicomedes(circa 180 BC). Nicomedes created a special device to use in his construction. As the upper part slide back and forth in its groove, the angle of the pointer changed so as to describe a curve known as a conchoid(as a function, y=K(x^2 + C)^(-1/2) is the simplest form).

47. Les Fondements De La Dakhiométrie. angle trisection is an old millenary problem until now geometrically About theangle trisection the possibility of which a solution remains uncertain. http://www.dakhi.com/livcome.htm

The dakhiometry Evolution of the universe Tan Tai NGUYEN Buy a first edition of report on the Diakhiometry researches (French version) : PLRASE CLICK HERE The following URL will work for put in an order with other operations for getting this Dakhiometry's book about new theorems and physical laws. Please have a glance on it and also mail your intention of getting this work e.g. "YES" for the book if it is the case. Note that no longer if there are no sufficient amount of book orders this means that the Dakhiometry is not interested at this time. thus, these book sale proposition will be stoped and no continuation follows it. The other Dakhiometry's contents are as vast as the Whole universe should be. The space is an absolute precise content. The summary is following : - The space calculator - The spatial proofs of the famous remarkable equations. For example : (a+b) or a + b and other ones. - The same concerning the remakable trigonometric ones. For example : x and other ones. - The actual four elementary transformation operations - Plus, minus, multiplication and division. - The distances and the angle spatial series - The complete spatial calculations for lengths, their power and reciprocal (the roots)

48. Angle Trisection Problem: Trisect An Angle With A Tomahawk Trisecting an angle is impossible with a straightedge and compass, but a specialtool called a tomahawk makes this construction possible. http://www.articlesforeducators.com/article.asp?aid=24

49. Morley's Trisection Theorem (Of course, it s impossible to trisect an arbitrary angle using Euclidean methods, First, it s worthwhile to review why an angle trisection cannot be http://www.mathpages.com/home/kmath376/kmath376.htm

Morley's Trisection Theorem In Proposition 4 of Book IV of the Elements, Euclid inscribes a circle inside an arbitrary triangle by showing that the bisectors of any two of the interior angles meet at a point equidistant from the three edges. Since there is only one such point, it follows that the bisectors of all three angles meet at the same point. Letting 2 a b , and 2 g denote the three interior angles of a triangle, the law of sines implies that the edge lengths are proportional to the sines of these angles, so we can scale the triangle to make the edge lengths equal to these sines as shown below. Since a b g p /2, the central angles are g p b p /2, and a p /2, and the law of sines gives the ratios Making use of the double-angle formula we can substitute for the sines of 2 g b and 2 a in the previous ratios and simplify to give Hence by the sine rule we see that the point of intersection is a distance of from each of the three edges, which confirms that this point is the center of the inscribed circle. Now, since the bisectors of a triangle meet at a single point (a fact which is not entirely self-evident), it seems natural to go on to consider how the tri sectors of a triangle meet. However, Euclid apparently didn't consider this question, nor did anyone else for over 2000 years. (Of course, it's impossible to trisect an arbitrary angle using Euclidean methods, i.e., by straight-edge and compass, so Euclid obviously couldn't have used trisectors in any constructions, but he could still have proven some interesting theorems about trisectors if he had wished.) It wasn't until 1899 that Frank Morley (one time was president of the American Mathematical Association) discovered that lines trisecting the angles of an arbitrary plane triangle meet at the vertices of an equilateral triangle as illustrated in the figure below, where the central triangle (in blue) is equilateral.

50. The Regular Nine-gon And Angle Trisection angle trisection and the regular ninegon. We know that it is impossible totrisect an angle using only a straight-edge and compass. http://www.nevada.edu/~baragar/geom/nine.html

Angle trisection and the regular nine-gon We know that it is impossible to trisect an angle using only a straight-edge and compass. Since Geometer's Sketchpad mimics such constructions, one cannot write a script or sketch that trisects an arbitrary angle (using only the buttons and construction pull down menu in sketchpad.) However, one can create a sketch that mimics Archimede's trisection using a notched straight-edge. Such a sketch is below. The one step that is not a valid construction must be done by hand. In the sketch at the right, select the (acute) angle CAB to be trisected by moving the point C . Now, move P so that the line PQ goes through C . The angle CQB is one third of the angle CAB Sorry, this page requires a Java-compatible web browser. The step that must be done by hand moving P so that PQ goes through C is the step which is not a valid construction. The regular pentagon is constructible. Thus, one can write a sketch which produces a regular pentagon inscribed in a given circle (see below). The regular nine-gon, on the other hand, cannot be constructed using only a straight-edge and compass. But, one can use Archimedes' construction. This is done below. Again, one can adjust the circle in which a regular nine-gon is to be constructed by moving A and B . Try doing this. Note how the figure is distorted. Now, adjust

51. Demonstration Of The Archimedes' Solution To The Trisection Problem The three problems are to trisect a given angle, to double a cube, and to square a The solution for the angle trisection can be presented in a more http://www.cut-the-knot.org/pythagoras/archi.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Angle Trisection by Archimedes of Syracuse (circa 287 - 212 B.C.) Archimedes of Syracuse is popularily known for the law he discovered on occasion of taking his bath . "Eurika" he exclaimed and made it into the history. (Along with Newton and Gauss he is counted among the greatest mathematicians of all times. As an engineer he frustrated numerous attempts by the Romans to capture the city of Syracuse.) The problem of constructing an angle equal to the one third of the given one has been pondered since the times of antiquity. Probably to make the notion of 'geometric construction' more exciting the Ancient Greeks have restricted the allowed operations to using a straightedge and a compass. It's thus specifically forbidden to use a ruler for the sake of measurement. Three famous construction problems lingered until early 19th century when it was shown that it's impossible to solve them with the help of only a straightedge and a compass. The three problems are: to trisect a given angle, to double a cube, and to square a circle . However, one illicit solution that has been found in the works of Archimedes is demonstrated below. (This is Proposition 8 from his

52. Trisection -- From MathWorld trisection is the division of a quantity, figure, etc. into three equal parts, SEE ALSO angle trisection, Bisection, Multisection, Trisected Perimeter http://mathworld.wolfram.com/Trisection.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Trigonometry Angles ... Geometric Construction Trisection Trisection is the division of a quantity, figure, etc. into three equal parts, i.e., multisection with SEE ALSO: Angle Trisection Bisection Multisection Trisected Perimeter Point ... [Pages Linking Here] CITE THIS AS: Eric W. Weisstein. "Trisection." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/Trisection.html Wolfram Research, Inc.

53. Math Forum: Ask Dr. Math FAQ: "Impossible" Geometric Constructions Trisecting an angle, squaring the circle, doubling the cube. Trisecting anglesand Squaring Circles Using Geometric Tools http://mathforum.org/dr.math/faq/faq.impossible.construct.html

Ask Dr. Math: FAQ I mpossible G eometric C onstructions Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Doubling the cube 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 and third problems were proved impossible by Wantzel in 1837, although their impossibility was already known to Gauss around 1800. The second problem was proved to be impossible by Lindemann in 1882.

54. Euclid Challenge - Trisection Of Any Angle By Straightedge And Compass - Page 4 Basic range of angles, Adjustment before trisection, Range of angles after adjustment 4. 90º 180º, Take ¼ of angle, 22½º - 45º, trisection X 4 http://www.euclidchallenge.org/pg_04.htm

EUCLID CHALLENGE Successful Response by Milton Mintz May 10, 2002 Page 4: Trisection of Any Angle by Straightedge and Compass Note 1: Basic range of angles Adjustment before trisection Range of angles after adjustment Adjustment after trisection Between: Between: Add 22 Trisection minus 7 None None Bisect angle Trisection X 2 Take ¼ of angle Trisection X 4 Note 2: EXAMPLE ANGLE: 30 Since 60º is a frequent test angle, the above 30º example was used so that the resulting trisection could be doubled. Previous Page Top of Page Next Page

55. Euclid Challenge - Trisection Of Any Angle By Straightedge And Page 9 trisection of Any angle by Straightedge and Compass. Proof Of The trisection.Radius BD’ = 4. Radius BD = 3. http://www.euclidchallenge.org/pg_09.htm

56. Trisecting An Angle There are a number of ways in which the problem of trisecting an angle differs The problem is therefore to trisect an arbitrary angle and the aim is to http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trisecting_an_angle.html

Trisecting an angle Ancient Greek index History Topics Index Version for printing There are three classical problems in Greek mathematics which were extremely influential in the development of geometry. These problems were those of squaring the circle, doubling the cube and trisecting an angle. Although these are closely linked, we choose to examine them in separate articles. The present article studies the problem of trisecting an arbitrary angle. In some sense this is the least famous of the three problems. Certainly in ancient Greek times doubling of the cube was the most famous, then in more modern times the problem of squaring the circle became the more famous, especially among amateur mathematicians. The problem of trisecting an arbitrary angle, which we examine here, is the one for which I [EFR] have been sent the largest number of false proofs during my career. It is an easy task to tell that a 'proof' one has been sent 'showing' that the trisector of an arbitrary angle can be constructed using ruler and compasses must be incorrect since no such construction is possible. Of course knowing that a proof is incorrect and finding the error in it are two different matters and often the error is subtle and hard to find. There are a number of ways in which the problem of trisecting an angle differs from the other two classical Greek problems. Firstly it has no real history relating to the way that the problem first came to be studied. Secondly it is a problem of a rather different type. One cannot square any circle, nor can one double any cube. However, it is possible to trisect certain angles. For example there is a fairly straightforward method to trisect a right angle. For given the right angle

57. Trisection De L Angle angle en troisparts égales? avec une équerre, c est oui! http://membres.lycos.fr/villemingerard/Histoire/Trisangl.htm

58. Akolad News| Romain Leon Romain has devised a theorem for trisecting any angle, The trisectionof an angle is one of the infamous three problems of antiquity which have 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."

59. Archimedes And Latitude This construction is Archimedes trisection of the angle by compass and straightedge.One thing which makes this construction remarkable is that in 1832, http://www.sonoma.edu/users/w/wilsonst/Courses/Math_150/projects/trisection.html

Math 150 Projects Archimedes Trisection of the Angle Dr. Wilson The angle at A is one third of the central angle. This construction is Archimedes' trisection of the angle by compass and straightedge. One thing which makes this construction remarkable is that in 1832, the French mathematician Evariste Galois proved that it was impossible to trisect an angle with a compass and straightedge. Every year, Mathematics Department chairs get trisections of the angle with compass and straightedge, and they throw them in the wastebasket because it is very well known that Galois proved that it can't be done. In fact, Galois' proof is more well known than Archimedes' construction. Whenever someone proves that something can't be done, one should examine the proof for unconscious, limiting assumptions. and this construction can be done using only a compass and a straightedge. I have found that it comes out about as accurately as any of my other compass and straightedge constructions. top Math 100 Steve Wilson

60. How To Trisect An Angle Resources And Help How to trisect an angle recommended resources, facts, fun help. http://www.kidsexperimentshelp.info/howtotrisectanangle.html

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