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Trisecting the General Angle
by Euclidean Means
By David E. Cochrane
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Presentation Outline
 Introduction
 Field Extensions
 Cubic Equations and Irreducibility Over
the Field of Rational Numbers
 DeMoivre’s Theorem
 Construction of a 20° Angle
 Trisection by Non-Euclidean Means
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Introduction
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The Tools
 Compasses
 Straightedges without
markings
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With These Tools We Can Construct
 A straight line between two points
 A circle with a given center and a given
radius
 The points of intersection of two
straight lines, two circles, or a straight
line and a circle
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Possible Constructions
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Possible Constructions
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Possible Constructions
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Possible Constructions
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Possible Constructions
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Other Constructions
The roots of the equations
and
Where a,b,c are given line segments
and a2>4b
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Q:
What can a given construction
be composed of?

Rational operations:
- multiplication
- division
- addition
- subtraction
 Extraction of square roots
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Field Extensions
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The Field of Rational Numbers
The Field Axioms
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Construct a Sequence of Fields
Let
be the field of rational
numbers. Form
ab k
and
show that these elements form a
new field,
F1
, where
a, b, k 
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is Closed Under Addition
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is Closed Under Subtraction
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is Closed Under Multiplication
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is Closed Under Division
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Some Examples
and is therefore constructible
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Some Examples
and is therefore constructible
and is therefore not constructible
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Theorem:
All the numbers in the fields
are constructible and,
conversely, any constructible
number must be in one of the
above fields.
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Cubic Equations
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When Are the Roots of a Cubic
Equation Constructible?
Theorem: If a cubic equation
with rational coefficients has no
rational root, then none of its
roots is constructible. In other
words, show that no root of an
irreducible cubic equation can
be constructed.
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The General Cubic Equation
x  px  qx  r  0
3
2
p, q, r 
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Assumptions

 root
is irreducible over
is constructible
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Sketch of Proof
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Conclusions
 If a cubic with rational coefficients has a
constructible root, it also has a rational root
 The roots of a cubic with rational coefficients
are constructible if and only if the equation
has a rational root
 If the equation is irreducible, then none of its
roots can be constructed with a straightedge
and compasses
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DeMoivre’s Theorem
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DeMoivre’s Theorem
For our purposes,
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Equate Real Parts
cos3x
 cos x  3cos x sin x
3
2
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Equate Real Parts
cos3x
 cos x  3cos x sin x
3
2
 cos x  3cos x 1  cos x 
3
2
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Equate Real Parts
cos3x
 cos x  3cos x sin x
3
2
 cos x  3cos x 1  cos x 
3
2
 4 cos x  3cos x
3
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Theorem:
An angle can be constructed if and only if
its cosine can be constructed.
1
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Show a 20° Cannot be Constructed
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Show a 20° Cannot be Constructed
Let
, then
.
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Show a 20° Cannot be Constructed
Let
Let
, then
.
.
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Show a 20° Cannot be Constructed
Let
, then
Let
Then
.
.
, or,
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Show
is Irreducible
 Suppose x=a/b, where a and b are





integers and gcd(a,b)=1, b ≠ 0
Then (a3/b3) – (3a/b) – 1 = 0
a3 – 3ab2 = b3
b3 = a(a2 – 3b2)
a divides b3
Since gcd(a,b) = 1, a must be +1 or –1
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Show
is Irreducible
 Similarly, a3 = b2(b+3ab)
 b2 divides a3
 Therefore, b is +1 or -1
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Show
is Irreducible
 So the only possible rational roots
of our equation are +1 or –1
 (1)3 – 3(1) – (1) = -3 ≠ 0
 (-1)3 – 3(-1) – (-1) = 3 ≠ 0
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Show
is Irreducible
 Therefore, the equation is irreducible
over the field of rational numbers
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Show
is Irreducible
 Therefore, the equation is irreducible
over the field of rational numbers
 None of its roots can be constructed
with a straightedge and compasses
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Show
is Irreducible
 Therefore, the equation is irreducible
over the field of rational numbers
 None of its roots can be constructed
with a straightedge and compasses
 cos 20 cannot be constructed
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Show
is Irreducible
 Therefore, the equation is irreducible
over the field of rational numbers
 None of its roots can be constructed
with a straightedge and compasses
 cos 20 cannot be constructed
 60 cannot be trisected
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Show
is Irreducible
 Therefore, the equation is irreducible
over the field of rational numbers
 None of its roots can be constructed
with a straightedge and compasses
 cos 20 cannot be constructed
 60 cannot be trisected
 The general angle cannot be trisected
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Trisecting the General Angle
by Non-Euclidean Means
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Method 1
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Method 1
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Method 1
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Method 1
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Method 1
1 1
1
 

4 16 64
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Method 1
a1
S
1 r
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Method 1
a1
1
1
S
, a1  , r 
1 r
4
4
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Method 1
1
4
1
a1
1
4
S



1
3
1 r
3
1
4
4
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Method 2: The Quadratrix
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Method 2: The Quadratrix
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Method 2: The Quadratrix
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Method 3: Insertion Principle
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Method 3: Insertion Principle
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Method 3: Insertion Principle
Equal
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Method 3: Insertion Principle
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Other Methods
 Trisectrix of MacLaurin
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Other Methods
 Conchoid of Nicomedes
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Other Methods
 Spiral of Archimedes
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Other Methods
 “Hatchet” Tool
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