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Topic: Constructible polygon

Related Topics

Regular polygon

Fermat prime

Star polygon

Heptadecagon

Triskaidecagon

Polygon

Simple polygon

Galois theory

Heptagonal number

Pentagon

Galois connection

Heptagon

Cyclic polygon

Hexagon

Symmetry group

	NationMaster - Encyclopedia: Exact trigonometric constants
		Example polygon and its fundamental right triangle File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version.
		Constructibility of 3, 4, 5, and 15 sided polygons are the basis, and angle bisectors allow multiples of two to also be derived.
		All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions:,,,,, In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...
	www.nationmaster.com /encyclopedia/Exact_trigonometric_constants/External_links (1409 words)

	polygon - Encyclopedia.com
		The simplest regular polygons are the equilateral triangle, the square, the regular pentagon (of 5 sides), and the regular hexagon (of 6 sides).
		Although the Greeks had developed methods of constructing these four polygons using only a straightedge and compass, they were unable to do the same for the regular heptagon (of 7 sides).
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory) of the form p = 2
	www.encyclopedia.com /doc/1E1-polygon.html (1108 words)

	ipedia.com: Constructible polygon Article (Site not responding. Last check: )
		For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.
		Compass and straightedge constructions are known for all constructible polygons.
		Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150).
	www.ipedia.com /constructible_polygon.html (987 words)

	math lessons - Constructible polygon
		For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.
		Compass-and-straightedge constructions are known for all constructible polygons.
		Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150).
	www.mathdaily.com /lessons/Constructible_polygon (911 words)

	Polygon
		A polygon (literally "many angle", see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments.
		If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
		Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g.
	www.brainyencyclopedia.com /encyclopedia/p/po/polygon.html (1024 words)

	Constructible polygon
		For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.
		Compass and straightedge constructions are known for all constructible polygons.
		Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150).
	www.xasa.com /wiki/en/wikipedia/c/co/constructible_polygon.html (938 words)

	Polygon - ExampleProblems.com
		If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
		Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g.
		A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal; for each number of sides, all regular polygons with the same number of sides are similar.
	www.exampleproblems.com /wiki/index.php/Polygon (926 words)

	Constructible polygon - Definition, explanation
		For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.
		Compass-and-straightedge constructions are known for all constructible polygons.
		Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150).
	www.calsky.com /lexikon/en/txt/c/co/constructible_polygon.php (946 words)

	Untitled Document
		The three best known were the trisection of the angle; the duplication of the cube (i.e., given the side of a cube, construct the side of a cube with double the volume); and the quadrature of the circle (given a circle, construct a square of equal area).
		Gauss's amazing construction must have given renewed hope to the angle trisectors and circle squarers, who could reasonably argue that if a regular 17-gon were constructible, then a dose of Gaussian ingenuity might lead to a successful trisection or circle squaring as well.
		Unfortunately for polygon constructors, is not prime, for it has a factor of 641 as the incomparable Euler had observed 50 years earlier.
	www.acs.appstate.edu /orgs/centroid/whatsnew.htm (1425 words)

	The Angles of a Regular Polygon
		Description: Give the program the number of sides of a regular polygon and it will give you the name of the polygon, the sum of the measure of the interior angles, the measure of each interior angle, and the measure of each exterior angle.
		A polygon is a closed plane figure bounded by straight line segments as sides.
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number of the form p = 22n + 1 or a product of such primes.
	www.homeworkhotline.com /AnglesofaRegularPolygon.htm (287 words)

	Talk:Constructible polygon - Biocrawler (Site not responding. Last check: )
		Question: the article claims that specific concrete constructions are known for ALL constructible polygons.
		In principle, getting from an explicit quadratic equation in terms of constructible reals, having real roots, to an explicit geometric construction of the length of a root, is nothing genuinely deep.
		It seems you're saying there's a definite algorithm to give you the construction, which I interpret to mean that explicit constructions are "known".
	www.biocrawler.com /encyclopedia/Talk:Constructible_polygon (313 words)

	Math Forum - Ask Dr. Math Archives: High School Constructions
		Given a circle with two points inside it, construct another circle that passes through the given points and is tangent to the given circle.
		Construct a tangent to a circle through a given point not on the circle.
		Given the altitude from vertex A, angle BAC, and the radius of the circumscribed circle, construct triangle ABC.
	mathforum.org /library/drmath/sets/high_constructions.html (794 words)

	polygon — FactMonster.com
		In a regular polygon the sides are of equal length and meet at equal angles; all other polygons are not regular, although either their sides or their angles may be equal, as in the cases of the rhombus and the rectangle.
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides
		pyramid, in geometry - pyramid pyramid, in geometry, solid figure bounded by a polygon (the base, or directrix) and the...
	www.factmonster.com /ce6/sci/A0839589.html (302 words)

	AllRefer.com - polygon (Mathematics) - Encyclopedia
		In a regular polygon the sides are of equal length and meet at equal angles; all other polygons are not regular, although either their sides or their angles may be equal, as in the cases of the rhombus and the rectangle.
		The simplest regular polygons are the equilateral triangle, the square, the regular pentagon (of 5 sides), and the regular hexagon (of 6 sides).
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory) of the form p = 2
	reference.allrefer.com /encyclopedia/P/polygon.html (314 words)

	Regular Polygon Definition (Site not responding. Last check: )
		Regular polygon - A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length).
		Constructible polygon - In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge.
		Star polygon - In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again.
	pillow.vvvvvv3.com /regularpolygondefinition.html (607 words)

	constructible
		The Greeks were adept at constructing polygons, but the question of proving which regular polygons are constructible and which are not had to wait for the genius of Carl Gauss.
		The reverse is also true, since Jakob Steiner showed that all constructions possible with straightedge and compass can be done using only a straightedge, as long as a fixed circle and its center (or two intersecting circles without their centers, or three nonintersecting circles) have been drawn beforehand.
		The Greeks were unable to achieve certain constructions, such as squaring the circle, duplicating the cube, and trisecting an angle, despite numerous attempts, but it wasn't until hundreds of years later that the problems were proved to be actually impossible under the limitations imposed.
	www.daviddarling.info /encyclopedia/C/constructible.html (319 words)

	Constructible polygon (Site not responding. Last check: )
		This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons: :A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes.
		As for the construction of Gauss, when the Galois group is 2-group it follows that it has a sequence of subgroups of orders :1, 2, 4, 8,...
		It should be stressed that the concept of constructibility as discussed in this article applies specifically to compass-and-straightedge constructions.
	constructible-polygon.iqnaut.net (917 words)

	2D Regularized Boolean Set-Operations
		The exact definition of the obtained polygon with holes as a result of a Boolean set-operation or a sequence of such operations is closely related to the definition of regularized Boolean set-operations, being the closure of the interior of the corresponding ordinary operation as explained next.
		Once constructed, it is possible to insert new polygons (or polygons with holes) into the set using the insert() method, as long as the inserted polygons and the existing polygons in the set are disjoint.
		Construction for a general polygon that represent the outer boundary and a range of general polygons that represent the holes.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Boolean_set_operations_2/Chapter_main.html (4756 words)

	Polygon: Free Encyclopedia Articles at Questia.com Online Library
		Two sides of a triangle, or polygon, are adjacent in case they share a common...length from the center of a regular polygon to a side.
		Numerical simulation of polygon boundary displacements was used to propagate...by overlay depends on the sinuosity of polygon boundaries, as well as the magnitude of...
		A circle is inscribed in a polygon if each side of the polygon is tangent to the circle; a circle is circumscribed about a polygon if all the vertices of the polygon lie on the circumference.
	www.questia.com /library/encyclopedia/polygon.jsp?l=P&p=5 (1643 words)

	List and Descriptions of Mathematical Pages.
		Construction of a regular polygon with 257 sides.
		Another constructible polygon is a regular polygon with 257 sides.
		Construction with ruler and compass is not so easy for this case.
	www.literka.addr.com /mathcountry/index.html (994 words)

	Regular n-gons
		In summary, an mn-gon is constructible iff an m-gon and an n-gon are constructible, for m and n coprime.
		Tweak the earlier proof, and z is constructible iff it lies at the top of a tower of field extensions, each of dimension two over the previous, with Q(i) at the base.
		Therefore a constructible p-gon forces p to be a fermat prime.
	www.mathreference.com /fld-cs,ngon.html (960 words)

	The construction of arctan(1/2)/Pi
		The answer is obvious: any line constructed on the plane that crosses the unit circle somewhere defines a point from which the binary expansion can be calculated.
		Note : the construction is done on a plain white paper and could be done on the sand in fact with small precision.
		In this context it means that we can't construct an arc length of 1 radian with the ruler and compass.
	www.lacim.uqam.ca /~plouffe/compass.html (1035 words)

	Geometry and Topology - Numericana
		Note that with any shape of constant width you can construct infinitely many new ones: The (convex hull of the) envelope of the circles of radius R centered on a curve of constant width is also a curve of constant width.
		One can construct figures of constant diameter [constant width] from a regular polygon (with an odd number of vertices) by drawing small circles of radii R around each vertex and then drawing arcs from each vertex as to connect the two opposite circles at a tangent.
		The construction(s) outlined at the end of the previous article seem to remain valid to obtain a symmetrical shape of constant width in N+1 dimensions from one in N dimensions.
	home.att.net /~numericana/answer/geometry.htm (7688 words)

	Problem E - Regular Polygon
		A regular polygon is an n-sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral).
		Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.
		These outputs should be sorted in ascending order of S. The regular polygons, which are not formed by the input points, should not be reported.
	acm.uva.es /p/v108/10824.html (391 words)

	Math Forum - Ask Dr. Math
		Date: 04/20/98 at 11:56:10 From: Doctor Wilkinson Subject: Re: Euclidean Construction of Polygons It was very clever of you to notice the connection between construction of regular polygons and construction of angles.
		The problem of which regular polygons can be constructed was solved completely by Gauss around 1800, although there are still some mysteries connected with it.
		The construction of a regular polygon with 15 sides was known to the ancient Greeks.
	www.mathforum.org /library/drmath/view/55007.html (298 words)

	polygon - Information from Reference.com
		A regular polygon has sides equal in length, and all the interior angles at the vertices are equal.
		A regular polygon with three sides is an equilateral triangle; one with four sides is a square.
		He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory) of the form pÃÂ =ÃÂ 2
	www.reference.com /browse/all/polygon (403 words)

	Zef Damen Constructions with ruler and compass
		It can be transferred to a third point to create a new point at an equal distance, or it can be used to draw a circle or arc with the distance as the radius.
		Every construction starts with two arbitrary points, used to draw a line or a circle (or both).
		From a mathematical law, it has been known, that a regular n-sided polygon can be constructed by ruler-and-compass, if and only if n is a finite product of different numbers from the set of powers of 2 (2, 4, 8, 16,...) and primes of Fermat (3, 5, 17, 257, 65537,...).
	www.zefdamen.nl /CropCircles/Constructions/Constructions_en.htm (503 words)

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