A few angles

Rulers. Note marks — our ruler must be un-marked

A compass

The problem of trisecting the angle is a classic problem of the compass and straightedge constructions of ancient Greek mathematics. Two tools are allowed: Image File history File links Angle_obtuse_acute_straight. ... Image File history File links Angle_obtuse_acute_straight. ... This article is about angles in geometry. ... Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... A variety of rulers A 2 metre carpenters rule Retractable flexible rule A ruler or rule is an instrument used in geometry, technical drawing and engineering/building to measure distances and/or to rule straight lines. ... Image File history File linksMetadata Download high resolution version (590x786, 38 KB) Compass (drafting) in mathematics and drafting, a device known as a compass (or pair of compasses) is used by mathematicians and craftsmen in geometry to draw or inscribe a circle or arc. ... Image File history File linksMetadata Download high resolution version (590x786, 38 KB) Compass (drafting) in mathematics and drafting, a device known as a compass (or pair of compasses) is used by mathematicians and craftsmen in geometry to draw or inscribe a circle or arc. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ...

1. An un-marked straightedge, and
2. a compass,

Problem: construct an angle one-third a given arbitrary angle. A straightedge is a tool similar to a ruler, but without markings. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... This article is about angles in geometry. ...

With such tools, it is in general impossible. This requires taking a cube root; see below. A proof of impossibility, sometimes called a negative proof or negative result, is a proof demonstrating that a particular problem cannot be solved, or cannot be solved in general. ... Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ...

Perspective and relationship to other problems

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. A straightedge is a tool similar to a ruler, but without markings. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Line redirects here. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... This article is about angles in geometry. ... Look up polygon in Wiktionary, the free dictionary. ... For other uses, see Square. ...

Nevertheless, three problems proved elusive, specifically:

• Trisecting the angle,
• Doubling the cube, and
• Squaring the circle

This article is about angles in geometry. ... Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... Squaring the circle: the areas of this square and this circle are equal. ...

Angles may not in general be trisected

Note, a number constructible in one step from a field K is a solution of a second-order polynomial; again, see constructible number. A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... Look up field in Wiktionary, the free dictionary A green field or paddock Field may refer to: A field is an open land area, used for growing agricultural crops. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ...

For example, the angle of π / 3 radians (60 degrees, notation 60°) cannot be trisected to 20°. Note cos(60°) = 1 / 2: Some common angles, measured in radians. ... This article describes the unit of angle. ...

If 60° could be trisected, the minimal polynomial of cos(20°) over the rationals would be second order. Note the trigonometric identity cos(3α) = 4cos³(α) − 3cos(α). Now let y = cos(20°).

By the trig identity, cos(60°) = 1 / 2 = 4y³ − 3y. So 4y³ − 3y − 1 / 2 = 0.

Multiply by two, and 8y³ − 6y − 1 = 0, or (2y)³ − 3 * (2y) − 1 = 0

Now substitute x = 2y, and x³ − 3x − 1 = 0. Let p(x) = x³ − 3x − 1.

The minimal polynomial for x (hence cos(20°)) is a factor of p(x). If p(x) has a rational root, by the rational root theorem, it must be 1 or −1, both clearly not roots. Therefore the p(x) is an irreducible polynomial, and the minimal polynomial for cos(20°) is degree 3.

So 60° = π / 3 radians cannot be trisected. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In algebra, the rational root theorem (or rational root test to find the zeros) states a constraint on solutions (or roots) to the polynomial equation an xn + an−1 xn −1 + ... + a1 x + a0 = 0 with integer coefficients. ... In algebra, the rational root theorem (or rational root test to find the zeros) states a constraint on solutions (or roots) to the polynomial equation an xn + an−1 xn −1 + ... + a1 x + a0 = 0 with integer coefficients. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ... In mathematics and physics, the radian is a unit of angle measure. ...

Some angles may be trisected

However, some angles may be trisected. For example, 2π / 5 radians (72°) may be constructed [1], and may be trisected, [2]. Also there are some angles, that are not-constructable, but (if somehow given) trisectable, e.g. 3Π / 7. Some common angles, measured in radians. ...

One general theorem

Denote the rational numbers Q: In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Theorem: The angle θ may be trisected if and only if q(t) = 4t³ − 3t − cos(θ) is reducible over the field extension Q(θ). Look up theorem in Wiktionary, the free dictionary. ... ↔ ⇔ ≡ logical symbols representing iff. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...

Proof. The proof would take us afield, but it may be derived from the above trig identity and is the content of problem 5.7 of Stewart, Ian (1989). Galois Theory. Chapman and Hall Mathematics, pg. 58. ISBN 0412345501.  Look up proof in Wiktionary, the free dictionary. ... In mathematics, trigonometric identities (or trig identities for short) are equations involving trigonometric functions that are true for all values of the occurring variables. ... Ian Stewart, FRS (b. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ...

Means to trisect angles by going outside the Greek framework

Origami

Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) operations of paper folding, or origami. Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots). See mathematics of paper folding. This article is about paper folding. ... Huzitas axioms are a set of rules related to the mathematical principles of paper folding. ... The art of paper folding or origami has received a considerable amount of mathematical study. ...

With a marked ruler

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. For other uses, see Archimedes (disambiguation). ... Neusis construction The neusis is a geometric construction method that was used in Antiquity by Greek mathematicians. ...

This requires three un-proven facts from geometry (at right):

1. Any full set of angles on a straight line add to 180°,
2. The sum of angles of any triangle is 180°, and,
3. Any two equal sides of an isosceles triangle meet the third in the same angle.

Look to the diagram at right; note angle a on left-hand side. Image File history File links Size of this preview: 800 × 243 pixelsFull resolution‎ (872 × 265 pixels, file size: 21 KB, MIME type: image/jpeg) I wrote it with Dia (software), and license it under GFDL Permission is granted to copy, distribute and/or modify this document under the terms of... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Image File history File links Size of this preview: 800 × 463 pixelsFull resolution‎ (836 × 484 pixels, file size: 25 KB, MIME type: image/jpeg) I made this with Dia (software) and release it as GFDL. File historyClick on a date/time to view the file as it appeared at that...

First, a ruler has two marks distance AB apart. Extend the lines of the angle and draw a circle of radius AB. This article is about an authentication, authorization, and accounting protocol. ...

"Anchor" the ruler at point A, and move it until one mark is at point C, one at point D, i.e., CD = AB. A radius BC is drawn as obvious.

That is to say, line segments AB, BC, and CD all have equal length. Segment AC is irrelevant.

Now: Triangles ABC and BCD are isosceles, so by Fact 3 above, re-draw the diagram, and label two more angles: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Hypothesis: Given AD is a straight line, and AB, BC, and CD are all equal length, Image File history File links Size of this preview: 800 × 463 pixelsFull resolution‎ (836 × 484 pixels, file size: 26 KB, MIME type: image/jpeg) I wrote it with Dia (software), and license it under GFDL I, the copyright holder of this work, hereby release it into the public domain. ... Look up Hypothesis in Wiktionary, the free dictionary. ...

Conclusion: angle b = 1 / 3a. A conclusion is a final proposition, which is arrived at after the consideration of evidence, arguments or premises. ...

Proof: Look up proof in Wiktionary, the free dictionary. ...

Steps:

1. From Fact 1) above, e + c = 180°.
2. Looking at triangle BCD, from Fact 2) e + 2b = 180°.
3. From the last two equations, c = 2b.
4. From Fact 2), d + 2c = 180°, thus d = 180° − 2c, so from last, d = 180° − 4b.
5. From Fact 1) above, a + d + b = 180°, thus a + (180° − 4b) + b = 180°.

Clearing, a − 3b = 0, or a = 3b, and the theorem is proved. Look up theorem in Wiktionary, the free dictionary. ... Look up QED in Wiktionary, the free dictionary. ...

Again: this construction stepped outside the framework of allowed constructions by using a marked straightedge. There is an unavoidable element of inaccuracy in placing the straightedge. Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...

There are other constructions (references).

See also

• Bisection
• Constructible number
• Constructible polygon
• Doubling the cube
• Euclidean geometry
• Galois theory
• History of geometry
• Intercept theorem
• List of geometry topics
• Neusis construction
• Squaring the circle
• Tomahawk (geometric shape)
• Trisectrix

For the bisection theorem, see ham sandwich theorem. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ... Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Table of Geometry, from the 1728 Cyclopaedia. ... The intercept theorem is an important theorem in elementary geometry about the ratios of various line segments, that are created if 2 intersecting lines are intercepted by a pair of parallels. ... This is list of geometry topics, by Wikipedia page. ... Neusis construction The neusis is a geometric construction method that was used in Antiquity by Greek mathematicians. ... Squaring the circle: the areas of this square and this circle are equal. ... The tomahawk is a geometric shape with an unknown inventor. ... Trisectrix A trisectrix is a curve which is a variety of the Limaçon of Pascal, and named from its property of angle trisection. ...

External references

• MathWorld site
• One site
• One link of marked ruler construction
• Another, mentioning Archimedes