ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Angle trisection</title>

</head>

Angle trisection

</h1>

</header>

Angle trisection is the construction of an <a href="Angle" title="Angle">angle</a> equal to one third of a given arbitrary angle, using only two tools: an unmarked <a href="Straightedge" title="Straightedge">straightedge</a> and a <a href="Compass_(drawing_tool)" title="Compass (drawing tool)">compass</a>. It is a classical problem of <a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">straightedge and compass construction</a> of ancient <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Greek mathematics</a>.

In 1837, <a href="Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> proved that the problem, as stated, is <a href="Proof_of_impossibility" title="Proof of impossibility">impossible</a> to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a <a href="Right_angle" title="Right angle">right angle</a>.

It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, <a href="Neusis_construction" title="Neusis construction">neusis construction</a>, 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 <a href="Pseudomathematics" title="Pseudomathematics">pseudomathematical</a> attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.<a href="#cite_note-trisectors-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Background_and_problem_statement">Background and problem statement</h2></div>

Using only an unmarked <a href="Straightedge" title="Straightedge">straightedge</a> and a compass, <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Greek mathematicians</a> found means to divide a <a href="Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> into an arbitrary set of equal segments, to draw <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a> lines, to <a href="Bisection#Angle_bisector" title="Bisection">bisect angles</a>, to construct many <a href="Polygon" title="Polygon">polygons</a>, and to construct <a href="Square_(geometry)" class="mw-redirect" title="Square (geometry)">squares</a> of equal or twice the area of a given polygon.

Three problems proved elusive, specifically, trisecting the angle, <a href="Doubling_the_cube" title="Doubling the cube">doubling the cube</a>, and <a href="Squaring_the_circle" title="Squaring the circle">squaring the circle</a>. The problem of angle trisection reads:

Construct an <a href="Angle" title="Angle">angle</a> equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools:

<ol><li>an unmarked straightedge, and</li>

<li>a compass.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Proof_of_impossibility">Proof of impossibility</h2></div>

<a href="Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> published a proof of the impossibility of classically trisecting an arbitrary angle in 1837.<a href="#cite_note-2">[2]</a> Wantzel's proof, restated in modern terminology, uses the concept of <a href="Field_extension" title="Field extension">field extensions</a>, a topic now typically combined with <a href="Galois_theory" title="Galois theory">Galois theory</a>. However, Wantzel published these results earlier than <a href="%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> (whose work, written in 1830, was published only in 1846) and did not use the concepts introduced by Galois.<a href="#cite_note-3">[3]</a>

The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. 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 <a href="Triple-angle_formula" class="mw-redirect" title="Triple-angle formula">triple-angle formula</a> gives an expression relating the cosines of the original angle and its trisection: cos θ = 4 cos3 ⁠θ/3⁠ − 3 cos ⁠θ/3⁠.

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 <a href="Cubic_polynomial" class="mw-redirect" title="Cubic polynomial">cubic polynomial</a>. This equivalence reduces the original geometric problem to a purely algebraic problem.

Every rational number is constructible. Every <a href="Irrational_number" title="Irrational number">irrational number</a> that is <a href="Constructible_number" title="Constructible number">constructible</a> in a single step from some given numbers is a root of a <a href="Polynomial" title="Polynomial">polynomial</a> of degree 2 with coefficients in the <a href="Field_(mathematics)" title="Field (mathematics)">field</a> generated by these numbers. Therefore, any number that is constructible by a sequence of steps is a root of a <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> whose degree is a <a href="Power_of_two" title="Power of two">power of two</a>. The angle ⁠π/3⁠ <a href="Radian" title="Radian">radians</a> (60 <a href="Degree_(angle)" title="Degree (angle)">degrees</a>, written 60°) is <a href="Equilateral_triangle" title="Equilateral triangle">constructible</a>. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.

Denote the set of <a href="Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a> by Q. If 60° could be trisected, the degree of a minimal polynomial of cos 20° over Q would be a power of two. Now let x = cos 20°. Note that cos 60° = cos ⁠π/3⁠ = ⁠1/2⁠. Then by the triple-angle formula, cos ⁠π/3⁠ = 4x3 − 3x and so 4x3 − 3x = ⁠1/2⁠. Thus 8x3 − 6x − 1 = 0. Define p(t) to be the polynomial p(t) = 8t3 − 6t − 1.

Since x = cos 20° is a root of p(t), the minimal polynomial for cos 20° is a factor of p(t). Because p(t) has degree 3, if it is reducible over by Q then it has a <a href="Rational_root" class="mw-redirect" title="Rational root">rational root</a>. By the <a href="Rational_root_theorem" title="Rational root theorem">rational root theorem</a>, this root must be ±1, ±⁠1/2⁠, ±⁠1/4⁠ or ±⁠1/8⁠, but none of these is a root. Therefore, p(t) is <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over by Q, and the minimal polynomial for cos 20° is of degree 3.

So an angle of measure 60° cannot be trisected.

<div class="mw-heading mw-heading2"><h2 id="Angles_which_can_be_trisected">Angles which can be trisected</h2></div>

However, some angles can be trisected. For example, for any <a href="Constructible_number" title="Constructible number">constructible</a> angle θ, an angle of measure 3θ can be trivially trisected by ignoring the given angle and directly constructing an angle of measure θ. There are angles that are not constructible but are trisectible (despite the one-third angle itself being non-constructible). For example, ⁠3π/7⁠ is such an angle: five angles of measure ⁠3π/7⁠ combine to make an angle of measure ⁠15π/7⁠, which is a full circle plus the desired ⁠π/7⁠.

For a <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> N, an angle of measure ⁠2π/N⁠ is trisectible if and only if 3 does not divide N.<a href="#cite_note-4">[4]</a><a href="#cite_note-McLean-5">[5]</a> In contrast, ⁠2π/N⁠ is constructible if and only if N is a power of 2 or the product of a power of 2 with the product of one or more distinct <a href="Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat primes</a>.

<div class="mw-heading mw-heading3"><h3 id="Algebraic_characterization">Algebraic characterization</h3></div>

Again, denote the set of <a href="Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a> by Q.

<a href="Theorem" title="Theorem">Theorem</a>: An angle of measure θ may be trisected <a href="If_and_only_if" title="If and only if">if and only if</a> q(t) = 4t3 − 3t − cos(θ) is reducible over the <a href="Field_extension" title="Field extension">field extension</a> Q(cos(θ)).

The <a href="Mathematical_proof" title="Mathematical proof">proof</a> is a relatively straightforward generalization of the proof given above that a 60° angle is not trisectible.<a href="#cite_note-Stewart-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Other_numbers_of_parts">Other numbers of parts</h3></div>

For any nonzero integer N, an angle of measure <style data-mw-deduplicate="TemplateStyles:r1154941027">

/* start https://en.wikipedia.org/ */

.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style>2π⁄N radians can be divided into n equal parts with straightedge and compass if and only if n is either a power of 2 or is a power of 2 multiplied by the product of one or more distinct Fermat primes, none of which divides N. In the case of trisection (n = 3, which is a Fermat prime), this condition becomes the above-mentioned requirement that N not be divisible by 3.<a href="#cite_note-McLean-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Other_methods">Other methods</h2></div>

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 <a href="Underwood_Dudley" title="Underwood Dudley">Underwood Dudley</a> has detailed some of these failed attempts in his book The Trisectors.<a href="#cite_note-trisectors-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="Approximation_by_successive_bisections">Approximation by successive bisections</h3></div>

Trisection can be approximated by repetition of the compass and straightedge method for bisecting an angle. The geometric series ⁠1/3⁠ = <a href="1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">⁠1/4⁠ + ⁠1/16⁠ + ⁠1/64⁠ + ⁠1/256⁠ + ⋯</a> or ⁠1/3⁠ = ⁠1/2⁠ − ⁠1/4⁠ + ⁠1/8⁠ − ⁠1/16⁠ + ⋯ 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.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Using_origami">Using origami</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mathematics_of_origami" class="mw-redirect" title="Mathematics of origami">Mathematics of origami § Trisecting an angle</a></div>

Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the operations of paper folding, or <a href="Origami" title="Origami">origami</a>. <a href="Huzita's_axioms" class="mw-redirect" title="Huzita's axioms">Huzita's axioms</a> (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots).

<div class="mw-heading mw-heading3"><h3 id="Using_a_linkage">Using a linkage</h3></div>

There are a number of simple <a href="Linkage_(mechanical)" title="Linkage (mechanical)">linkages</a> which can be used to make an instrument to trisect angles including Kempe's Trisector and Sylvester's Link Fan or Isoklinostat.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="With_a_right_triangular_ruler">With a right triangular ruler</h3></div>

In 1932, <a href="Ludwig_Bieberbach" title="Ludwig Bieberbach">Ludwig Bieberbach</a> published in <a href="Crelle's_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a> his work Zur Lehre von den kubischen Konstruktionen.<a href="#cite_note-Ludwig_Bieberbach-9">[9]</a> He states therein (free translation):

<dl><dd>"As is known ... every cubic construction can be traced back to the trisection of the angle and to the multiplication of the cube, that is, the extraction of the third root. I need only to show how these two classical tasks can be solved by means of the right angle hook."</dd></dl>

The construction begins with drawing a <a href="Circle" title="Circle">circle</a> passing through the vertex P of the angle to be trisected, centered at A on an edge of this angle, and having B as its second intersection with the edge. A circle centered at P and of the same radius intersects the line supporting the edge in A and O.

Now the <a href="Set_square" title="Set square">right triangular ruler</a> is placed on the drawing in the following manner: one <a href="Cathetus" title="Cathetus">leg</a> of its right angle passes through O; the vertex of its right angle is placed at a point S on the line PC in such a way that the second leg of the ruler is tangent at E to the circle centered at A. It follows that the original angle is trisected by the line PE, and the line PD perpendicular to SE and passing through P. This line can be drawn either by using again the right triangular ruler, or by using a traditional <a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">straightedge and compass construction</a>. With a similar construction, one can improve the location of E, by using that it is the intersection of the line SE and its perpendicular passing through A.

Proof: One has to prove the angle equalities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {EPD}}={\widehat {DPS}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {EPD}}={\widehat {DPS}}}</annotation>

</semantics>

</math><img src="./7fa35fa29dd632a3405c5873fada88dcb970bafc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.713ex; height:3.176ex;" alt="{\displaystyle {\widehat {EPD}}={\widehat {DPS}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {BPE}}={\widehat {EPD}}.}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {BPE}}={\widehat {EPD}}.}</annotation>

</semantics>

</math><img src="./0ca29eed1e4809ab7d2173b31b0f0144e57e365f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.476ex; height:3.176ex;" alt="{\displaystyle {\widehat {BPE}}={\widehat {EPD}}.}" loading="lazy"> The three lines OS, PD, and AE are parallel. As the <a href="Line_segment" title="Line segment">line segments</a> OP and PA are equal, these three parallel lines delimit two equal segments on every other secant line, and in particular on their common perpendicular SE. Thus SD' = D'E, where D' is the intersection of the lines PD and SE. It follows that the <a href="Right_triangle" title="Right triangle">right triangles</a> PD'S and PD'E are congruent, and thus that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {EPD}}={\widehat {DPS}},}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {EPD}}={\widehat {DPS}},}</annotation>

</semantics>

</math><img src="./65291402945032b471697b67a831f152c95f3d65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.36ex; height:3.509ex;" alt="{\displaystyle {\widehat {EPD}}={\widehat {DPS}},}" loading="lazy"> the first desired equality. On the other hand, the triangle PAE is <a href="Isosceles_triangle" title="Isosceles triangle">isosceles</a>, since all <a href="Radius" title="Radius">radiuses</a> of a circle are equal; this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {APE}}={\widehat {AEP}}.}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {APE}}={\widehat {AEP}}.}</annotation>

</semantics>

</math><img src="./5f44464c39aba063b40e6d265821d90a90ef8141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.274ex; height:3.176ex;" alt="{\displaystyle {\widehat {APE}}={\widehat {AEP}}.}" loading="lazy"> One has also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {AEP}}={\widehat {EPD}},}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {AEP}}={\widehat {EPD}},}</annotation>

</semantics>

</math><img src="./b20037450b1c660e33caacd54cc9829bf99de2b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.455ex; height:3.509ex;" alt="{\displaystyle {\widehat {AEP}}={\widehat {EPD}},}" loading="lazy"> since these two angles are <a href="Alternate_angles" class="mw-redirect" title="Alternate angles">alternate angles</a> of a transversal to two parallel lines. This proves the second desired equality, and thus the correctness of the construction.

<div class="mw-heading mw-heading3"><h3 id="With_an_auxiliary_curve">With an auxiliary curve</h3></div>

<div class="gallerytext">Trisection using the Archimedean spiral</div>

</li>

<div class="gallerytext">Trisection using the Maclaurin trisectrix</div>

</li>

</ul>There are certain curves called <a href="Trisectrix" title="Trisectrix">trisectrices</a> which, if drawn on the plane using other methods, can be used to trisect arbitrary angles.<a href="#cite_note-10">[10]</a> Examples include the <a href="Trisectrix_of_Maclaurin" title="Trisectrix of Maclaurin">trisectrix of Colin Maclaurin</a>, given in <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> by the <a href="Implicit_curve" title="Implicit curve">implicit equation</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2}),}</annotation>

</semantics>

</math><img src="./6cc590600a4c4a60ccaad26b1617507c288fcc86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.126ex; height:3.176ex;" alt="{\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2}),}" loading="lazy"></dd></dl>

and the <a href="Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a>. The spiral can, in fact, be used to divide an angle into any number of equal parts.

Archimedes described how to trisect an angle using the Archimedean spiral in <a href="On_Spirals#Trisecting_an_angle" title="On Spirals">On Spirals</a> around 225 BC.

<div class="mw-heading mw-heading3"><h3 id="With_a_marked_ruler">With a marked ruler</h3></div>

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 <a href="Archimedes" title="Archimedes">Archimedes</a>, called a <a href="Neusis_construction" title="Neusis construction">Neusis construction</a>, 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):

<ol><li>Any full set of angles on a straight line add to 180°,</li>

<li>The sum of angles of any triangle is 180°, and,</li>

<li>Any two equal sides of an <a href="Isosceles_triangle" title="Isosceles triangle">isosceles triangle</a> will <a href="Pons_asinorum" title="Pons asinorum">meet the third side at the same angle</a>.</li></ol>

Let l be the horizontal line in the adjacent diagram. Angle a (left of point B) is the subject of trisection. First, a point A is drawn at an angle's <a href="Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">ray</a>, one unit apart from B. A circle of <a href="Radius" title="Radius">radius</a> AB is drawn. Then, the markedness of the ruler comes into play: one mark of the ruler is placed at A and the other at B. While keeping the ruler (but not the mark) touching A, the ruler is slid and rotated until one mark is on the circle and the other is on the line l. The mark on the circle is labeled C and the mark on the line is labeled D. This ensures that CD = AB. A radius BC is drawn to make it obvious that line segments AB, BC, and CD all have equal length. Now, triangles ABC and BCD are <a href="Isosceles_triangle" title="Isosceles triangle">isosceles</a>, thus (by Fact 3 above) each has two equal angles.

<a href="Hypothesis" title="Hypothesis">Hypothesis</a>: Given AD is a straight line, and AB, BC, and CD all have equal length,

<a href="Logical_consequence" title="Logical consequence">Conclusion</a>: angle b = ⁠a/3⁠.

<a href="Mathematical_proof" title="Mathematical proof">Proof</a>:

<ol><li>From Fact 1) above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e+c=180}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e+c=180}</annotation>

</semantics>

</math><img src="./2439a6f06c4338926d8745e0cd268a38067ec067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.517ex; height:2.343ex;" alt="{\displaystyle e+c=180}" loading="lazy">°.</li>

<li>Looking at triangle BCD, from Fact 2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e+2b=180}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e+2b=180}</annotation>

</semantics>

</math><img src="./d8eaa2466fa49dfe332dbb0bc8ffa2f5d5f474f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.67ex; height:2.343ex;" alt="{\displaystyle e+2b=180}" loading="lazy">°.</li>

<li>From the last two equations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=2b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=2b}</annotation>

</semantics>

</math><img src="./7870ea53f23640a56e2da808bff82c66c8b75ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.265ex; height:2.176ex;" alt="{\displaystyle c=2b}" loading="lazy">.</li>

<li>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=c+b=2b+b=3b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=c+b=2b+b=3b}</annotation>

</semantics>

</math><img src="./f41c5b47dcae05a14e3fed22df5e965ac0d19be0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.528ex; height:2.343ex;" alt="{\displaystyle a=c+b=2b+b=3b}" loading="lazy">.</li></ol>

and the <a href="Theorem" title="Theorem">theorem</a> is proved.

Again, this construction stepped outside the <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">framework</a> of <a href="Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">allowed constructions</a> by using a marked straightedge.

<div class="mw-heading mw-heading3"><h3 id="With_a_string">With a string</h3></div>

Thomas Hutcheson published an article in the <a href="Mathematics_Teacher" class="mw-redirect" title="Mathematics Teacher">Mathematics Teacher</a><a href="#cite_note-11">[11]</a> 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.

<div class="mw-heading mw-heading3"><h3 id="With_a_"tomahawk"">With a "tomahawk"</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tomahawk_(geometry)" title="Tomahawk (geometry)">Tomahawk (geometry)</a></div>

A "<a href="Tomahawk_(geometry)" title="Tomahawk (geometry)">tomahawk</a>" is a geometric shape consisting of a semicircle and two orthogonal line segments, such that the length of the shorter segment is equal to the circle radius. Trisection is executed by leaning the end of the tomahawk's shorter segment on one ray, the circle's edge on the other, so that the "handle" (longer segment) crosses the angle's vertex; the trisection line runs between the vertex and the center of the semicircle.

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.

As a tomahawk can be used as a <a href="Set_square" title="Set square">set square</a>, it can be also used for trisection angles by the method described in <a href="#With_a_right_triangular_ruler">§ With a right triangular ruler</a>.

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 (<a href="Steel_square#Carpenter's_square" title="Steel square">Carpenter's Square</a>).

<div class="mw-heading mw-heading3"><h3 id="With_interconnected_compasses">With interconnected compasses</h3></div>

An angle can be trisected with a device that is essentially a four-pronged version of a compass, with linkages between the prongs designed to keep the three angles between adjacent prongs equal.<a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Uses_of_angle_trisection">Uses of angle trisection</h2></div>

A <a href="Cubic_equation" title="Cubic equation">cubic equation</a> with real coefficients can be solved geometrically with compass, straightedge, and an angle trisector if and only if it has three <a href="Real_number" title="Real number">real</a> <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a>.<a href="#cite_note-Gleason-13">[13]</a>: Thm. 1

A <a href="Regular_polygon" title="Regular polygon">regular polygon</a> with n sides can be constructed with ruler, compass, and angle trisector if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}</annotation>

</semantics>

</math><img src="./7ae328cd2cda9ef3946fd59e24bf6ce275b3a532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.645ex; height:2.676ex;" alt="{\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}" loading="lazy"> where r, s, k ≥ 0 and where the pi are distinct primes greater than 3 of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{t}3^{u}+1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{t}3^{u}+1}</annotation>

</semantics>

</math><img src="./80b1a1e7ef9108da38a6c6e78e2c9bbcc5e88b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.326ex; height:2.676ex;" alt="{\displaystyle 2^{t}3^{u}+1}" loading="lazy"> (i.e. <a href="Pierpont_prime" title="Pierpont prime">Pierpont primes</a> greater than 3).<a href="#cite_note-Gleason-13">[13]</a>: Thm. 2

<ul><li><a href="Bisection" title="Bisection">Bisection</a></li>

<li><a href="Constructible_number" title="Constructible number">Constructible number</a></li>

<li><a href="Constructible_polygon" title="Constructible polygon">Constructible polygon</a></li>

<li><a href="Morley's_trisector_theorem" title="Morley's trisector theorem">Morley's trisector theorem</a></li>

<li><a href="Trisectrix" title="Trisectrix">Trisectrix</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFDudley1994" class="citation cs2"><a href="Underwood_Dudley" title="Underwood Dudley">Dudley, Underwood</a> (1994), The trisectors, <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-514-0</bdi></cite> (Originally published 1987 as A Budget of Trisections)

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWantzel1837" class="citation journal cs1">Wantzel, P M L (1837). <a rel="nofollow" class="external text" href="http://math-doc.ujf-grenoble.fr/JMPA/PDF/JMPA_1837_1_2_A31_0.pdf#2">"Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas"</a> (PDF). Journal de Mathématiques Pures et Appliquées. 1. 2: 366–372. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://math-doc.ujf-grenoble.fr/JMPA/PDF/JMPA_1837_1_2_A31_0.pdf#2">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 3 March 2014.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> For the historical basis of Wantzel's proof in the earlier work of Ruffini and Abel, and its timing vis-a-vis Galois, see <cite id="CITEREFSmorynski2007" class="citation cs2">Smorynski, Craig (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_zliInaOM8UC&pg=PA130">History of Mathematics: A Supplement</a>, Springer, p. 130, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387754802</bdi></cite>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> MacHale, Desmond. "Constructing integer angles", Mathematical Gazette 66, June 1982, 144–145.

</li>

<li id="cite_note-McLean-5">^ <a href="#cite_ref-McLean_5-0">a</a> <a href="#cite_ref-McLean_5-1">b</a> <cite id="CITEREFMcLean,_K._Robin2008" class="citation journal cs1">McLean, K. Robin (July 2008). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/mathematical-gazette/article/9252-trisecting-angles-with-ruler-and-compasses/FD6933F81AC55AF2225AF75568E2103E">"Trisecting angles with ruler and compasses"</a>. Mathematical Gazette. 92: 320–323. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0025557200183317">10.1017/S0025557200183317</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126351853">126351853</a>. <q>See also Feedback on this article in vol. 93, March 2009, p. 156.</q></cite>

</li>

<li id="cite_note-Stewart-6"><a href="#cite_ref-Stewart_6-0">^</a> <cite id="CITEREFStewart1989" class="citation book cs1"><a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (1989). <a href="Galois_Theory" class="mw-redirect" title="Galois Theory">Galois Theory</a>. Chapman and Hall Mathematics. pp. g. 58. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-412-34550-0</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFJim_Loy2003" class="citation web cs1">Jim Loy (2003) [1997]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120225124232/http://www.jimloy.com/geometry/trisect.htm">"Trisection of an Angle"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.jimloy.com/geometry/trisect.htm">the original</a> on February 25, 2012. Retrieved 30 March 2012.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFYates1942" class="citation book cs1">Yates, Robert C (1942). <a rel="nofollow" class="external text" href="http://files.eric.ed.gov/fulltext/ED058058.pdf#53">The Trisection Problem</a> (PDF). The National Council of Teachers of Mathematics. pp. 39–42. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://files.eric.ed.gov/fulltext/ED058058.pdf#53">Archived</a> (PDF) from the original on 2022-10-09.</cite>

</li>

<li id="cite_note-Ludwig_Bieberbach-9"><a href="#cite_ref-Ludwig_Bieberbach_9-0">^</a> Ludwig Bieberbach (1932) "Zur Lehre von den kubischen Konstruktionen", Journal für die reine und angewandte Mathematik, H. Hasse und L. Schlesinger, Band 167 Berlin, p. 142–146 <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0167&DMDID=DMDLOG_0020">online-copie (GDZ)</a>. Retrieved on June 2, 2017.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Jim Loy <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131104113041/http://www.jimloy.com/geometry/trisect.htm">"Trisection of an Angle"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.jimloy.com/geometry/trisect.htm">the original</a> on November 4, 2013. Retrieved 2013-11-04.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFHutcheson2001" class="citation journal cs1">Hutcheson, Thomas W. (May 2001). "Dividing Any Angle into Any Number of Equal Parts". Mathematics Teacher. 94 (5): 400–405. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2FMT.94.5.0400">10.5951/MT.94.5.0400</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> Isaac, Rufus, "Two mathematical papers without words", <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a> 48, 1975, p. 198. Reprinted in Mathematics Magazine 78, April 2005, p. 111.

</li>

<li id="cite_note-Gleason-13">^ <a href="#cite_ref-Gleason_13-0">a</a> <a href="#cite_ref-Gleason_13-1">b</a> <a href="#cite_ref-Gleason_13-2">c</a> <cite id="CITEREFGleason1988" class="citation journal cs1"><a href="Andrew_M._Gleason" title="Andrew M. Gleason">Gleason, Andrew Mattei</a> (March 1988). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141105205944/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#3">"Angle trisection, the heptagon, and the triskaidecagon"</a> (PDF). The American Mathematical Monthly. 95 (3): 185–194. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2323624">10.2307/2323624</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2323624">2323624</a>. Archived from <a rel="nofollow" class="external text" href="http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#3">the original</a> (PDF) on November 5, 2014.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: an elementary approach to ideas and methods, Oxford University Press US, 1996. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-510519-3</bdi>.</li>

<li>Dudley, Underwood. (1983) "What To Do When the Trisector Comes." THE MATHEMATICAL INTELLIGENCER. 5(1): 20–25. Springer-Verlag, New York. <a rel="nofollow" class="external text" href="https://www.ufv.ca/media/faculty/gregschlitt/information/WhatToDoWhenTrisectorComes.pdf">link</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/AngleTrisection.html">MathWorld site</a></li>

<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/GeometricProblemsofAntiquity.html">Geometric problems of antiquity, including angle trisection</a></li>

<li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trisecting_an_angle.html">Some history</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091227084452/http://www.uwgb.edu/dutchs/PSEUDOSC/trisect.HTM">One link of marked ruler construction</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/pythagoras/archi.shtml">Another, mentioning Archimedes</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131104113041/http://www.jimloy.com/geometry/trisect.htm">A long article with many approximations & means going outside the Greek framework</a></li>

<li><a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/docs/forum/angtri/">Geometry site</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Other_means_of_trisection">Other means of trisection</h3></div>

<ul><li><a href="https://commons.wikimedia.org/wiki/File:01-Trisection_of_angle_E-10_Animation.gif" class="extiw external" title="commons:File:01-Trisection of angle E-10 Animation.gif"> Approximate angle trisection as an animation, max. error of the angle ≈ ±4E-8°</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110831120043/http://trisectlimacon.webs.com/">Trisecting via</a> (<a rel="nofollow" class="external text" href="https://archive.today/20091025181718/http://www.geocities.com/trisect_limacon/">Archived</a> 2009-10-25) the <a href="Limacon" class="mw-redirect" title="Limacon">limacon</a> of <a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a>; see also <a href="Trisectrix" title="Trisectrix">Trisectrix</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091227084452/http://www.uwgb.edu/dutchs/PSEUDOSC/trisect.HTM">Trisecting via</a> an <a href="Archimedean_Spiral" class="mw-redirect" title="Archimedean Spiral">Archimedean Spiral</a></li>

<li><a rel="nofollow" class="external text" href="http://xahlee.org/SpecialPlaneCurves_dir/ConchoidOfNicomedes_dir/conchoidOfNicomedes.html">Trisecting via</a> the <a href="Conchoid_(mathematics)" title="Conchoid (mathematics)">Conchoid</a> of <a href="Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li>

<li><a rel="nofollow" class="external text" href="https://www.sciencenews.org/article/trisecting-angle-origami">sciencenews.org site</a> on using <a href="Origami" title="Origami">origami</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071208124719/http://www.song-of-songs.net/Star-of-David-Flower-of-Life.html">Hyperbolic trisection and the spectrum of regular polygons</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Ancient_Greek_mathematics550" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Ancient_Greek_mathematics550" style="font-size:114%;margin:0 4em"><a href="Ancient_Greek_mathematics" title="Ancient Greek mathematics">Ancient Greek mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_Greek_mathematicians" title="List of Greek mathematicians">Mathematicians</a> <a href="Timeline_of_ancient_Greek_mathematicians" title="Timeline of ancient Greek mathematicians">(timeline)</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anaxagoras" title="Anaxagoras">Anaxagoras</a></li>

<li><a href="Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius</a></li>

<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li>

<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>

<li><a href="Archytas" title="Archytas">Archytas</a></li>

<li><a href="Aristaeus_the_Elder" title="Aristaeus the Elder">Aristaeus the Elder</a></li>

<li><a href="Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li>

<li><a href="Autolycus_of_Pitane" title="Autolycus of Pitane">Autolycus</a></li>

<li><a href="Bryson_of_Heraclea" title="Bryson of Heraclea">Bryson</a></li>

<li><a href="Callippus" title="Callippus">Callippus</a></li>

<li><a href="Carpus_of_Antioch" title="Carpus of Antioch">Carpus</a></li>

<li><a href="Chrysippus" title="Chrysippus">Chrysippus</a></li>

<li><a href="Cleomedes" title="Cleomedes">Cleomedes</a></li>

<li><a href="Conon_of_Samos" title="Conon of Samos">Conon</a></li>

<li><a href="Ctesibius" title="Ctesibius">Ctesibius</a></li>

<li><a href="Democritus" title="Democritus">Democritus</a></li>

<li><a href="Dicaearchus" title="Dicaearchus">Dicaearchus</a></li>

<li><a href="Dinostratus" title="Dinostratus">Dinostratus</a></li>

<li><a href="Diocles_(mathematician)" title="Diocles (mathematician)">Diocles</a></li>

<li><a href="Dionysodorus" title="Dionysodorus">Dionysodorus of Caunus</a></li>

<li><a href="Dionysodorus_of_Amisene" title="Dionysodorus of Amisene">Dionysodorus of Amisene</a></li>

<li><a href="Diophantus" title="Diophantus">Diophantus</a></li>

<li><a href="Domninus_of_Larissa" title="Domninus of Larissa">Domninus</a></li>

<li><a href="Eratosthenes" title="Eratosthenes">Eratosthenes</a></li>

<li><a href="Euclid" title="Euclid">Euclid</a></li>

<li><a href="Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus</a></li>

<li><a href="Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li>

<li><a href="Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius</a></li>

<li><a href="Geminus" title="Geminus">Geminus</a></li>

<li><a href="Heliodorus_of_Larissa" title="Heliodorus of Larissa">Heliodorus</a></li>

<li><a href="Hero_of_Alexandria" title="Hero of Alexandria">Heron</a></li>

<li><a href="Hipparchus" title="Hipparchus">Hipparchus</a></li>

<li><a href="Hippasus" title="Hippasus">Hippasus</a></li>

<li><a href="Hippias" title="Hippias">Hippias</a></li>

<li><a href="Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates</a></li>

<li><a href="Hypatia" title="Hypatia">Hypatia</a></li>

<li><a href="Hypsicles" title="Hypsicles">Hypsicles</a></li>

<li><a href="Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a></li>

<li><a href="Marinus_of_Neapolis" title="Marinus of Neapolis">Marinus</a></li>

<li><a href="Menaechmus" title="Menaechmus">Menaechmus</a></li>

<li><a href="Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li>

<li><a href="Metrodorus_(grammarian)" title="Metrodorus (grammarian)">Metrodorus</a></li>

<li><a href="Nicomachus" title="Nicomachus">Nicomachus</a></li>

<li><a href="Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li>

<li><a href="Nicoteles_of_Cyrene" title="Nicoteles of Cyrene">Nicoteles</a></li>

<li><a href="Oenopides" title="Oenopides">Oenopides</a></li>

<li><a href="Pandrosion" title="Pandrosion">Pandrosion</a></li>

<li><a href="Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a></li>

<li><a href="Perseus_(geometer)" title="Perseus (geometer)">Perseus</a></li>

<li><a href="Philolaus" title="Philolaus">Philolaus</a></li>

<li><a href="Philon" title="Philon">Philon</a></li>

<li><a href="Philonides_of_Laodicea" title="Philonides of Laodicea">Philonides</a></li>

<li><a href="Porphyry_of_Tyre" title="Porphyry of Tyre">Porphyry of Tyre</a></li>

<li><a href="Posidonius" title="Posidonius">Posidonius</a></li>

<li><a href="Proclus" title="Proclus">Proclus</a></li>

<li><a href="Ptolemy" title="Ptolemy">Ptolemy</a></li>

<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>

<li><a href="Serenus_of_Antino%C3%B6polis" title="Serenus of Antinoöpolis">Serenus</a></li>

<li><a href="Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes</a></li>

<li><a href="Sporus_of_Nicaea" title="Sporus of Nicaea">Sporus</a></li>

<li><a href="Thales_of_Miletus" title="Thales of Miletus">Thales</a></li>

<li><a href="Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a></li>

<li><a href="Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a></li>

<li><a href="Theodosius_of_Bithynia" title="Theodosius of Bithynia">Theodosius</a></li>

<li><a href="Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a></li>

<li><a href="Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a></li>

<li><a href="Thymaridas" title="Thymaridas">Thymaridas</a></li>

<li><a href="Xenocrates" title="Xenocrates">Xenocrates</a></li>

<li><a href="Zeno_of_Sidon" title="Zeno of Sidon">Zeno of Sidon</a></li>

<li><a href="Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Treatises</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almagest" title="Almagest">Almagest</a></li>

<li><a href="Arithmetica" class="mw-redirect" title="Arithmetica">Arithmetica</a></li>

<li><a href="Apollonius_of_Perga#Conics" title="Apollonius of Perga">Conics (Apollonius)</a></li>

<li><a href="Catoptrics" title="Catoptrics">Catoptrics</a></li>

<li><a href="Data_(Euclid)" class="mw-redirect" title="Data (Euclid)">Data (Euclid)</a></li>

<li><a href="Euclid's_Elements" title="Euclid's Elements">Elements (Euclid)</a></li>

<li><a href="Little_Astronomy" title="Little Astronomy">Little Astronomy</a></li>

<li><a href="Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></li>

<li><a href="On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></li>

<li><a href="On_the_Sizes_and_Distances_(Aristarchus)" title="On the Sizes and Distances (Aristarchus)">On the Sizes and Distances (Aristarchus)</a></li>

<li><a href="On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)">On Sizes and Distances (Hipparchus)</a></li>

<li><a href="Autolycus_of_Pitane" title="Autolycus of Pitane">On the Moving Sphere (Autolycus)</a></li>

<li><a href="Euclid's_Optics" title="Euclid's Optics">Optics (Euclid)</a></li>

<li><a href="On_Spirals" title="On Spirals">On Spirals</a></li>

<li><a href="On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></li>

<li><a href="Ostomachion" title="Ostomachion">Ostomachion</a></li>

<li><a href="Euclid's_Phaenomena" title="Euclid's Phaenomena">Phaenomena (Euclid)</a></li>

<li><a href="Planisphaerium" title="Planisphaerium">Planisphaerium</a></li>

<li><a href="Theodosius'_Spherics" title="Theodosius' Spherics">Spherics (Theodosius)</a></li>

<li><a href="Menelaus_of_Alexandria" title="Menelaus of Alexandria">Spherics (Menelaus)</a></li>

<li><a href="The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a></li>

<li><a href="The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts and definitions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chord_(geometry)" title="Chord (geometry)">Chord</a></li>

<li><a href="Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a>

<ul><li><a href="Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li>

<li><a href="Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li>

<li><a href="Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</a></li></ul></li>

<li><a href="Commensurability_(mathematics)" title="Commensurability (mathematics)">Commensurability</a></li>

<li><a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Golden_ratio" title="Golden ratio">Golden ratio</a></li>

<li><a href="Lune_of_Hippocrates" title="Lune of Hippocrates">Lune of Hippocrates</a></li>

<li><a href="Method_of_exhaustion" title="Method of exhaustion">Method of exhaustion</a></li>

<li><a href="Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li>

<li><a href="Platonic_solid" title="Platonic solid">Platonic solid</a></li>

<li><a href="Regular_polygon" title="Regular polygon">Regular polygon</a></li>

<li><a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass construction</a>

<ul>

<li><a href="Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li>

<li><a href="Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li>

<li><a href="Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></li>

<li><a href="Neusis_construction" title="Neusis construction">Neusis construction</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="In_Elements37" scope="row" class="navbox-group" style="width:1%">In <a href="Euclid's_elements" class="mw-redirect" title="Euclid's elements">Elements</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li>

<li><a href="Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li>

<li><a href="Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li>

<li><a href="Euclid's_theorem" title="Euclid's theorem">Euclid's theorem</a></li>

<li><a href="Geometric_mean_theorem" title="Geometric mean theorem">Geometric mean theorem</a></li>

<li><a href="Hinge_theorem" title="Hinge theorem">Hinge theorem</a></li>

<li><a href="Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></li>

<li><a href="Intercept_theorem" title="Intercept theorem">Intercept theorem</a></li>

<li><a href="Intersecting_chords_theorem" title="Intersecting chords theorem">Intersecting chords theorem</a></li>

<li><a href="Intersecting_secants_theorem" title="Intersecting secants theorem">Intersecting secants theorem</a></li>

<li><a href="Law_of_cosines" title="Law of cosines">Law of cosines</a></li>

<li><a href="Pons_asinorum" title="Pons asinorum">Pons asinorum</a></li>

<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li>

<li><a href="Tangent-secant_theorem" class="mw-redirect" title="Tangent-secant theorem">Tangent-secant theorem</a></li>

<li><a href="Thales's_theorem" title="Thales's theorem">Thales's theorem</a></li>

<li><a href="Theorem_of_the_gnomon" title="Theorem of the gnomon">Theorem of the gnomon</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Apollonius's_theorem" title="Apollonius's theorem">Apollonius's theorem</a></li>

<li><a href="Aristarchus's_inequality" title="Aristarchus's inequality">Aristarchus's inequality</a></li>

<li><a href="Heron's_formula" title="Heron's formula">Heron's formula</a></li>

<li><a href="Law_of_sines" title="Law of sines">Law of sines</a></li>

<li><a href="Menelaus's_theorem" title="Menelaus's theorem">Menelaus's theorem</a></li>

<li><a href="Pappus's_area_theorem" title="Pappus's area theorem">Pappus's area theorem</a></li>

<li><a href="Diophantus_II.VIII" title="Diophantus II.VIII">Problem II.8 of Arithmetica</a></li>

<li><a href="Ptolemy's_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></li>

<li><a href="Ptolemy's_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li>

<li><a href="Ptolemy's_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a></li>

<li><a href="Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers/Schools</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyrene%2C_Libya" title="Cyrene, Libya">Cyrene</a></li>

<li><a href="Platonic_Academy" title="Platonic Academy">Platonic Academy</a></li>

<li><a href="Pythagoreanism" title="Pythagoreanism">Pythagoreanism</a></li>

<li><a href="School_of_Chios" title="School of Chios">School of Chios</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ancient_Greek_astronomy" title="Ancient Greek astronomy">Ancient Greek astronomy</a></li>

<li><a href="Attic_numerals" title="Attic numerals">Attic numerals</a></li>

<li><a href="Greek_numerals" title="Greek numerals">Greek numerals</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">History of</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a>

<ul><li>by <a href="Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li></ul></li>

<li><a href="Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li>

<li><a href="History_of_algebra" title="History of algebra">algebra</a>

<ul><li><a href="Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li>

<li><a href="History_of_arithmetic" class="mw-redirect" title="History of arithmetic">arithmetic</a>

<ul><li><a href="Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li>

<li><a href="History_of_calculus" title="History of calculus">calculus</a>

<ul><li><a href="Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li></ul></li>

<li><a href="History_of_geometry" title="History of geometry">geometry</a>

<ul><li><a href="Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li>

<li><a href="History_of_logic" title="History of logic">logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="History_of_mathematics" title="History of mathematics">mathematics</a>

<ul><li><a href="Timeline_of_mathematics" title="Timeline of mathematics">timeline</a></li></ul></li>

<li><a href="History_of_numbers" class="mw-redirect" title="History of numbers">numbers</a>

<ul><li><a href="Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">prehistoric counting</a></li></ul></li>

<li><a href="History_of_ancient_numeral_systems" title="History of ancient numeral systems">numeral systems</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other cultures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Arabian/Islamic</a></li>

<li><a href="Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a></li>

<li><a href="Chinese_mathematics" title="Chinese mathematics">Chinese</a></li>

<li><a href="Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Egyptian</a></li>

<li><a href="Mathematics_of_the_Incas" title="Mathematics of the Incas">Incan</a></li>

<li><a href="Indian_mathematics" title="Indian mathematics">Indian</a></li>

<li><a href="Japanese_mathematics" title="Japanese mathematics">Japanese</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <a href="Portal%3AAncient_Greece" title="Portal:Ancient Greece">Ancient Greece portal</a> • <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q733081#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata644" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q733081#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata644" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85137916">United States</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007551116505171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/34d695b5-1a80-452e-ae22-d884f43c32d7">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Angle_trisection&oldid=1304024353">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>