283 Sentences With "straightedge" | Random Sentence Generator

Use a business card as a straightedge for a perfect winged eye.

I think it was at AOL and it was a straightedge one.

You guys know I'm no prude, but I'm kind of like a straightedge.

Donald Trump is America's most famous straightedge man (eat your heart out, Ian MacKaye).

They ARE at an angle rather than strictly horizontal - you can hold a straightedge under to see.

It takes guts to let someone other than a barber go after your face with a straightedge.

I don't have time to take out a straightedge or a protractor to measure my dessert, sorry not sorry.

You guys know I'm no prude but I'm kind of like a straightedge, I thought he was kidding the first time.

Even when we were a five-piece and Aaron Dalbec [of Bane] was in our band, Aaron was more of a straightedge hardcore guy.

I'd never had a straightedge-razor shave on my head or my face, but I sat down in the chair and got my first one.

In the past, you could use a stylus as a digital pen or paintbrush on a touchscreen, and Adobe even developed a straightedge you could use on tablets.

"If you use a utility knife to trim the wallpaper at the edges, use a metal straightedge to prevent scoring the actual walls in the process," she added.

The 2000-year-old former straightedge skateboarder and roller derby star is an unlikely character at a racetrack, yet today she is killing it on the horseracing handicapping tournament scene.

You guys know I'm no prude but I'm kind of like a straightedge, I thought he was kidding the first time...the second time I do the show, same thing.

What makes things worse for Roderick is that before adopting the monastic precepts of straightedge hardcore music — no drugs, no booze, no fun — he used to be just like Tuffer.

Shawna Kenney: I liked putting together weird bills, like pairing local skinhead band Immoral Discipline with local emo band Moss Icon, or joke straightedge band Crucial Youth with drunken punks Murphy's Law.

Taiwanese artist and publisher Son Ni's comic Travel features a bouncing bowling ball (highlighted in gold) that pops in and out of a variety of scenes, each fastidiously rendered with pencil and straightedge.

Just in time for Halloween, we bring you a real-life horror story: An entire family ate a dinner of weed-laced chicken after their straightedge mother mistook her son's cannabutter for vegan butter.

Trump continued to make the case that he was a straightedge Nixon when he tweeted, then deleted, then tweeted again that the media was not his enemy, but the enemy of the United States.

At one point, while looking at "Stone Wash Freezer Burn" (2016), I imagined the artist using a straightedge razor to lay the paint down, methodically working it back and forth over the surface as he tries to smooth out the layers of colored paste, one atop another.

Whether he's a twenty-something Christian vegan straightedge in the midst of drugged-out teenage raves, or a rave star who makes a dyspeptic punk record record (143's Animal Rights) that nobody else likes and plays rock shows nobody attends, his memories and insights are pin-sharp.

On Monday, we got our first good look at Booksmart, Olivia Wilde's new movie about two straightedge girls—Molly (Feldstein) and Amy (Kaitlyn Dever)—who realize rules are for suckers when they find out they made it into the same colleges as the kids who partied all throughout high school.

Though his family name is known throughout town in association with prominent businesses, Anselmi's dad is the black sheep, a stoner dropout whose fuck-up demeanor pushes young Anselmi toward straightedge—Rock Springs is the type of place where an eighth grader can sincerely develop an identity based on sobriety.

The Amazon blurb for the book probably has the best, most concise explanation of the book, which captures one aspect of the search for identity among America's Muslim communities: A Muslim punk house in Buffalo, New York, inhabited by burqa-wearing riot girls, mohawked Sufis, straightedge Sunnis, Shi'a skinheads, Indonesian skaters, Sudanese rude boys, gay Muslims, drunk Muslims, and feminists.

The Poncelet–Steiner theorem should be contrasted with the Mohr–Mascheroni theorem, which states that any compass and straightedge construction can be performed with only a compass. It is not possible to construct everything that can be constructed with straightedge and compass with straightedge alone. If the centre of the only given circle is not provided, it cannot be obtained by a straightedge alone. Many constructions are impossible with straightedge alone.

Such a proof would require the formalization of what a straightedge and compass could construct. This groundwork was provided by Jean Victor Poncelet in 1822. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used once. The result that a straightedge and single circle with given centre is equivalent to a straightedge and compass was proved by Jakob Steiner in 1833.

The band's earliest recordings included anti-authoritarian statements and metaphysical themes. Although never explicitly a "straightedge band", all four original band members either identified as straightedge or led a drug-free lifestyle, independent of straightedge culture and politics. To this day both Danny Cipher and Moe Cipher continue to lead straightedge / drug-free lifestyles. Although not explicitly a "vegan or vegetarian band", three of the four original band members were either vegan or vegetarian.

Those in the subculture who gravitated toward animal-related causes but disagreed with some of hardline's finer points found themselves becoming vegan straightedge. Vegan straightedge band Earth Crisis initially wanted Muttaqi to release their debut EP, All Out War on his record label, Uprising Records. As hardline came into its own, many hardliners decided that their philosophy was so beyond the narrow scene politics of straightedge that the two were entirely different things. The "X" was removed from the crossed rifles logo, straightedge was harshly criticized, and hardliners were encouraged to leave the hardcore scene.

Only two other palettes, a straightedge, and a t-square hang upon the wall.

One algorithm for calculating 2-dimensional line of sight is given in the StraightEdge project.

Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube. The construction of the fold guaranteed by Axiom 6 requires "sliding" the paper, or neusis, which is not allowed in classical compass and straightedge constructions. Use of neusis together with a compass and straightedge does allow trisection of an arbitrary angle.

Motivated by the classical problems of constructions with straightedge and compass, the constructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.

For example, doubling the cube (the problem of constructing a cube of twice the volume of a given cube) cannot be done using only a straightedge and compass, but Menaechmus showed that the problem can be solved by using the intersections of two parabolas. Therefore, van Roomen's solution—which uses the intersection of two hyperbolas—did not determine if the problem satisfied the straightedge-and-compass property. Van Roomen's friend François Viète, who had urged van Roomen to work on Apollonius' problem in the first place, developed a method that used only compass and straightedge. Prior to Viète's solution, Regiomontanus doubted whether Apollonius' problem could be solved by straightedge and compass.

Angles may be trisected via a neusis construction using tools beyond an unmarked straightedge and a compass. The example shows trisection of any angle by a ruler with length equal to the radius of the circle, giving trisected angle . Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.

Construction of a regular pentagon Some regular polygons (e.g. a pentagon) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass? Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes.

To draw the parallel (h) to a diameter g through any given point P. Chose auxiliary point C anywhere on the straight line through B and P outside of BP. (Steiner) In Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions with additional restrictions. This result states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given.

He proposed that any construction possible by straightedge and compass could be done with straightedge alone. The one stipulation though is that a single circle with its center identified must be provided. The Poncelet- Steiner theorem was proved by Jakob Steiner eleven years later. This was a generalization of the proofs given by Ferrari and Cardano and several others in the 16th century where they demonstrated that all the constructions appearing in Euclid's Elements were possible with a straightedge and a "rusty" (fixed-width) compass.

The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction).

Construction of a regular pentagon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.

Flag using compass-and-straightedge construction Physical requirements for the Iranian flag, a simple construction sheet, and a compass-and-straightedge construction for the emblem and the takbir are described in the national Iranian standard ISIRI 1.ISIRI 1 , 1st revision. Retrieved 19 June 2012ISIRI 1 / IRANIAN ISLAMIC REPUBLIC FLAG, 1371 , 3rd edition, March 1993. Retrieved 19 June 2012.

Poncelet discovered the following theorem in 1822: Euclidean compass and straightedge constructions can be carried out using only a straightedge if a single circle and its center is given. Swiss mathematician Jakob Steiner proved this theorem in 1833, leading to the name of the theorem. The constructions that this theorem states are possible are known as Steiner constructions.

Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary.

Both trisecting the general angle and doubling the cube require taking cube roots, which are not constructible numbers by compass and straightedge.

Angle trisection with a parabola A parabola can be used as a trisectrix, that is it allows the exact trisection of an arbitrary angle with straightedge and compass. This is not in contradiction to the impossibility of an angle trisection with compass-and-straightedge constructions alone, as the use of parabolas is not allowed in the classic rules for compass-and-straightedge constructions. To trisect \angle AOB, place its leg OB on the x axis such that the vertex O is in the coordinate system's origin. The coordinate system also contains the parabola y = 2x^2.

Within ten years additional sets of solutions were obtained by Cardano, Tartaglia and Tartaglia's student Benedetti. During the next century these solutions were generally forgotten until, in 1673, Georg Mohr published (anonymously and in Dutch) Euclidis Curiosi containing his own solutions. Mohr had only heard about the existence of the earlier results and this led him to work on the problem. Showing that "all of Euclid" could be performed with straightedge and rusty compass is not the same as proving that all straightedge and compass constructions could be done with a straightedge and just a rusty compass.

Geometer's Sketchpad includes the traditional Euclidean tools of classical geometric constructions. If a figure (such as the pentadecagon) can be constructed with the compass and straightedge method, it can also be constructed in the program. The program allows "cheat" transformations to create figures impossible to construct under the compass and straightedge rules (such as the regular nonagon). Objects can be animated.

As 16 = 24 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians..

But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass.

In geometry, a ruler without any marks on it (a straightedge) may be used only for drawing straight lines between points. A straightedge is also used to help draw accurate graphs and tables. A ruler and compass construction refers to constructions using an unmarked ruler and a compass. It is possible to bisect an angle into two equal parts with a ruler and compass.

The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number . If is constructible, it follows from standard constructions that would also be constructible. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients. Thus, constructible lengths must be algebraic numbers.

An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in the Elements can be constructed using only a compass and straightedge. Codex Vaticanus 190 Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge.

It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not.

Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a alt=Construction of a regular pentagon using straightedge and compass Fermat primes are primes of the form :F_k = 2^{2^k}+1, with k a nonnegative integer. also include 2^0+1=2, which is not of this form. They are named after Pierre de Fermat, who conjectured that all such numbers are prime.

Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource. To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex.

Girih consists of geometric designs, often of stars and polygons, which can be constructed in a variety of ways. Girih star and polygon patterns with 5- and 10-fold rotational symmetry are known to have been made as early as the 13th century. Such figures can be drawn by compass and straightedge. The first girih patterns were made by copying a pattern template on a regular grid; the pattern was drawn with compass and straightedge.

The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.

False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (pseudomathematics). Origami may also be used to construct the cube root of two by folding paper.

An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge, doubling the cube and trisecting an angle.

Doubling the cube: PB/PA = cube root of 2 The classical problem of doubling the cube can be solved using origami. This construction is due to Peter Messer: A square of paper is first creased into three equal strips as shown in the diagram. Then the bottom edge is positioned so the corner point P is on the top edge and the crease mark on the edge meets the other crease mark Q. The length PB will then be the cube root of 2 times the length of AP. The edge with the crease mark is considered a marked straightedge, something which is not allowed in compass and straightedge constructions. Using a marked straightedge in this way is called a neusis construction in geometry.

Subsets of the axioms can be used to construct different sets of numbers. The first three can be used with three given points not on a line to do what Alperin calls Thalian constructions. The first four axioms with two given points define a system weaker than compass and straightedge constructions: every shape that can be folded with those axioms can be constructed with compass and straightedge, but some things can be constructed by compass and straightedge that cannot be folded with those axioms. The numbers that can be constructed are called the origami or pythagorean numbers, if the distance between the two given points is 1 then the constructible points are all of the form (\alpha,\beta) where \alpha and \beta are Pythagorean numbers.

In horseshoes, there are two ways to score: by throwing "ringers" or by throwing the horseshoe nearest to the stake. A ringer is a horseshoe that has been thrown in such a way as to completely encircle the stake. Disputes are settled by using a straightedge to touch the two points at the ends of the horseshoe, called "heel calks". If the straightedge does not touch the stake at any point, the throw is classified as a ringer.

In xkcd comic 866, Ferdinand von Lindemann is referenced in the alt-text, apparently having used a compass and straightedge to construct the greatest birthday party ever, to which nobody showed up.

The problem as stated is impossible to solve for arbitrary angles, as proved by Pierre Wantzel in 1837. However, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees). It is possible to trisect an arbitrary angle by using tools other than straightedge and compass.

Creating a regular hexagon with a straightedge and compass Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass; see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates.

In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. It must be understood that by "any geometric construction", we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as: : Any Euclidean construction, insofar as the given and required elements are points, may be completed with the compass alone if it can be completed with both the compass and the straightedge together.

For this release, Hope recorded two new songs, "Suck Factory" and "Almost Like", and covered the Malenkos' song "Jon's Gone Straightedge". Meanwhile, the Malenkos recorded two new songs and covered the Hope song "Raelienation".

Mark this on the straightedge; that's 10'. Make more marks to indicate 15, 30, etc., as desired. Now you have a "Tree Ruler" that can be used, on approximately level ground, to estimate tree heights.

The result is obtained by laying a straightedge across the known values on the scales and reading the unknown value from where it crosses the scale for that variable. The virtual or drawn line created by the straightedge is called an index line or isopleth. Nomograms flourished in many different contexts for roughly 75 years because they allowed quick and accurate computations before the age of pocket calculators. Results from a nomogram are obtained very quickly and reliably by simply drawing one or more lines.

As 28 = 22 × 7, the icosioctagon is not constructible with a compass and straightedge, since 7 is not a Fermat prime. However, it can be constructed with an angle trisector, because 7 is a Pierpont prime.

In the tenth century, the Persian mathematician Abu al-Wafa' Buzjani (940−998) considered geometric constructions using a straightedge and a compass with a fixed opening, a so-called rusty compass. Constructions of this type appeared to have some practical significance as they were used by artists Leonardo da Vinci and Albrecht Dürer in Europe in the late fifteenth century. A new viewpoint developed in the mid sixteenth century when the size of the opening was considered fixed but arbitrary and the question of how many of Euclid's constructions could be obtained was paramount. Renaissance mathematician Lodovico Ferrari, a student of Gerolamo Cardano in a "mathematical challenge" against Niccolò Fontana Tartaglia was able to show that "all of Euclid" (that is, the straightedge and compass constructions in the first six books of Euclid's Elements) could be accomplished with a straightedge and rusty compass.

Adjustments to the surface of the board are then made (e.g., with a hand plane). This process is repeated all across the piece until the piece is satisfactorily true. Longitudinally the piece is checked with a straightedge.

August Adler (24 January 1863, Opava, Austrian Silesia – 17 October 1923, Vienna) was a Czech and Austrian mathematician noted for using the theory of inversion to provide an alternate proof of Mascheroni's compass and straightedge construction theorem.

No continuous arc of a conic can be constructed with straightedge and compass. However, there are several straightedge-and-compass constructions for any number of individual points on an arc. One of them is based on the converse of Pascal's theorem, namely, if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic. Specifically, given five points, and a line passing through , say , \a point that lies on this line and is on the conic determined by the five points can be constructed.

The group featured on skate or bmx videos including ones on Volatile Visions. In 1986 Curran was replaced on guitar by Geoff Goddard. In June 1988, Where's the Pope? released an eight-track album, Straightedge Holocaust, on Reactor Records.

It can be proved, though, that it is impossible to divide an angle into three equal parts using only a compass and straightedge — the problem of angle trisection. However, should two marks be allowed on the ruler, the problem becomes solvable.

Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more. Thus, a regular -gon has a straightedge- and-compass construction if n is a product of distinct Fermat primes and any power of 2.

Some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number.

Because is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge. In addition to being irrational, is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as . The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or n-th roots (such as or ). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle".

The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: duplicating the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry (Plut., Quaestiones convivales VIII.

Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding. Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is usually treated alongside them. The Greeks knew how to construct regular -gons with (for any integer ) or the product of any two or three of these numbers, but other regular -gons eluded them.

It was first described by the German painter and mathematician Albrecht Dürer (1471–1528) in his book Underweysung der Messung (Instruction in Measurement with Compass and Straightedge p. 38), calling it Ein muschellini (Conchoid or Shell). Dürer only drew one branch of the curve.

Trisection can be approximated by repetition of the compass and straightedge method for bisecting an angle. The geometric series or can be used as a basis for the bisections. An approximation to any degree of accuracy can be obtained in a finite number of steps.

The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.

A chondrometer consists of a filling hopper, a measuring container, a straightedge, a weighing instrument. The filling hopper allows the grain to fall into the measuring container in a reproducible pattern as several measurements are taken, and all readings need to be within a strict degree of accuracy. The measuring cylinder has flat top edge to it can levelled using the straightedge (a strickle) to give a set volume. Today, the measuring instrument can be a set of digital scales with an accuracy greater than 0.1 g, though in the past it was a steelyard balance with the measuring cylinder hooking directly onto the scales.

Construction of equilateral triangle with compass and straightedge An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment An alternative method is to draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius.

Like most branches of mathematics, Euclidean geometry is concerned with proofs of general truths from a minimum of postulates. For example, a simple proof would show that at least two angles of an isosceles triangle are equal. One important type of proof in Euclidean geometry is to show that a geometrical object can be constructed with a compass and an unmarked straightedge; an object can be constructed if and only if (iff) (something about no higher than square roots are taken). Therefore, it is important to determine whether an object can be constructed with compass and straightedge and, if so, how it may be constructed.

Plotting the angle of the target was a simple process of taking the gonio reading and setting a rotating straightedge to that value. The problem was determining where along that straightedge the target lay; the radar measured the slant range straight-line distance to the target, not the distance over the ground. That distance was affected by the target's altitude, which had to be determined by taking the somewhat time- consuming altitude measurements. Additionally, that altitude was affected by the range, due to the curvature of the Earth, as well as any imperfections in the local environment, which caused the lobes to have different measurements depending on the target angle.

Bisection of arbitrary angles has long been solved. Using only an unmarked straightedge and a compass, Greek mathematicians found means to divide a line into an arbitrary set of equal segments, to draw parallel lines, to bisect angles, to construct many polygons, and to construct squares of equal or twice the area of a given polygon. Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads: Construct an angle equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools: # an unmarked straightedge, and # a compass.

Constructible Polygon Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three. It is not known if the regular hectogon is neusis constructible. However, a hectogon is constructible using an auxiliary curve such as an Archimedean spiral. A 72° angle is constructible with compass and straightedge, so a possible approach to constructing one side of a hectogon is to construct a 72° angle using compass and straightedge, use an Archimedean spiral to construct a 14.4° angle, and bisect one of the 14.4° angles twice.

Though squaring the circle with perfect accuracy is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to . It takes only minimal knowledge of elementary geometry to convert any given rational approximation of into a corresponding compass-and- straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly and informally as constructions that are particularly simple among other imaginable constructions that give similar precision.

During his tenure as a civil engineer, Lemoine wrote a treatise concerning compass and straightedge constructions entitled, La Géométrographie ou l'art des constructions géométriques, which he considered his greatest work, despite the fact that it was not well-received critically. The original title was De la mesure de la simplicité dans les sciences mathématiques, and the original idea for the text would have discussed the concepts Lemoine devised as concerning the entirety of mathematics. Time constraints, however, limited the scope of the paper. Instead of the original idea, Lemoine proposed a simplification of the construction process to a number of basic operations with the compass and straightedge.

Compass-and-straightedge, also known as ruler-and-compass construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass, see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates.

To construct a Reuleaux triangle The Reuleaux triangle may be constructed either directly from three circles, or by rounding the sides of an equilateral triangle.. The three-circle construction may be performed with a compass alone, not even needing a straightedge. By the Mohr–Mascheroni theorem the same is true more generally of any compass-and-straightedge construction,. but the construction for the Reuleaux triangle is particularly simple. The first step is to mark two arbitrary points of the plane (which will eventually become vertices of the triangle), and use the compass to draw a circle centered at one of the marked points, through the other marked point.

For the first time, the limits of mathematics were explored. Niels Henrik Abel (1802–1829), a Norwegian, and Évariste Galois, (1811–1832) a Frenchman, investigated the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four (Abel–Ruffini theorem). With these concepts, Pierre Wantzel (1837) proved that straightedge and compass alone cannot trisect an arbitrary angle nor double a cube. In 1882, Lindemann building on the work of Hermite showed that a straightedge and compass quadrature of the circle (construction of a square equal in area to a given circle) was also impossible by proving that is a transcendental number.

As 26 = 2 × 13, the icosihexagon can be constructed by truncating a regular tridecagon. However, the icosihexagon is not constructible with a compass and straightedge, since 13 is not a Fermat prime. It can be constructed with an angle trisector, since 13 is a Pierpont prime.

Jørgen Mohr (Latinised Georg(ius) Mohr; 1 April 1640 – 26 January 1697) was a Danish mathematician, known for being the first to prove the Mohr–Mascheroni theorem, which states that any geometric construction which can be done with compass and straightedge can also be done with compasses alone.

Various attempts have been made to restrict the allowable tools for constructions under various rules, in order to determine what is still constructable and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can.

As with all odd regular polygons and star polygons whose orders are not products of distinct Fermat primes, the regular hendecagrams cannot be constructed with compass and straightedge. However, describe folding patterns for making the hendecagrams {11/3}, {11/4}, and {11/5} out of strips of paper.

Axiom 5 may have 0, 1, or 2 solutions, while Axiom 6 may have 0, 1, 2, or 3 solutions. In this way, the resulting geometries of origami are stronger than the geometries of compass and straightedge, where the maximum number of solutions an axiom has is 2.

They were provided with large maps of their operational area printed on light paper so they could be stored for future reference. A rotating straightedge with the centrepoint at the radar's location on the map was fixed on top, so when the operator called an angle the plotter would rotate the straightedge to that angle, look along it to pick off the range, and plot a point. The range called from the operator is the line- of-sight range, or slant range, not the over-ground distance from the station. To calculate the actual location over the ground, the altitude also had to be measured (see below) and then calculated using simple trigonometry.

As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible using only a compass and straightedge, but even in ancient times solutions were known that employed other tools. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem.Plato's Republic Book VII "if any whole city should hold these things honourable and take a united lead and supervise, they would obey, and solution sought constantly and earnestly would become clear." However, the nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837.

We begin with the unit line segment defined by points (0,0) and (1,0) in the plane. We are required to construct a line segment defined by two points separated by a distance of . It is easily shown that compass and straightedge constructions would allow such a line segment to be freely moved to touch the origin, parallel with the unit line segment - so equivalently we may consider the task of constructing a line segment from (0,0) to (, 0), which entails constructing the point (, 0). Respectively, the tools of a compass and straightedge allow us to create circles centred on one previously defined point and passing through another, and to create lines passing through two previously defined points.

The Lune of Hippocrates. Partial solution of the "Squaring the circle" task, suggested by Hippocrates. The area of the shaded figure is equal to the area of the triangle ABC. This is not a complete solution of the task (the complete solution is proven to be impossible with compass and straightedge).

Similar to other Northwest Coast artists, Seaweed did not sign his artwork. However, it can be identified through his specific technique. Seaweed used tools, such as a compass and straightedge, in order to make precise and symmetrical compositions. His carving was so accurate that the surfaces are smooth without discrepancies.

The concept of constructibility as discussed in this article applies specifically to compass and straightedge construction. More constructions become possible if other tools are allowed. The so-called neusis constructions, for example, make use of a marked ruler. The constructions are a mathematical idealization and are assumed to be done exactly.

Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid. Let a and b be given. It is required to find u so that u3=a2b, giving u and v=u2/a as the mean proportionals.

Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points.

For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites. Origami can be used to construct various geometrical designs not possible with compass and straightedge constructions. For instance paper folding may be used for angle trisection and doubling the cube.

Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969). Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle; but these methods also cannot be followed with just straightedge and compass.

It is also associated with Celtic peoples, most notably Galicians and Asturians. It can be constructed with a compass and straightedge, beginning with the formation of a square template; each head can be drawn from a neighboring vertex of this template with two compass settings, with one radius half the length of the other.

Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable. Fields serve as foundational notions in several mathematical domains.

The geometric mean theorem asserts that . Choosing allows construction of the square root of a given constructible number . In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is in general impossible to trisect a given angle in this way.

As 22 = 2 × 11, the icosidigon can be constructed by truncating a regular hendecagon. However, the icosidigon is not constructible with a compass and straightedge, since 11 is not a Fermat prime. Consequently, the icosidigon cannot be constructed even with an angle trisector, because 11 is not a Pierpont prime. It can, however, be constructed with the neusis method.

In the 5th century BC, Hippocrates of Chios showed that Lune of Hippocrates and two other lunes could be exactly squared (converted into a square having the same area) by straightedge and compass. In the 19th century two more squarable lunes were found, and in the 20th century it was shown that these five are the only squarable lunes.

This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.

It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the algebraic proof of the previous section viewed geometrically in yet another way. Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem, .

Kids Like Us is an American hardcore punk band from north Florida that formed in 2003. Their music was characterized as skate-punk, with lyrics about skateboarding, eating burritos, straightedge lifestyle, and drinking Coca- Cola. They released three full-length albums, two split albums, and toured with bands that included Down To Nothing, Casey Jones, and the Mongoloids.

The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructed if n is not a power of 2..

65535 is the product of the first four Fermat primes: 65535 = (2 + 1)(4 + 1)(16 + 1)(256 + 1). Because of this property, it is possible to construct with compass and straightedge a regular polygon with 65535 sides. See constructible polygon. 65535 is the 15th 626-gonal number, the 5th 6555-gonal number, and the 3rd 21846-gonal number.

A golden rectangle can be constructed with only a straightedge and compass in four simple steps: # Draw a simple square. # Draw a line from the midpoint of one side of the square to an opposite corner. # Use that line as the radius to draw an arc that defines the height of the rectangle. # Complete the golden rectangle.

Geometric Exercises in Paper Folding shows how to construct various geometric figures using paper- folding in place of the classical Greek Straightedge and compass constructions. The book begins by constructing regular polygons beyond the classical constructible polygons of 3, 4, or 5 sides, or of any power of two times these numbers, and the construction by Carl Friedrich Gauss of the heptadecagon, it also provides a paper-folding construction of the regular nonagon, not possible with compass and straightedge. The nonagon construction involves angle trisection, but Rao is vague about how this can be performed using folding; an exact and rigorous method for folding-based trisection would have to wait until the work in the 1930s of Margherita Piazzola Beloch. The construction of the square also includes a discussion of the Pythagorean theorem.

The band was always incredibly tight due to their die-hard work ethic and practice schedule, and their team oriented attitude that really showed in their live shows. The members have all continued to play music since the band split in 1997. The last show for Screw 32 was at Boarderline Warehouse near Twain Harte, California in April 1997 run by a friend of theirs named Mark Kirkman (Who ran the now famous BullPen skateboard shop in Danville, California where the San Ramon and Danville straightedge scene was born and in extension the Northern California straightedge hardcore scene was also born with such bands as Unit Pride, Breakaway, Rabid Lassie, No Reason and many others) with Citizen Fish (ex-Subhumans), The Criminals, and Fury 66 from Santa Cruz, CA.

Quadratic surd: An algebraic number that is the root of a quadratic equation. Such a number can be expressed as the sum of a rational number and the square root of a rational. Constructible number: A number representing a length that can be constructed using a compass and straightedge. These are a subset of the algebraic numbers, and include the quadratic surds.

Cyclopaedia. Geometry (from the ; geo- "earth", -metron "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic). Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today.

Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2π/5 radians (72° = 360°/5) can be trisected, but the angle of π/3 radians (60°) cannot be trisected.Instructions for trisecting a 72˚ angle.

What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with such a tool.Gleason, Andrew: "Angle trisection, the heptagon, and the triskaidecagon", Amer. Math.

The mathematical theory of origami is more powerful than straightedge and compass construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, origami can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems.

One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs. Mechanism of a scroll compressor The Archimedean spiral has a variety of real-world applications.

This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which n-gons (that is polygons with n edges) are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.

Although a regular nonagon is not constructible with compass and straightedge (as 9 = 32, which is not a product of distinct Fermat primes), there are very old methods of construction that produce very close approximations.J. L. Berggren, "Episodes in the Mathematics of Medieval Islam", p. 82 - 85 Springer-Verlag New York, Inc. 1st edition 1986, retrieved on 11 December 2015.

First, divide the right angle into five congruent angles. An arbitrary point P is selected on the first ray counterclockwise. For the radius of the circle inscribed in the decagram, one half of the segment created from the third ray, segment AM, is selected. The following figure illustrates a step-by-step compass-straightedge visual solution to the problem by the author.

Monster X was a straight edge grindcore band from Albany, New York. The band formed in 1992 out of the ashes of Intent, and PNA. Members of MX Went on to play in Dropdead, Devoid Of Faith, Das Oath, and By The Throat. The band was one of the few straightedge bands not playing the typical mosh core sound of the time period.

Reprinted as The Trisectors. The two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. Like squaring the circle, these cannot be solved by compass-and-straightedge methods. However, unlike squaring the circle, they can be solved by the slightly more powerful construction method of origami, as described at mathematics of paper folding.

Compass-straightedge constructions allow only those with 2^a\phi\ge3 sides, where \phi is a product of distinct Fermat primes. (Fermat primes are a subset of Pierpont primes.) The seventh axiom does not allow construction of further axioms. The seven axioms give all the single-fold constructions that can be done rather than being a minimal set of axioms.

One way of specifying a real number uses geometric techniques. A real number r is a constructible number if there is a method to construct a line segment of length r using a compass and straightedge, beginning with a fixed line segment of length 1. Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible.

68-71 (German) DeTemple used in 1989 and 1991 Carlyle circles to devise Compass- and-straightedge constructions for regular polygons, in particular the pentagon, the heptadecagon, the 257-gon and the 65537-gon. Ladislav Beran described in 1999, how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.Ladislav Beran: The Complex Roots of a Quadratic from a Circle.

Its degree is a power of two, if and only if is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular -gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the casus irreducibilis, that is, every expression of the roots in terms of radicals involves nonreal radicals.

Euclid developed numerous constructions with compass and straightedge. Examples include: regular polygons such as the pentagon and hexagon, a line parallel to another that passes through a given point, etc. Many rose windows in Gothic Cathedrals, as well as some Celtic knots, can be designed using only Euclidean constructions. However, some geometrical constructions are not possible with those tools, including the heptagon and trisecting an angle.

The latter three methods can be found in Ramanujan's theory of elliptic functions to alternative bases. The inversion applied in high-precision calculations of elliptic function periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting compass and straightedge constructions).

Halliday pleaded no contest to "criminal contempt" on July 27, 2010. He was sentenced on November 3, 2010 to 10 months in prison and 3 years probation upon release. He filed an appeal with the Tenth Circuit Court of Appeals which was denied. He was accused of violating his terms by allegedly associating with the vegan straightedge after allegedly giving an interview to an apparel company.

In 1882, Lindemann published the result for which he is best known, the transcendence of . His methods were similar to those used nine years earlier by Charles Hermite to show that e, the base of natural logarithms, is transcendental. Before the publication of Lindemann's proof, it was known that if was transcendental, then it would be impossible to square the circle by compass and straightedge.

In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible. , p. 185.

Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked ruler or a protractor, because these were a workman's tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 BC), Plato's greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful.P. Hummel, "Solid constructions using ellipses", The Pi Mu Epsilon Journal, 11(8), 429 -- 435 (2003) The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions.

The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but F_5 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular n-gon is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes. Likewise, a regular n-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2^a3^b+1. It is possible to partition any convex polygon into n smaller convex polygons of equal area and equal perimeter, when n is a power of a prime number, but this is not known for other values of n.

Section IV develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary quadratic forms. Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible, i.e., can be constructed with a compass and unmarked straightedge alone.

Although successful in solving Apollonius' problem, van Roomen's method has a drawback. A prized property in classical Euclidean geometry is the ability to solve problems using only a compass and a straightedge. Many constructions are impossible using only these tools, such as dividing an angle in three equal parts. However, many such "impossible" problems can be solved by intersecting curves such as hyperbolas, ellipses and parabolas (conic sections).

According to Berthiaume, "we don't want to act a straightedge band. It wouldn't be possible because we have members who like drinking and eating meat. But the most important thing is that people wake up and think about this topic". "Eternal Destination" reflects on how mankind has ruined Earth, the environment, the animals and itself in the past decades, but the track also shows a glimmer of hope.

Where's the Pope? were a hardcore punk band from Adelaide, South Australia formed in 1985 with mainstays Frank Pappagalo on vocals and Robert Stafford on bass guitar. In June 1988 they issued Straightedge Holocaust on Reactor Records and in March 1990, after some line-up changes, they released Sunday Afternoon BBQ's on Greasy Pop Records. In March 1999 their final album, PSI appeared on Resist Records before they disbanded that year.

T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (ca. 440 BC) was the first to put compass- and-straightedge constructions above neuseis. The principle to avoid neuseis whenever possible may have been spread by Hippocrates of Chios (ca. 430 BC), who originated from the same island as Oenopides, and who was--as far as we know--the first to write a systematically ordered geometry textbook.

He published a systematic treatment of geometrical constructions (with straightedge and compass) in 1880. A French translation was reprinted in 1990. A special issue of Discrete Mathematics has been dedicated to the 150th birthday of Petersen. Petersen, as he claimed, had a very independent way of thinking. In order to preserve this independence he made a habit to read as little as possible of other people’s mathematics, pushing it to extremes.

For any nonzero integer , an angle of measure radians can be divided into equal parts with straightedge and compass if and only if is either a power of or is a power of multiplied by the product of one or more distinct Fermat primes, none of which divides . In the case of trisection (, which is a Fermat prime), this condition becomes the above- mentioned requirement that not be divisible by .

The charge is a unique one in that it is sui generis, meaning it is neither a felony nor a misdemeanor. He was sentenced on November 3, 2010, to 10 months in prison with 3 years of probation upon release. He filed an appeal with the Tenth Circuit Court of Appeals which was denied. He was accused of violating his terms by allegedly associating with the vegan straightedge.

Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).

A T-square is a straightedge which uses the edge of the drawing board as a support. It is used with the drafting board to draw horizontal lines and to align other drawing instruments. Wooden, metal, or plastic triangles with 30° and 60° angles or with two 45° angles are used to speed drawing of lines at these commonly used angles. A continuously adjustable 0–90° protractor is also in use.

The Ancient Tradition of Geometric Problems is a book on ancient Greek mathematics, focusing on three problems now known to be impossible if one uses only the straightedge and compass constructions favored by the Greek mathematicians: squaring the circle, doubling the cube, and trisecting the angle. It was written by Wilbur Knorr (1945–1997), a historian of mathematics, and published in 1986 by Birkhäuser. Dover Publications reprinted it in 1993.

Johnson, p. 112U.S. Army, Map Reading and Land Navigation, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993), ch. 11, pp. 1–3: Any 'floating card' type compass with a straightedge or centerline axis can be used to read a map bearing by orienting the map to magnetic north using a drawn magnetic azimuth, but the process is far simpler with a protractor compass.

Because the cotangent function is invariant under negation of its argument, and has a simple pole at each multiple of , the quadratrix has reflection symmetry across the y axis, and similarly has a pole for each value of x of the form x = 2na, for integer values of n, except at x = 0 where the pole in the cotangent is canceled by the factor of x in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the y axis has y = 2a/; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of 1/, leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge.

To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a compass alone, as these are the foundations of, or elementary steps for, all other constructions. These are: #Creating the line through two existing points #Creating the circle through one point with centre another point #Creating the point which is the intersection of two existing, non-parallel lines #Creating the one or two points in the intersection of a line and a circle (if they intersect) #Creating the one or two points in the intersection of two circles (if they intersect). #1 - A line through two points It is understood that a straight line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it.

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.

The general problem of angle trisection is solvable by using additional tools, and thus going outside of the original Greek framework of compass and straightedge. Many incorrect methods of trisecting the general angle have been proposed. Some of these methods provide reasonable approximations; others (some of which are mentioned below) involve tools not permitted in the classical problem. The mathematician Underwood Dudley has detailed some of these failed attempts in his book The Trisectors.

Five is the third prime number. Because it can be written as , five is classified as a Fermat prime; therefore, a regular polygon with 5 sides (a regular pentagon) is constructible with compass and an unmarked straightedge. Five is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third factorial prime, also an alternating factorial.

Lorenzo Mascheroni Lorenzo Mascheroni (May 13, 1750 – July 14, 1800) was an Italian mathematician. He was born near Bergamo, Lombardy. At first mainly interested in the humanities (poetry and Greek language), he eventually became professor of mathematics at Pavia. La geometria del compasso, 1797 In his Geometria del Compasso (Pavia, 1797), he proved that any geometrical construction which can be done with compass and straightedge, can also be done with compasses alone.

That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon. A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes.

A scoring knife or scoring tool is a handheld tool used to cut a groove in a sheet of material. The cutting edge of the knife is often made of hard material such tungsten carbide. A Glass cutter is a type of scoring tool specialized for cutting glass. The scoring knife is drawn across the material in a straight line (with the help of a straightedge), creating a scratch or score in the sheet.

Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle. He proved that the lune bounded by the arcs labeled E and F in the figure has the same area as triangle ABO. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles.

The International Superheroes of Hardcore is a side project of all members of the band and features Gilbert on vocals and Pundik on guitar, with the remaining members playing the same instruments they play in New Found Glory. All the members use pseudonyms for their "characters" in the band (e.g. Gilbert is known as "Captain Straightedge"). The band also recorded an internet-only music video for "Dig My Own Grave" with director Joseph Pattisall.

Additionally, it was also ported to Microsoft Windows. KSEG is a tool designed to let you easily visualize dynamic properties of compass and straightedge construction and to make geometric exploration as fast and easy as possible. KSEG was inspired by the Geometer's Sketchpad, but it goes beyond the functionality that Sketchpad provides. KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX.

The 'blueprint-like' "tomb-plan" is not a draft plan of the tomb construction, but is a post-construction record. Notes in Hieratic name rooms, with dimensions. The composition of straight lines, (from a straightedge/device) use mostly 90 degree angles, but the design layout also conforms to the linear shape of the sherd, (thus requiring deviations from the 90 degree right angles). As a linear sherd, the ostracon is broken into four contiguous pieces.

All of these are problems in Euclidean construction, and Euclidean constructions can be done only if they involve only Euclidean numbers (by definition of the latter) (Hardy and Wright p. 159). Irrational numbers can be Euclidean. A good example is the irrational number the square root of 2. It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be constructed with straightedge and compass.

For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.

In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g.

Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness. It is known that is a transcendental number: it is not the root of any polynomial with rational coefficients. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations.

Squaring the circle, proposed by ancient geometers, is the problem of constructing a square with the same area as a given circle, by using only a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a transcendental number rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

Lemoine's system of constructions, the Géométrographie, attempted to create a methodological system by which constructions could be judged. This system enabled a more direct process for simplifying existing constructions. In his description, he listed five main operations: placing a compass's end on a given point, placing it on a given line, drawing a circle with the compass placed upon the aforementioned point or line, placing a straightedge on a given line, and extending the line with the straightedge.Lemoine, Émile.

Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been, and remains, a rich source of inspiration. Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically.

These problems can be settled using the field of constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within .

In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem. Another example are Gaussian integers, that is, numbers of the form , where and are integers, which can be used to classify sums of squares.

Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.

Doubling the cube is the construction, using only a straight-edge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. 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. This follows because its minimal polynomial over the rationals has degree 3\. This construction is possible using a straightedge with two marks on it and a compass.

Finding the area under a curve, known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag. 4 for squaring Curve lines Geometrically" (emphasis added).

Any doubts about this construction would equally apply to traditional constructions that do involve a straightedge. #5 - Intersection of two circles This construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle (given below), it can always be assumed that any circle is described by its center and radius. Indeed, some authors include this in their descriptions of the basic constructions.

As described below, Apollonius' problem has ten special cases, depending on the nature of the three given objects, which may be a circle (C), line (L) or point (P). By custom, these ten cases are distinguished by three letter codes such as CCP. Viète solved all ten of these cases using only compass and straightedge constructions, and used the solutions of simpler cases to solve the more complex cases. Figure 4: Tangency between circles is preserved if their radii are changed by equal amounts.

X-Acto knife Graphical and model-making scalpels tend to have round handles, with textured grips (either knurled metal or soft plastic). The blade is usually flat and straight, allowing it to be run easily against a straightedge to produce straight cuts. There are many kinds of graphic arts blades; the most common around the graphic design studio is the #11 blade which is very similar to a #11 surgical blade (q.v.). Other blade shapes are used for wood carving, cutting leather and heavy fabric, etc.

Today, artisans using traditional techniques use a pair of dividers to leave an incision mark on a paper sheet that has been left in the sun to make it brittle. Straight lines are drawn with a pencil and an unmarked straightedge. Girih patterns made this way are based on tessellations, tiling the plane with a unit cell and leaving no gaps. Because the tiling makes use of translation and rotation operations, the unit cells need to have 2-, 3-, 4- or 6-fold rotational symmetry.

If the problem of the quadrature of the circle could be solved using only compass and straightedge, then would have to be an algebraic number. Johann Heinrich Lambert conjectured that was not algebraic, that is, a transcendental number, in 1761. He did this in the same paper in which he proved its irrationality, even before the general existence of transcendental numbers had been proven. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of and so showed the impossibility of this construction.

It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that is a transcendental number. The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.

A finger placed at a point along the drawing implement can be used to compare that dimension with other parts of the image. A ruler can be used both as a straightedge and a device to compute proportions. Variation of proportion with age When attempting to draw a complicated shape such as a human figure, it is helpful at first to represent the form with a set of primitive volumes. Almost any form can be represented by some combination of the cube, sphere, cylinder, and cone.

Some classical construction problems of geometry — namely trisecting an arbitrary angle or doubling the cube — are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds. Paper fold strips can be constructed to solve equations up to degree 4. The Huzita–Justin axioms or Huzita–Hatori axioms are an important contribution to this field of study. These describe what can be constructed using a sequence of creases with at most two point or line alignments at once.

He also breaks with the Greek tradition of associating powers with geometric referents, with an area, with a volume and so on, and treats them all as possible lengths of line segments. These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass. The bulk of the remainder of this book is occupied by Descartes's solution to "the locus problems of Pappus."Pappus discussed the problems in his commentary on the Conics of Apollonius.

257 is a prime number of the form 2^{2^n}+1, specifically with n = 3, and therefore a Fermat prime. Thus a regular polygon with 257 sides is constructible with compass and unmarked straightedge. It is currently the second largest known Fermat prime.. It is also a balanced prime, an irregular prime, a prime that is one more than a square, and a Jacobsthal–Lucas number. There are exactly 257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes).

The book consists of six chapters, the first of which introduces the problem, sets it in the context of the investigation of the mathematical strength of straightedge and compass constructions, and introduces one of the major themes of the book, the relegation of paper folding to recreational mathematics as this sort of investigation fell out of favor among professional mathematicians, and its more recent resurrection as a serious topic of investigation. As a work of history, the book follows Hans-Jörg Rheinberger in making a distinction between epistemic objects, the not-yet-fully-defined subjects of scientific investigation, and technical objects, the tools used in these investigations, and it links the perceived technicality of folding with its fall from mathematical favor. The remaining chapters are organized chronologically, beginning in the 16th century and the second chapter. This chapter includes the work of Albrecht Dürer on polyhedral nets, arrangements of polygons in the plane that can be folded to form a given polyhedron, and of Luca Pacioli on the use of folding to replace the compass and straightedge in geometric constructions; it also discusses the history of paper, and paper folding in the context of bookbinding.

Using the labeling in the illustration, construct the segments , , and a semicircle over (center at the midpoint ), which intersects the perpendicular line through in a point , at a distance of exactly h=\sqrt p from when has length one. Not all real numbers are constructible. It can be shown that \sqrt[3] 2 is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks.

In the mid-1970s, advances in the technologies of astronomical observations – radio, infrared, and X-ray astronomy – opened up the Universe of exploration. New tools became necessary. In this book, Hawking and Ellis attempt to establish the axiomatic foundation for the geometry of four-dimensional spacetime as described by Albert Einstein's general theory of relativity and to derive its physical consequences for singularities, horizons, and causality. Whereas the tools for studying Euclidean geometry were a straightedge and a compass, those needed to investigate curved spacetime are test particles and light rays.

In geometry, the Philo line is a line segment defined from an angle and a point. The Philo line for a point P that lies inside an angle with edges d and e is the shortest line segment that passes through P and has its endpoints on d and e. Also known as the Philon line, it is named after Philo of Byzantium, a Greek writer on mechanical devices, who lived probably during the 1st or 2nd century BC. The Philo line is not, in general, constructible by compass and straightedge.

One hundred years after him Euclid too shunned neuseis in his very influential textbook, The Elements. The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were: #constructions with straight lines and circles only (compass and straightedge); # constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas); # constructions that needed yet other means of construction, for example neuseis.

Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red) Construction of the regular 17-gon Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).

This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons. Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem: :A regular n-gon (that is, a polygon with n sides) can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes (including none).

The last name Páez Vilaró is recognized worldwide for the work of her father Carlos, and because her brother Carlos is a survivor of the 1972 Andes flight disaster. She is the granddaughter of an Argentine woman from Rosario. She has taken the Camino de Santiago road several times, as well as the Inca Trail, and has hiked to the Interior of Uruguay on foot. Together with her two brothers, she helped her father to build Casapueblo, which would be his home, with wavy lines, without straightedge, plumb bob, or level.

Note that while a tomahawk is constructible with compass and straightedge, it is not generally possible to construct a tomahawk in any desired position. Thus, the above construction does not contradict the nontrisectibility of angles with ruler and compass alone. The tomahawk produces the same geometric effect as the paper-folding method: the distance between circle center and the tip of the shorter segment is twice the distance of the radius, which is guaranteed to contact the angle. It is also equivalent to the use of an architects L-Ruler (Carpenter's Square).

Pouring a Slab-on-grade foundation A concrete finisher is a skilled tradesperson who works with concrete by placing, finishing, protecting and repairing concrete in engineering and construction projects. Concrete finishers are often responsible for setting the concrete forms, ensuring they have the correct depth and pitch. Concrete finishers place the concrete either directly from the concrete wagon chute, concrete pump, concrete skip or wheelbarrow. They spread the concrete using shovels and rakes, sometimes using a straightedge back and forth across the top of the forms to screed or level the freshly placed concrete.

In 2007, the physicists Peter J. Lu and Paul J. Steinhardt suggested that girih tilings possess properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings, predating them by five centuries.Supplemental figures This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th-century Persian scrolls. There is no indication of how much more the architects may have known about the mathematics involved. It is generally believed that such designs were constructed by drafting zigzag outlines with only a straightedge and a compass.

This can be done by placing a straightedge across the front of the wheels and adjusting until each wheel touches. Rotate the wheels with the blade in position and properly tensioned and check that the tracking is correct. Now install the blade guide rollers and leave a gap of about 1 mm between the back of the blade and the guide flange. The teeth of blades that have become narrow through repeated sharpening will foul the front edge of the guide rollers due to their kerf set and force the blade out of alignment.

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 (including none). (A Fermat prime is a prime number of the form 2^{(2^n)}+1.) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.

The compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass.

Chalk line tool. A chalk line or chalk box is a tool for marking long, straight lines on relatively flat surfaces, much farther than is practical by hand or with a straightedge. They may be used to lay out straight lines between two points, or vertical lines by using the weight of the line reel as a plumb line. It is an important tool in carpentry, the working of timber in a rough and unplaned state, as it does not require the timber to have a straight or squared edge formed onto it beforehand.

Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and- straightedge operations or by performing the operations in certain non- Euclidean geometries also makes squaring the circle possible in some sense. For example, the quadratrix of Hippias provides the means to square the circle and also to trisect an arbitrary angle, as does the Archimedean spiral. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms.

Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling the cube and squaring the circle.

Friedrich Julius Richelot (6 November 1808 – 31 March 1875) was a German mathematician, born in Königsberg. He was a student of Carl Gustav Jacob Jacobi. He was promoted in 1831 at the Philosophical Faculty of the University of Königsberg with a dissertation on the division of the circle into 257 equal parts (see references) and was a professor there. Richelot authored numerous publications in German, French and Latin, among them — with his 1832 dissertation — the first known guide to the Euclidean construction of the regular 257-gon with compass and straightedge.

This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number that is not zero or one, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem.

The effort quickly failed due to personality conflicts (especially the ongoing debate among group members as to whether or not cooked food was natural enough) and a distinct lack of required agricultural and engineering skills. This self-defeating arc reflects tensions in similarly idealistic communes of the 1960s and 1970s during the History of the hippie movement and the back-to-the-land movement. Hardline grew out of straight edge. The original logo of the movement was an outline of a large "X" (a sign associated with straightedge) with two crossed M16 rifles inside it.

The bill, written by the crank Edward J. Goodwin, does imply various incorrect values of , such as 3.2. The bill never became law, due to the intervention of Professor C. A. Waldo of Purdue University, who happened to be present in the legislature on the day it went up for a vote. The impossibility of squaring the circle using only compass and straightedge constructions, suspected since ancient times, was rigorously proven in 1882 by Ferdinand von Lindemann. Better approximations of than those implied by the bill have been known since ancient times.

A putto sits atop a millstone (or grindstone) with a chip in it. He scribbles on a tablet, or perhaps a burin used for engraving; he is generally the only active element of the picture.e.g., Klibansky, Panofsky & Saxl, 321 Attached to the structure is a balance scale above the putto, and above Melancholy is a bell and an hourglass with a sundial at the top. Numerous unused tools and mathematical instruments are scattered around, including a hammer and nails, a saw, a plane, pincers, a straightedge, a molder's form, and either the nozzle of a bellows or an enema syringe (clyster).

Impossibility theorems are usually expressible as negative existential propositions, or universal propositions in logic (see universal quantification for more). Perhaps one of the oldest proofs of impossibility is that of the irrationality of square root of 2, which shows that it is impossible to express the square root of 2 as a ratio of integers. Another famous proof of impossibility was the 1882 proof of Ferdinand von Lindemann, showing that the ancient problem of squaring the circle cannot be solved, because the number is transcendental (i.e., non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge.

Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems. Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results.

Squaring the circle: the areas of this square and this circle are both equal to . In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. Ukrainian copyright certificate for a "proof" of Fermat's Last Theorem Pseudomathematics, or mathematical crankery, is a form of mathematics-like activity that aims at advancing a set of questionable beliefs that do not adhere to the framework of rigor of formal mathematical practice. Pseudomathematics has equivalents in other scientific fields, such as pseudophysics, and overlaps with these to some extent.

Restating the Gauss- Wantzel theorem: :A regular n-gon is constructible with straightedge and compass if and only if n = 2kp1p2...pt where k and t are non-negative integers, and the pi's (when t > 0) are distinct Fermat primes. The five known Fermat primes are: :F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 . Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides. The next twenty-eight Fermat numbers, F5 through F32, are known to be composite.

Thus the regular chiliagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three. Therefore, construction of a chiliagon requires other techniques such as the quadratrix of Hippias, Archimedean spiral, or other auxiliary curves. For example, a 9° angle can first be constructed with compass and straightedge, which can then be quintisected (divided into five equal parts) twice using an auxiliary curve to produce the 0.36° internal angle required.

It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as part of his solution of the Delian problem (Duplicating the cube).According to Plutarch this solution was rejected by Plato on the grounds that it could not be achieved using only straightedge and compass, however this interpretation of Plutarch's statement has come under criticism. His work did not survive, not even the names he used for these curves, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today.

The problem of constructing an angle of a given measure is equivalent to constructing two segments such that the ratio of their length is . From a solution to one of these two problems, one may pass to a solution of the other by a compass and straightedge construction. The triple-angle formula gives an expression relating the cosines of the original angle and its trisection: = . It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a cubic polynomial.

Richeson is the author of the book Euler's Gem: The Polyhedron Formula and the Birth of Topology (Princeton University Press, 2008; paperback, 2012), on the Euler characteristic of polyhedra. The book won the 2010 Euler Book Prize of the Mathematical Association of America. His second book, Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity (Princeton University Press, 2019), concerns four famous problems of straightedge and compass construction, unsolved by the ancient Greek mathematicians and now known to be impossible: doubling the cube, squaring the circle, constructing regular polygons of any order, and trisecting the angle.

Fiberglass duct board panels provide built-in thermal insulation and the interior surface absorbs [sound], helping to provide quiet operation of the HVAC system. The duct board is formed by sliding a specially-designed knife along the board using a straightedge as a guide. The knife automatically trims out a groove with 45° sides which does not quite penetrate the entire depth of the duct board, thus providing a thin section acting as a hinge. The duct board can then be folded along the groove to produce 90° folds, making the rectangular duct shape in the fabricator's desired size.

Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up. The problem was even mentioned in Aristophanes's play The Birds. It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).

In the second book, called On the Nature of Curved Lines, Descartes described two kinds of curves, called by him geometrical and mechanical. Geometrical curves are those which are now described by algebraic equations in two variables, however, Descartes described them kinematically and an essential feature was that all of their points could be obtained by construction from lower order curves. This represented an expansion beyond what was permitted by straightedge and compass constructions. Other curves like the quadratrix and spiral, where only some of whose points could be constructed, were termed mechanical and were not considered suitable for mathematical study.

Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions.

Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.

Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction. The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more challenging to locate the midpoint using only a compass, but it is still possible according to the Mohr-Mascheroni theorem.

It is now known that none of these problems can be solved by compass and straightedge, but Knorr argues that emphasizing these impossibility results is an anachronism due in part to the foundational crisis in 1930s mathematics.Review of both The Ancient Tradition of Geometric Problems and Textual Studies in Ancient and Medieval Geometry by Thomas Drucker (1991), Isis 82: 718–720, . Instead, Knorr argues, the Greek mathematicians were primarily interested in how to solve these problems by whatever means they could, and viewed theorem and proofs as tools for problem- solving more than as ends in their own right. ;Textual Studies in Ancient and Medieval Geometry.

The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,... Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.

This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2.

The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories.T.L. Heath, "A History of Greek Mathematics, Volume I" This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above) has a planar construction.

Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular -gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if is a prime number, then a regular -gon can be constructed if and only if is a Fermat prime; in other words if \varphi(p)=p-1=2^k is a power of 2. For and primitive roots of unity admit a simple expression via square root of three, namely: : , Hence, both corresponding cyclotomic fields are identical to the quadratic field Q().

Rulers. The displayed ones are marked — an ideal straightedge is un-marked Compasses Pierre Wantzel published a proof of the impossibility of classically trisecting an arbitrary angle in 1837. Wantzel's proof, restated in modern terminology, uses the abstract algebra of field extensions, a topic now typically combined with Galois theory. However Wantzel published these results earlier than Galois (whose work was published in 1846) and did not use the connection between field extensions and groups that is the subject of Galois theory itself.For the historical basis of Wantzel's proof in the earlier work of Ruffini and Abel, and its timing vis-a-vis Galois, see .

Shah-i-Zinda in Samarkand, Uzbekistan Girih (, "knot", also written gereh) are decorative Islamic geometric patterns used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern. Girih decoration is believed to have been inspired by Syrian Roman knotwork patterns from the second century. The earliest girih dates from around 1000 CE, and the artform flourished until the 15th century. Girih patterns can be created in a variety of ways, including the traditional straightedge and compass construction; the construction of a grid of polygons; and the use of a set of girih tiles with lines drawn on them: the lines form the pattern.

By the 15th century, some girih patterns were no longer periodic, and may have been constructed using girih tiles. This method is based on a set of five tiles with lines drawn on them; when used to tile the plane with no gaps, the lines on the tiles form a girih pattern. It is not yet known when girih tiles were first used for architectural decoration instead of compass and straightedge, but it was probably at the start of the 13th century.Lu and Steinhardt, Supplementary figures Methods of ornamentation were extremely diverse, however, and the idea that one method was used for all of them has been criticised as anachronistic.

The girih patterns on the Darb-e Imam shrine built in 1453 at Isfahan had a much more complex pattern than any previously seen. The details of the pattern indicate that girih tiles, rather than compass and straightedge, were used for decorating the shrine. The patterns appear aperiodic; within the area on the wall where they are displayed, they do not form a regularly repeating pattern; and they are drawn at two different scales. A large-scale pattern is discernible when the building is viewed from a distance, and a smaller-scale pattern forming part of the larger one can be seen from closer up.

As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge- bisection of a regular pentagon.. An alternative (but similar) method is as follows: #Construct a pentagon in a circle by one of the methods shown in constructing a pentagon. #Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. #The five corners of the pentagon constitute alternate corners of the decagon.

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi () is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

A chalk line or chalk box is a tool for marking long, straight lines on relatively flat surfaces, much farther than is practical by hand or with a straightedge. It is an important tool in construction and carpentry, the working of timber in a rough and unplaned state, as it does not require the timber to have a straight or squared edge formed onto it beforehand. A chalk line draws straight lines by the action of a taut nylon or similar string that has been previously coated with a loose dye, usually chalk. The string is then laid across the surface to be marked and pulled tight.

Then, in 1796, an eighteen-year-old student named Carl Friedrich Gauss announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass. Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular -gon. Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein, attributed this part of the proof to him as well.

The Canadian Loonie dollar coin uses another Reuleaux polygon with 11 sides.. Similar methods can be used to enclose an arbitrary simple polygon within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge.. Although the regular-polygon based Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes based on irregular polygons that have an even number of sides..

Bisection of an angle using a compass and straightedge An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The interior or internal bisector of an angle is the line, half-line, or line segment that divides an angle of less than 180° into two equal angles. The exterior or external bisector is the line that divides the supplementary angle (of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.

Given the lack of surviving objects, we cannot know how common the techniques employed were, but the quality of the execution suggests that the binder was experienced in them.Regemorter, 44–45 At the same time, an analysis by Robert Stevick suggests that the designs for both covers were intended to follow a sophisticated geometric scheme of compass and straightedge constructions using the "two true measures of geometry", the ratio between Pythagoras' constant and one, and the golden section. However slips in the complicated process of production, some detailed below, mean that the finished covers do not quite exhibit the intended proportions, and are both slightly out of true in some respects.

Growing up in the East Bay straightedge and hardcore community in San Ramon and Berkeley, he was involved in the San Francisco scene of the early eighties prior to the East Bay becoming popular in the mid to late eighties as well. Working as a volunteer in almost every capacity at the Gilman Street Project, Andrew was closely associated with the club. He attended the first show and then regularly for the next ten years. A close friend of Tim Armstrong of Operation Ivy, Andrew would later live with him and Brett Reed at the Adeline Street house just as Screw 32 was becoming popular and Rancid was increasing its base in the early and mid nineties.

360px In this figure, segments PU and VT are of equal length, and RV is perpendicular to TU. These properties can be used as part of an equivalent alternative definition for the Philo line for a point P and angle edges d and e: it is a line segment connecting d to e through P such that the distance along the segment from P to d is equal to the distance along the segment from V to e, where V is the closest point on the segment to the corner point of the angle. Since doubling the cube is impossible with compass and straightedge, it is similarly impossible to construct the Philo line with these tools.

The spiral of Theodorus: A construction for line segments with lengths whose ratios are the square root of a positive integer One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number) can be constructed using a straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement.

The ancient Greek mathematicians first attempted straightedge and compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and a regular polygon with 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides.Bold, Benjamin.

Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them.Conway, John H. and Richard Guy: The Book of Numbers The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some quintics that are not solvable using radicals.A. Baragar, "Constructions using a Twice-Notched Straightedge", The American Mathematical Monthly, 109 (2), 151 -- 164 (2002).

In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, , the cube root of two. This is because a cube of side length 1 has a volume of , and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2. The impossibility of doubling the cube is therefore equivalent to the statement that is not a constructible number. This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge are roots of polynomials over the field generated by the coordinates of previous points, of no greater degree than a quadratic.

Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a Neusis construction, i.e., that uses tools other than an un-marked straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees. This requires three facts from geometry (at right): # Any full set of angles on a straight line add to 180°, # The sum of angles of any triangle is 180°, and, # Any two equal sides of an isosceles triangle will meet the third in the same angle.

Parabolic compass designed by Leonardo da Vinci The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola. The name "parabola" is due to Apollonius, who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved.

A drafting table Old-fashioned technical drawing instruments Stencils for lettering technical drawings to DIN standards The basic drafting procedure is to place a piece of paper (or other material) on a smooth surface with right-angle corners and straight sides—typically a drawing board. A sliding straightedge known as a T-square is then placed on one of the sides, allowing it to be slid across the side of the table, and over the surface of the paper. "Parallel lines" can be drawn simply by moving the T-square and running a pencil or technical pen along the T-square's edge. The T-square is used to hold other devices such as set squares or triangles.

The mathematical proof that the quadrature of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway. Having squared the circle is a famous crank assertion. (See also pseudomathematics.) In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim that was refuted by John Wallis as part of the Hobbes–Wallis controversy. During the 18th and 19th century, the notion that the problem of squaring the circle was somehow related to the longitude problem seems to have become prevalent among would-be circle squarers.

A geometrical hexafoil. One common form of the hexafoil has a ring of six tangent circles circumscribed by a larger circle. It may be constructed by compass and straightedge, by drawing six circles at the centers of a regular hexagon, with diameter equal to the side of the hexagon.. The inner circles of the hexafoil have radius 1/3 that of the outer circle containing them, from which it is possible to derive the area and perimeter of the figure as a mathematical exercise.. Hexafoil in leftHexafoil framing in 236x236px Another method of drawing a hexafoil is the vesica piscis method. To do so one takes a ruler and uses it to draw a line.

In 1954, she joined the first cohort of mathematics students in the Faculty of Philosophy in Novi Sad, now part of the University of Novi Sad. She finished her studies there in 1958 and became a high school mathematics teacher in Zrenjanin. She returned to Novi Sad (newly founded as a university) in 1960, as an assistant to geometer Mileva Prvanović and in the same year was responsible for the university's first mathematics publication, of her lecture notes on straightedge and compass constructions. She traveled to Rome in 1961 to work with Lucio Lombardo-Radice then, returning to Novi Sad in 1963, defended her Ph.D., the first mathematical doctorate at Novi Sad.

In 1842 Clausen was hired by the staff of the Tartu Observatory, becoming its director in 1866-1872. Works by Clausen include studies on the stability of Solar system, comet movement, ABC telegraph code and calculation of 250 decimals of Pi (later, only 248 were confirmed to be correct). In 1840 he discovered the Von Staudt–Clausen theorem. Also in 1840 he also found two compass and straightedge constructions of lunes with equal area to a square, adding to three (including the lune of Hippocrates) known to the ancient Greek mathematician Hippocrates of Chios; it was later shown that these five lunes are the only possible solutions to this problem.. Translated from Postnikov's 1963 Russian book on Galois theory.

The ability to translate, or copy, a circle to a new center is vital in these proofs and fundamental to establishing the veracity of the theorem. The creation of a new circle with the same radius as the first, but centered at a different point, is the key feature distinguishing the collapsing compass from the modern, rigid compass. The equivalence of a collapsing compass and a rigid compass was proved by Euclid (Book I Proposition 2 of The Elements) using straightedge and collapsing compass when he, essentially, constructs a copy of a circle with a different center. This equivalence can also be established with compass alone, a proof of which can be found in the main article.

Compasses-and-straightedge constructions are used to illustrate principles of plane geometry. Although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles. The most rigorous definition of this abstract tool is the "collapsing compass"; having drawn a circle from a given point with a given radius, it disappears; it cannot simply be moved to another point and used to draw another circle of equal radius (unlike a real pair of compasses). Euclid showed in his second proposition (Book I of the Elements) that such a collapsing compass could be used to transfer a distance, proving that a collapsing compass could do anything a real compass can do.

Godfried Toussaint Godfried Theodore Patrick Toussaint (1944 – July 2019) was a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD)New York University Abu Dhabi in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition (k-nearest neighbor algorithm, cluster analysis), motion planning, visualization (computer graphics), knot theory (stuck unknot problem), linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality (unimodal function), and others. Other interests included meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory.

No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of n sides to be constructible. In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary. Then in 1882 Lindemann showed that \pi is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.

Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.

The vesica piscis is the intersection of two congruent disks, each centered on the perimeter of the other. The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other.. In Latin, "vesica piscis" literally means "bladder of a fish", reflecting the shape's resemblance to the conjoined dual air bladders ("swim bladder") found in most fish. In Italian, the shape's name is mandorla ("almond"). Euclid's Elements This figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge.

In the last section of the DisquisitionesGauss, DA. The 7th § is arts. 336–366Gauss proved if satisfies certain conditions then the -gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the -gon is constructible, then must satisfy Gauss's conditions Gauss provesGauss, DA, art 366 that a regular -gon can be constructed with straightedge and compass if is a power of 2. If is a power of an odd prime number the formula for the totient says its totient can be a power of two only if is a first power and is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537.

Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.

Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red) Carl Friedrich Gauss developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons. Gauss stated that this condition was also necessary, but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem: : An n-sided regular polygon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and distinct Fermat primes: in other words, if and only if n is of the form n = 2kp1p2…ps, where k is a nonnegative integer and the pi are distinct Fermat primes.

In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two,Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover, while doubling a cube requires the solution of a third-order equation. Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).

Euclides Danicus (the Danish Euclid) is one of three books of mathematics written by Georg Mohr.. It was published in 1672 simultaneously in Copenhagen and Amsterdam, in Danish and Dutch respectively. It contains the first proof of the Mohr–Mascheroni theorem, which states that every geometric construction that can be performed using a compass and straightedge can also be done with compass alone.. The book is divided into two parts. In the first part, Mohr shows how to perform all of the constructions of Euclid's Elements using a compass alone. In the second part, he includes some other specific constructions, including some related to the mathematics of the sundial.. Euclides Danicus languished in obscurity, possibly caused by its choice of language, until its rediscovery in 1928 in a bookshop in Copenhagen.

In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw circles of equal radii and different centers. The segment is bisected by drawing intersecting circles of equal radius, whose centers are the endpoints of the segment and such that each circle goes through one endpoint. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment, since it crosses the segment at its center. This construction is in fact used when constructing a line perpendicular to a given line at a given point: drawing an arbitrary circle whose center is that point, it intersects the line in two more points, and the perpendicular to be constructed is the one bisecting the segment defined by these two points.

In isometric projection, the most commonly used form of axonometric projection in engineering drawing, the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. As the distortion caused by foreshortening is uniform, the proportionality between lengths is preserved, and the axes share a common scale; this eases the ability to take measurements directly from the drawing. Another advantage is that 120° angles are easily constructed using only a compass and straightedge. In dimetric projection, the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction is determined separately.

The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century: The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and easily expressed as an algorithm. Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction.

During his process, he observes a variety of oddities of Vermeer's work that he attributes to the theory of Vermeer having mechanical help: He notes Vermeer's hyper-accurate recreation of diffuse lighting would be impossible to recreate by simple eyesight because of color constancy. He also observes that some of Vermeer's work features chromatic aberration and depth of field, two distinct features of a photographic lens but not of the human eye. While painting the virginal, he accidentally notices that while he used a straightedge to roughly sketch out the outline of the instrument, the curvature of the lens almost caused him to add a slight curvature to the virginal's seahorse-pattern itself. Curious, he looks at a print of the original painting and notices that the original painting has the same curvature in the pattern.

Once the finite set of points on the cissoid have been drawn, then line PC will probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes). One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical -- axiomatic -- consistency.

A schema for horizontal dials is a set of instructions used to construct horizontal sundials using compass and straightedge construction techniques, which were widely used in Europe from the late fifteenth century to the late nineteenth century. The common horizontal sundial is a geometric projection of an equatorial sundial onto a horizontal plane. The special properties of the polar-pointing gnomon (axial gnomon) were first known to the Moorish astronomer Abdul Hassan Ali in the early thirteenth century and this led the way to the dial-plates, with which we are familiar, dial plates where the style and hour lines have a common root. Through the centuries artisans have used different methods to markup the hour lines sundials using the methods that were familiar to them, in addition the topic has fascinated mathematicians and become a topic of study.

Goodwin's main goal was not to measure lengths in the circle but to square it, which he interpreted literally as finding a square with the same area as the circle. He knew that Archimedes' formula for the area of a circle, which calls for multiplying the diameter by one fourth of the circumference, is not considered a solution to the ancient problem of squaring the circle. This is because the problem is to construct the area using compass and straightedge only, and Archimedes did not give a method for constructing a straight line with the same length as the circumference. Apparently, Goodwin was unaware of this central requirement; he believed that the problem with the Archimedean formula is that it gives wrong numerical results, and that a solution of the ancient problem should consist of replacing it with a "correct" formula.

Fifty-one is a pentagonal number as well as a centered pentagonal number and an 18-gonal number and a Perrin number. It is also the 6th Motzkin number, telling the number of ways to draw non-intersecting chords between any six points on a circle's boundary, no matter where the points may be located on the boundary. Since the greatest prime factor of 512 \+ 1 = 2602 is 1301, which is substantially more than 51 twice, 51 is a Størmer number. There are 51 different cyclic Gilbreath permutations on 10 elements, and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set.. Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass and straightedge, the angle is constructible, and the number cos is expressible in terms of square roots.

Threat (2006) is an independent film about a straightedge "hardcore kid" and a hip hop revolutionary whose friendship is doomed by the intolerance of their respective street tribes. It is an ensemble film of kids living in New York City in the aftermath of 9-11, each of them suffering from a sense of doom brought on by dealing with HIV, racism, sexism, class struggle, and general nihilism. The intellectual issues are played out amid an aesthetic of raw ultraviolence that has earned director Matt Pizzolo both accolades and condemnations (such as Film Threat's rave review stating "great art should assail the status quo, and that is what Pizzolo and Nisa’s film has skillfully accomplished" in contrast to Montreal Film Journal's scathing review saying the film "openly glorifies murderous revolt, literally telling the audience to go out and beat up random people, just because"). Unlike past urban dramas, the film does not outright condemn its characters' violent outbursts.

A 2-metre carpenter's ruler A ruler or rule is a tool used in, for example, geometry, technical drawing, engineering, and carpentry, to measure lengths or distances or to draw straight lines. Strictly speaking, the ruler is the instrument used to rule straight lines and the calibrated instrument used for determining length is called a measure, however common usage calls both instruments rulers and the special name straightedge is used for an unmarked rule. The use of the word measure, in the sense of a measuring instrument, only survives in the phrase tape measure, an instrument that can be used to measure but cannot be used to draw straight lines. As can be seen in the photographs on this page, a two-metre carpenter's rule can be folded down to a length of only 20 centimetres, to easily fit in a pocket, and a five-metre-long tape measure easily retracts to fit within a small housing.

Then setting one end at the top left corner (A), he lays out the dividers along the ESE ray (= two quarter-winds below the East ray, or horizontal top of the grid) and marks the spot (point B on the diagram). Then using a straightedge ruler draws a line up to the East ray, and marks the corresponding spot C. It is easy to see immediately that a right-angled triangle ABC has been created. The length BC is the alargar (distance from intended course), which can be measured as 46 miles (this can be visually seen as two grid squares plus a bit, that is 20m + 20m and a little bit which can be assessed as 6m by using the dividers and the 20m bar scale). The length AC is the avanzar (distance made good), which is 111 miles – visually, five grid squares and a bit, or (20 × 5) + 11, measured by dividers and scale again.

While Oenopides's innovations as an astronomer mainly concern practical issues, as a geometer he seems to have been rather a theorist and methodologist, who set himself the task to make geometry comply with higher standards of theoretical purity. Thus he introduced the distinction between 'theorems' and 'problems': though both are involved with the solution of an exercise, a theorem is meant to be a theoretical building block to be used as the fundament of further theory, while a problem is only an isolated exercise without further follow-up or importance. Oenopides apparently also was the author of the rule that geometrical constructions should use no other means than compass and straightedge. In this context his name adheres to two specific elementary constructions of plane geometry: first, to draw from a given point a straight line perpendicular to a given straight line; and second, on a given straight line and at a given point on it, to construct a rectilineal angle equal to a given rectilineal angle.

1040) showed that two lunes, formed on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes added together are equal to the area of the triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen.Hippocrates' Squaring of the Lune at cut-the- knot, accessed 2012-01-12.. The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle.. In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. All such lunes can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°, 180°).

Since the prime factors of 2 − 1 are exactly the five known Fermat primes, this number is the largest known odd value n for which a regular n-sided polygon is constructible using compass and straightedge. Equivalently, it is the largest known odd number n for which the angle 2\pi/n can be constructed, or for which \cos(2\pi/n) can be expressed in terms of square roots. Not only is 4,294,967,295 the largest known odd number of sides of a constructible polygon, but since constructibility is related to factorization, the list of odd numbers n for which an n-sided polygon is constructible begins with the list of factors of 4,294,967,295. If there are no more Fermat primes, then the two lists are identical. Namely (assuming 65537 is the largest Fermat prime), an odd-sided polygon is constructible if and only if it has 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, or 4294967295 sides.

Fermat factoring status by Wilfrid Keller. Thus a regular n-gon is constructible if :n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ... , while a regular n-gon is not constructible with compass and straightedge if :n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ... .