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Linear Geometric Constructions
Holli Tatum, Andrew Chapple,Minesha Estell, and Maxalan Vickers
Friday, July 8th, 2011.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 1 / 24
Introduction
What is a Geometric Construction?
Types of Geometric Constructions
Mathematicians
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 2 / 24
Introduction
What is a Geometric Construction?
Types of Geometric Constructions
Mathematicians
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 2 / 24
Introduction
What is a Geometric Construction?
Types of Geometric Constructions
Mathematicians
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 2 / 24
Definitions and Rules for Basic Construcitons
Tools
Compass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))Constructions of lines and circlesUse of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Definitions and Rules for Basic Construcitons
ToolsCompass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))Constructions of lines and circlesUse of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Definitions and Rules for Basic Construcitons
ToolsCompass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))Constructions of lines and circlesUse of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Definitions and Rules for Basic Construcitons
ToolsCompass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))
Constructions of lines and circlesUse of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Definitions and Rules for Basic Construcitons
ToolsCompass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))Constructions of lines and circles
Use of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Definitions and Rules for Basic Construcitons
ToolsCompass Straightedge
Postulates for Basic Constructions
Assume we can construct two points (the origin and (1,0))Constructions of lines and circlesUse of the intersections of those lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 3 / 24
Basic constructions
Perpendicular Lines
Parallel Lines
Squareroots
Bisecting an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 4 / 24
Basic constructions
Perpendicular Lines
Parallel Lines
Squareroots
Bisecting an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 4 / 24
Basic constructions
Perpendicular Lines
Parallel Lines
Squareroots
Bisecting an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 4 / 24
Basic constructions
Perpendicular Lines
Parallel Lines
Squareroots
Bisecting an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 4 / 24
Examples of Basic Constructions
Theorem
Given a line L and a point P on the line, we can draw a line perpendicularto L that passes through P.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 5 / 24
Examples of Basic Constructions
Theorem
Given a line L and a point P on the line, we can draw a line perpendicularto L that passes through P.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 6 / 24
Examples of Basic Constructions
Theorem
Given a line L and a point P on the line, we can draw a line perpendicularto L that passes through P.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 7 / 24
Examples of Basic Constructions
Theorem
Given a constructible number a, we can construct√
a.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 8 / 24
Examples of Basic Constructions
Theorem
Given an angle BAC , we can bisect the angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 9 / 24
Examples of Basic Constructions
Theorem
Given an angle BAC , we can bisect the angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 10 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Fields
What is a field?
Operations
AdditionSubtractionMultiplicationDivision
Properties
AssociativeCommunitiveDistributive
Identities
Additive identityMultiplicative identity
Inverses
Additive inverseMultiplicative inverse
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 11 / 24
Examples of Fields
Q, the rational numbers
R, the real numbers
C, the complex numbers
E, the constructible numbers
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 12 / 24
Examples of Fields
Q, the rational numbers
R, the real numbers
C, the complex numbers
E, the constructible numbers
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 12 / 24
Examples of Fields
Q, the rational numbers
R, the real numbers
C, the complex numbers
E, the constructible numbers
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 12 / 24
Examples of Fields
Q, the rational numbers
R, the real numbers
C, the complex numbers
E, the constructible numbers
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 12 / 24
Two-tower over Q Theorem
Theorem
The number α ∈ R is constructible with straight edge and compass(α ∈ constructiblenumbers) if and only there is a sequence of fieldextensions
Q = F0 ⊂ F1 ⊂ F2 ⊂......⊂ Fn so that [Fi : Fi−1] = 2 or 1 for
i = 1, ......., n (i.e. Fi = Fi−1(√βi )), βi ∈ Fi−1 and α ∈ Fn
The Theorem states
(Q) is constructible with only a straight edge and compass.√p
qis constructible with only a straight edge and compass
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 13 / 24
Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge andcompass?
Trisecting an angle
Taking the cube root of a constructible number
Why is this?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 14 / 24
Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge andcompass?
Trisecting an angle
Taking the cube root of a constructible number
Why is this?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 14 / 24
Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge andcompass?
Trisecting an angle
Taking the cube root of a constructible number
Why is this?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 14 / 24
Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge andcompass?
Trisecting an angle
Taking the cube root of a constructible number
Why is this?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 14 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
Tools
Compass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis Constructions
Using a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))
Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a point
Slide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Definitions and Rules for Neusis Construcitons
What is a Neusis Construction?
ToolsCompass Twice-Notched Straightedge
Postulates for Neusis ConstructionsUsing a straightedge and compass we can:
Assume we can construct two points (the origin and (1,0))Construct lines and circles
Using a twice notched straightedge and compass we can:
Pivot the twice notched straightedge around a pointSlide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge andthose previously constructed lines and cirlces to constuct new points
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 15 / 24
Examples of Neusis Constructions
Trisection of an angle
Cube roots
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 16 / 24
Examples of Neusis Constructions
Trisection of an angle
Cube roots
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 16 / 24
Neusis Constuction for trisection of an angle
Theorem
A triscetum construction using Neusis and given angle A’BC’
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 17 / 24
Neusis Constuction for trisection of an angle
Theorem
A triscetum construction using Neusis and given angle A’BC’
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 18 / 24
Neusis Constuction for trisection of an angle
Theorem
A triscetum construction using Neusis and given angle A’BC’
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 19 / 24
Neusis Constuction for trisection of an angle
Theorem
A triscetum construction using Neusis and given angle A’BC’
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 20 / 24
Neusis Constuction for cube roots
Theorem
Given a constructible length a, it is possible to find 3√
a using a compassand twice-notched straightedge.
Claim: We can calculate x = 2 3√
a by using similar triangles andproportions.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 21 / 24
2-3 tower over Q
Theorem
If a number α ∈ (R) is in Fn so that there is a sequence of field extensions
(Q) = F0 ⊂ F1 ⊂ ...... ⊂ Fn with [Fi : Fi−1] = 1, 2, 3
for i = 1, ......, n, then α is constructible with straight edge with twonotches and compass (i.e. α ∈ the constructible numbers with twonotches.)
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 22 / 24
Conclusion
Basic Constructions versus Neusis Constructions
Fields and field extensions
Numbers we can construct
Is there a real number not degree 2p3q over Q that can beconstructed using a twice notched straight edge?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 23 / 24
Conclusion
Basic Constructions versus Neusis Constructions
Fields and field extensions
Numbers we can construct
Is there a real number not degree 2p3q over Q that can beconstructed using a twice notched straight edge?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 23 / 24
Conclusion
Basic Constructions versus Neusis Constructions
Fields and field extensions
Numbers we can construct
Is there a real number not degree 2p3q over Q that can beconstructed using a twice notched straight edge?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 23 / 24
Conclusion
Basic Constructions versus Neusis Constructions
Fields and field extensions
Numbers we can construct
Is there a real number not degree 2p3q over Q that can beconstructed using a twice notched straight edge?
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 23 / 24
Acknowledgements
We would like to thank our graduate mentor LauraRider for her help on the project and Professor Smolinskyteaching the class. We would also like to thank ProfessorDavidson for allowing us to participate in this program.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers ()Linear Geometric Constructions Friday, July 8th , 2011. 24 / 24