Angle trisection - Citizendia

Rulers. The displayed ones are marked — an ideal straightedge is un-marked

Rulers. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A ruler, or rule, is an instrument used in Geometry, Technical drawing and engineering/building to measure distances and/or to rule straight The displayed ones are marked — an ideal straightedge is un-marked

A compass

The problem of trisecting the angle is a classic problem of compass and straightedge constructions of ancient Greek mathematics. A straightedge is a tool with an accurately straight edge used for drawing or cutting straight lines or checking the straightness of lines A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles 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 Two tools are allowed

An un-marked straightedge, and
a compass,

Problem: construct an angle one-third a given arbitrary angle. A straightedge is a tool with an accurately straight edge used for drawing or cutting straight lines or checking the straightness of lines A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

With such tools, it is generally impossible. 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 This requires taking a cube root, impossible with the given tools; see below. In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 = x A broader definition of a tool is an entity used to interface between two or more domains that facilitates more effective action of one domain upon the other

1 A common misunderstanding
2 Perspective and relationship to other problems
3 Angles may not in general be trisected
4 Some angles may be trisected
- 4.1 One general theorem
5 Means to trisect angles by going outside the Greek framework
6 See also
7 Notes
8 External references
- 8.1 Other means of trisection

A common misunderstanding

It is common to hear "It is impossible to trisect an angle! Q.E.D." Leaving aside the lack of proof, this statement is false: it is only impossible to solve in general using only an un-marked straightedge and a compass, i. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" e. it may be done by using other tools, and, some angles may be trisected with a straightedge and a compass.

Perspective and relationship to other problems

Bisection of arbitrary angles has long been solved. In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector For the concept of arbitrariness in trademark law see Trademark distinctiveness. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

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 with an accurately straight edge used for drawing or cutting straight lines or checking the straightness of lines A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as 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 In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

Three problems proved elusive, specifically:

Angles may not in general be trisected

Denote the rational numbers $\mathbb{Q}$ . In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Doubling the cube (also known as The Delian Problem) is one of the three most famous geometric problems unsolvable by Compass and straightedge construction Squaring the circle is a problem proposed by ancient Geometers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

Note that 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 In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations A point in the Euclidean plane is a constructible point if given a fixed Coordinate system (or a fixed Line segment of unit Length Note also that $π / 3$ radians (60 degrees, written 60°) is constructible. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 This article describes the unit of angle For other meanings see Degree. Properties The area of an equilateral triangle with sides of length a\\!

However, the angle of $π / 3$ radians (60 degrees) cannot be trisected. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 This article describes the unit of angle For other meanings see Degree. Note $\cos(\pi/3) = \cos(60^\circ) = 1/2$ .

If 60° could be trisected, the minimal polynomial of $\cos(20^\circ)$ over $\mathbb{Q}$ would be of second order. Note the trigonometric identity $cos(3α) = 4cos 3 (α) - 3cos(α)$ . In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables Now let $y = \cos(20^\circ)$ .

By the above identity, $\cos(60^\circ) = 1/2 = 4y^{3} - 3y$ . So $4 y 3 - 3 y - 1 / 2 = 0$ . Multiplying by two yields $8 y 3 - 6 y - 1 = 0$ , or $(2 y) 3 - 3(2 y) - 1 = 0$ . Now substitute $x = 2 y$ , so that $x 3 - 3 x - 1 = 0$ . Let $p (x) = x 3 - 3 x - 1$ .

The minimal polynomial for x (hence $\cos(20^\circ)$ ) is a factor of $p (x)$ . In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial If $p (x)$ has a rational root, by the rational root theorem, it must be 1 or −1, both clearly not roots. 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 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 Therefore $p (x)$ is irreducible over $\mathbb{Q}$ , and the minimal polynomial for $\cos(20^\circ)$ is of degree 3. 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 set

So an angle of $60^\circ = \pi/3$ radians cannot be trisected. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

Some angles may be trisected

However, some angles may be trisected. Given angle $θ$ , angle $3θ$ trivially trisects to $θ$ . More notably, $2π / 5$ radians (72°) may be constructed, and may be trisected, [1]. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 Also there are angles, while non-constructable, but (if somehow given) are trisectable, for example $3π / 7$

One general theorem

Again, denote the rational numbers $Q$ :

Theorem: The angle $θ$ may be trisected if and only if $q (t) = 4 t 3 - 3 t - c o s (θ)$ is reducible over the field extension $Q (c o s (θ))$ . In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements ↔ 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. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables ^[1]

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. (from oru meaning "folding" and kami meaning "paper" is the ancient Japanese Art of Paper folding. 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). The Huzita-Hatori axioms or Huzita-Justin axioms are a set of rules related to the mathematical principles of paper folding, describing the operations that can be See mathematics 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. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The neusis is a geometric construction method that was used in antiquity by Greek mathematicians e. , that uses tools other than an un-marked straightedge.

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 meet the third in the same angle. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Pons Asinorum ( Latin for "Bridge of Asses" is the name given to Euclid 's fifth proposition in Book 1 of his Elements of

Look to the diagram at right; note angle a left of point B. We trisect angle a.

First, a ruler has two marks distance AB apart. Extend the lines of the angle and draw a circle of radius AB. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

"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. Triangle BCD has two equal sides, thus is isosceles.

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, thus by Fact 3 each has two equal angles. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Now re-draw the diagram, and label all angles:

Hypothesis: Given AD is a straight line, and AB, BC, and CD are all equal length,

Conclusion: angle $b = 1 / 3 a$ . A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible A conclusion is a Proposition, which is arrived at after the consideration of Evidence, Arguments or Premises Logic

Proof:

Steps:

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

Clearing, $a - 3 b = 0$ , or $a = 3 b$ , and the theorem is proved. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated"

Again: this construction stepped outside the framework of allowed constructions by using a marked 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 Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles There is an unavoidable element of inaccuracy in placing the straightedge.

With a string

Hutcheson published an article in Mathematics Teacher, vol. 94, No. 5, May, 2001 that used a string instead of a compass and straight edge. A string can be used as either a straight edge (by stretching it) or a compass (by fixing one point and identifying another), but can also wrap around a cylinder, the key to Hutcheson's solution.

Hutcheson constructed a cylinder from the angle to be trisected by drawing an arc across the angle, completing it as a circle, and constructing from that circle a cylinder on which a, say, equilateral triangle was inscribed (a 360-degree angle divided in three). This was then "mapped" onto the angle to be trisected, with a simple proof of similar triangles.

For the detailed proof and its generalization, see the article cited: Mathematics Teacher, vol. 94, No. 5, May, 2001, pp. 400-405.

There are other constructions (references).

Notes

^ Stewart, Ian (1989). In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector A point in the Euclidean plane is a constructible point if given a fixed Coordinate system (or a fixed Line segment of unit Length In mathematics a constructible polygon is a Regular polygon that can be constructed with compass and straightedge. Doubling the cube (also known as The Delian Problem) is one of the three most famous geometric problems unsolvable by Compass and straightedge construction 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 Geometry ( Greek γεωμετρία; geo = earth metria = measure arose as the field of knowledge dealing with spatial relationships 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 This is list of Geometry topics, by Wikipedia page Geometric shape covers standard terms for plane shapes List of mathematical shapes In Plane geometry, Morley's trisector theorem states that in any Triangle, the three points of intersection of the adjacent angle trisectors form an Equilateral The neusis is a geometric construction method that was used in antiquity by Greek mathematicians In Mathematics, a quadratrix (from the Latin word quadrator squarer is a curve having Ordinates which are a measure of the area (or quadrature Squaring the circle is a problem proposed by ancient Geometers. The tomahawk is a "tool" in Geometry consisting in essence of a semi-circle and two line segments In Geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle Ian Nicholas Stewart (born 1945) is a professor of Mathematics at University of Warwick, England and a widely known popular-science and science-fiction Galois Theory. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory Chapman and Hall Mathematics, pg. 58. ISBN 0412345501.

External references

Other means of trisection

Trisecting via the limacon of Pascal; see also Trisectrix
Trisecting via an Archimedean Spiral
Trisecting via the Conchoid of Nicomedes
sciencenews.org site on using origami

In Geometry, limaçons (pronounced with a soft c) also known as limaçons of Pascal, are heart-shaped mathematical curves Blaise Pascal (blɛz paskal (June 19 1623 &ndash August 19 1662 was a French Mathematician, Physicist, and religious Philosopher In Geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician A conchoid is a Curve derived from a fixed point O, another curve and a length d. Nicomedes (ca 280 BCE - ca 210 BCE was an ancient Greek mathematician (from oru meaning "folding" and kami meaning "paper" is the ancient Japanese Art of Paper folding.

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