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Topic: Trisection of an angle

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	Ruler-and-compass construction - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-14)
		Trisecting the angle: Dividing a given angle into three smaller angles all of the same size.
		Angle trisection: using only ruler and compass, construct an angle that is one-third of a given arbitrary angle.
		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.
	encyclopedia.learnthis.info /r/ru/ruler_and_compass_construction.html (1320 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.
		When told that it is impossible to trisect an angle with a straightedge and compass people then often believe it is impossible to trisect an angle.
		First take the angle to be trisected, angle ABC, and construct a line parallel to BC at point A. Next use the compass to create a circle of radius AB centered at A. Now comes the part where the marked straightedge is used.
	www.geom.uiuc.edu /docs/forum/angtri (1278 words)

	Ruler - Wikipedia, the free encyclopedia
		A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines.
		Strictly speaking, the ruler is the instrument used to rule and calibrated stick for measurement is called a measure.
		This is resolved by referring to an instrument that can only rule as a straightedge.
	en.wikipedia.org /wiki/Ruler (223 words)

	Angle trisection. Who is Angle trisection? What is Angle trisection? Where is Angle trisection? Definition of Angle ...
		A number of ancient problems in geometry involve the construction of lengths or angles using only an idealised ruler and compass.
		The ruler is indeed a straightedge, and may not be marked; the compass may only be set to already constructed distances, and used to describe circular arcs.
		This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.
	www.knowledgerush.com /kr/encyclopedia/Angle_trisection (605 words)

	[No title]
		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.
		Suppose that AE is the arc to be trisected, and that ED is the diameter through E of the circle o f which AE is an arc.
		In his discussion of the trisection of an angle, he said, "Nicomedes trisected any rectilineal angle by means of the conchoidal curves, the construction, order and prope rties of which he handed down, being himself the discoverer of their peculiar character.
	www.math.rutgers.edu /courses/436/436-s00/Papers2000/jackter.html (2256 words)

	Archimedes and Latitude (Site not responding. Last check: 2007-10-14)
		The angle at A is one third of the central angle.
		This construction is Archimedes' trisection of the angle by 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.
	www.sonoma.edu /users/w/wilsonst/courses/math_100/projects/3-17.html (273 words)

	The Trisection of an Angle (Site not responding. Last check: 2007-10-14)
		It turns out that trisecting the angle is equivalent to solving a cubic equation.
		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.
	db.uwaterloo.ca /~alopez-o/math-faq/node57.html (229 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		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.
		Arbitrary $n$-section (particularly trisection) of an angle can be done, given an Archimedean spiral.
		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)

	Angle trisection
		Let T be the measure of the angle to be trisected, that is BOA = T. Our purpose is to calculate the size of the construction error, that is, the difference between T/3 and the measure of the constructed angle E'OA.
		This is an increasing function of T. The worst error occurs at T = Pi and equals e(Pi,Pi/4) = 0.00299 radians.
		We proceed by determining several angles in the diagram in terms of T and a.
	www.math.umbc.edu /~rouben/Geometry/trisect-stark-proof.html (548 words)

	Trisection Unraveled (Site not responding. Last check: 2007-10-14)
		Trisection of an Angle Unraveled (by Quadratic Equations) manuscript was developed over a period of thirty years, the main body prior to 1972.
		Their angles, when trisected are of the highest importance, for they occupy a precise position of the squared circle and incorporate a distance of pi.
		These angles in particular need exploring – they are both bound together with their trisection.
	www.trisectionunraveled.com /unraveled.htm (705 words)

	Angle Trisection (Site not responding. Last check: 2007-10-14)
		This is a simple straightedge and compass construction of a third part of an angle.
		Angle ODE is equal to angle OED, and their sum is equal to the sum of DOC and FOE.
		The angle DOC is exactly one third of the angle AOB.
	mentock.home.mindspring.com /trisect.htm (197 words)

	Angle Trisection by Hippocrates
		As the trisection of Archimedes, this one, too, is not done by straightedge and compass, which we know is impossible.
		For a given angle BAC, let D be the foot of the perpendicular from C to AB.
		The external angle AGC is twice as large.
	www.cut-the-knot.com /Curriculum/Geometry/Hippocrates.shtml (228 words)

	Clearing up the market cycle... best Trisection (Site not responding. Last check: 2007-10-14)
		An angle "trisection" A highly accurate approximate construction by Mark Stark.
		Angle Trisection -- from MathWorld Angle Trisection -- from MathWorld Angle trisection is the division of an arbitrary angle into three equal angles.
		Trisection -- from MathWorld Trisection -- from MathWorld Trisection is the division of a quantity, figure, etc. into three equal parts, i.e., k-multisection with k = 3.
	ascot.pl /th/Fourier5/Trisection.htm (598 words)

	sci.math FAQ: The Trisection of an Angle (Site not responding. Last check: 2007-10-14)
		The trisection of the angle by an unmarked ruler and compass alone is in general not possible.
		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.
		Then angle BOA = theta, and angle XBO = angle BXO = 2 theta Since the sum of the internal angles of a triangle equals pi radians (180 degrees) we have angle XBO + angle BXO + angle BOX = pi, implying 4 theta + angle BOX = pi.
	omicron.felk.cvut.cz /FAQ/articles/a3254.html (399 words)

	Demonstration of the Archimedes' solution to the Trisection problem
		The problem of constructing an angle equal to the one third of the given one has been pondered since the times of antiquity.
		The three problems are: to trisect a given angle, to double a cube, and to square a circle.
		One (B) is dragged to specify the angle (trisect.
	www.cut-the-knot.org /pythagoras/archi.shtml (1008 words)

	Newsgroup FAQ's: sci.answers/sci-math-faq/trisection
		The trisection of the angle by an unmarked ruler and
		It is possible to trisect an angle using a compass and a ruler marked
		Figure 7.1: Trisection of the Angle with a marked ruler
	newsville.com /cgi-bin/getfaq?file=sci.answers/sci-math-faq/trisection (327 words)

	Trisection (Site not responding. Last check: 2007-10-14)
		Angle trisection is the division of an arbitrary
		Angles are geometrically trisectable, but cannot be constructed in the first place, such as
		Dörrie, H. ``Trisection of an Angle.'' §36 in
	www.itu.dk /edu/documentation/mathworks/math/math/t/t383.htm (172 words)

	The regular nine-gon and angle trisection
		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.
		The angle CQB is one third of the angle CAB.
	www.nevada.edu /~baragar/geom/nine.html (277 words)

	An angle trisection (Site not responding. Last check: 2007-10-14)
		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:
		Start with an unknown angle <90 deg., label the vertex O. 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.
	www.math.umbc.edu /~rouben/Geometry/trisect-stark.html (225 words)

	The Math Forum - Math Library - Web Interactive/Java (Site not responding. Last check: 2007-10-14)
		An interactive page illustrating how the relation between the vertices and the sides of a triangle can be investigated by rotating the triangle.
		An interactive study of coprime increasing number-pairs of different parity and their relations to divisors of n2 + 1.
		An Applet able to process symbolic and numeric simplifications of entered mathematical expressions, including symbolic differentiation, integration (standard integrals), and the development of Taylor as well as Fourier series.
	www.mathforum.org /library/resource_types/applets (2116 words)

	Trisection by Successive Approximation (Site not responding. Last check: 2007-10-14)
		I have proved the trisection of an angle using successive approximations!
		Then extend the middle segment one third the length of each of the two lines you just drew and the end segments two thirds the length of their corresponding short segments.
		You have just trisected an angle using successive approximations.
	ugrad-www.cs.colorado.edu /~tstone/trisection.html (235 words)

	AllRefer.com - trisection of an angle (Mathematics) - Encyclopedia
		AllRefer.com - trisection of an angle (Mathematics) - Encyclopedia
		trisection of an angle: see geometric problems of antiquity.
		More articles from AllRefer Reference on trisection of an angle
	reference.allrefer.com /encyclopedia/X/X-trisecti.html (114 words)

	Trisection of an Angle Unraveled (Site not responding. Last check: 2007-10-14)
		The problem of geometrically constructing an angle that is one third of a given angle has been pondered since the times of the ancient Greek mathematicians of Pythagoras and Archimedes.
		It sits alongside the "squaring of a circle" and the "doubling of a cube" as the the most famous of geometric problems.
		His analysis demonstrates unequivocally that a trisection - based algebraically and geometrically - on a quadratic equation, is possible.
	www.trisectionunraveled.com (215 words)

	The Geometry Junkyard: All Topics
		An aperiodic set of Wang cubes, J. Culik and Kari describe how to increase the dimension of sets of aperiodic tilings, turning a 13-square set of tiles into a 21-cube set.
		There is only one parameter to optimize, the angle of the triangle to the lattice vectors; my answer is that the densest packing occurs when this angle is 15 or 45 degrees, shown below.
		An 8x8 (64 unit) square is cut into pieces which (seemingly) can be rearranged to form a 5x13 (65 unit) rectangle.
	www.ics.uci.edu /~eppstein/junkyard/all.html (9717 words)

	trisection of an angle on Encyclopedia.com (Site not responding. Last check: 2007-10-14)
		Magazines and Newspapers for: trisection of an angle
		A survey of paper cutting, folding and tearing in mathematics textbooks for prospective elementary school teachers.
		Pictures and Maps for: trisection of an angle
	www.encyclopedia.com /html/X/X-trisecti.asp (82 words)

	definition of trisection
		The division of a thing into three parts, Specifically: (Geom.) the division of an angle into three equal parts.
		An, Angle, Division, Equal, Into, Of, Specifically, The, Thing, Three
		An, Angle, Division, Equal, Into, Of, Parts, Specifically, The, Thing, Three
	www.brainydictionary.com /words/tr/trisection232632.html (54 words)

	perplexus.info :: Geometry : Angle Trisection
		You may only use a straightedge and a compass to achieve this.
		This was an old problem long time ago.
		This is an unsolved problem from the past.
	perplexus.info /show.php?pid=117&cid=1401 (89 words)

	Geometric Cryptography: Identification by Angle Trisection (ResearchIndex)
		Abstract: We propose the field of "geometric cryptography," where messages and ciphertexts may be represented by geometric quantities such as angles or intervals, and where computation is performed by ruler and compass constructions.
		We describe a elegant little zero-knowledge identification scheme, based on the impossibility of trisecting an angle using ruler and compass operations.
		6 An elementary approach to ideas and methods (context) - Courant, Robbins et al.
	citeseer.ist.psu.edu /burmester97geometric.html (296 words)

	Angle trisection (Site not responding. Last check: 2007-10-14)
		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:
	www.math.lsu.edu /~verrill/origami/trisect (371 words)

	Find in a Library
		The impossible in mathematics; the trisection of an angle, the fifteen puzzle, and other problems.
		To find a library, type in a postal code, state, province, or country.
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	www.worldcatlibraries.org /wcpa/ow/3d44b9a36fc3fd1a.html (44 words)

	Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the ... (Site not responding. Last check: 2007-10-14)
		Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle
		Widely regarded as a classic of modern mathematics, this expanded version of Felix Klein's celebrated 1894 lectures uses contemporary techniques to examine three famous problems of antiquity: doubling the volume of a cube, trisecting an angle, and squaring a circle.
		No warranties are made express or implied about the accuracy, timeliness, merit, or value of the information provided.
	isbn.nu /0486495515 (471 words)

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