Linear Geometric Constructions
Linear Geometric Constructions Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers
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Introduction
What is a Geometric Construction?
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Introduction
What is a Geometric Construction? Types of Geometric Constructions
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Introduction
What is a Geometric Construction? Types of Geometric Constructions Mathematicians
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Definitions and Rules for Basic Construcitons
Tools
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
Postulates for Basic Constructions
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
Postulates for Basic Constructions Assume we can construct two points (the origin and (1,0))
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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 circles
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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 circles Use of the intersections of those lines and cirlces to constuct new points
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Basic constructions
Perpendicular Lines
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Basic constructions
Perpendicular Lines Parallel Lines
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Basic constructions
Perpendicular Lines Parallel Lines Squareroots
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Basic constructions
Perpendicular Lines Parallel Lines Squareroots Bisecting an angle
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a constructible number a, we can construct
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√
a.
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Examples of Basic Constructions
Theorem Given an angle BAC , we can bisect the angle
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Examples of Basic Constructions
Theorem Given an angle BAC , we can bisect the angle
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Fields What is a field?
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Fields What is a field? Operations
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Fields What is a field? Operations Addition Subtraction Multiplication Division
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
Inverses
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
Inverses Additive inverse Multiplicative inverse Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Examples of Fields
Q, the rational numbers
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Examples of Fields
Q, the rational numbers R, the real numbers
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Examples of Fields
Q, the rational numbers R, the real numbers C, the complex numbers
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Examples of Fields
Q, the rational numbers R, the real numbers C, the complex numbers E, the constructible numbers
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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 field extensions 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) r is constructible with only a straight edge and compass. p is constructible with only a straight edge and compass q
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass?
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle Taking the cube root of a constructible number
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle Taking the cube root of a constructible number Why is this?
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction?
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
Postulates for Neusis Constructions
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
Postulates for Neusis Constructions Using a straightedge and compass we can:
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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))
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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:
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point Slide the twice notched straightedge along lines and circles
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point Slide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge and those previously constructed lines and cirlces to constuct new points Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Examples of Neusis Constructions
Trisection of an angle
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Examples of Neusis Constructions
Trisection of an angle Cube roots
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Neusis Constuction for trisection of an angle
Theorem A triscetum construction using Neusis and given angle A’BC’
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Neusis Constuction for trisection of an angle
Theorem A triscetum construction using Neusis and given angle A’BC’
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Neusis Constuction for trisection of an angle
Theorem A triscetum construction using Neusis and given angle A’BC’
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Neusis Constuction for trisection of an angle
Theorem A triscetum construction using Neusis and given angle A’BC’
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Neusis Constuction for cube roots
Theorem
√ Given a constructible length a, it is possible to find 3 a using a compass and twice-notched straightedge. √ Claim: We can calculate x = 2 3 a by using similar triangles and proportions. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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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 two notches and compass (i.e. α ∈ the constructible numbers with two notches.)
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Conclusion
Basic Constructions versus Neusis Constructions
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions Numbers we can construct
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions Numbers we can construct Is there a real number not degree 2p 3q over Q that can be constructed using a twice notched straight edge?
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Acknowledgements
We would like to thank our graduate mentor Laura Rider for her help on the project and Professor Smolinsky teaching the class. We would also like to thank Professor Davidson for allowing us to participate in this program.
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Friday, July 8th , 2011.
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Introduction
What is a Geometric Construction?
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Introduction
What is a Geometric Construction? Types of Geometric Constructions
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Introduction
What is a Geometric Construction? Types of Geometric Constructions Mathematicians
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Definitions and Rules for Basic Construcitons
Tools
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
Postulates for Basic Constructions
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Definitions and Rules for Basic Construcitons
Tools Compass
Straightedge
Postulates for Basic Constructions Assume we can construct two points (the origin and (1,0))
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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 circles
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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 circles Use of the intersections of those lines and cirlces to constuct new points
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Basic constructions
Perpendicular Lines
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Basic constructions
Perpendicular Lines Parallel Lines
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Basic constructions
Perpendicular Lines Parallel Lines Squareroots
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Basic constructions
Perpendicular Lines Parallel Lines Squareroots Bisecting an angle
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a line L and a point P on the line, we can draw a line perpendicular to L that passes through P.
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Examples of Basic Constructions
Theorem Given a constructible number a, we can construct
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√
a.
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Examples of Basic Constructions
Theorem Given an angle BAC , we can bisect the angle
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Examples of Basic Constructions
Theorem Given an angle BAC , we can bisect the angle
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Fields What is a field?
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Fields What is a field? Operations
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Fields What is a field? Operations Addition Subtraction Multiplication Division
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
Inverses
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Fields What is a field? Operations Addition Subtraction Multiplication Division
Properties Associative Communitive Distributive
Identities Additive identity Multiplicative identity
Inverses Additive inverse Multiplicative inverse Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Examples of Fields
Q, the rational numbers
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Examples of Fields
Q, the rational numbers R, the real numbers
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Examples of Fields
Q, the rational numbers R, the real numbers C, the complex numbers
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Examples of Fields
Q, the rational numbers R, the real numbers C, the complex numbers E, the constructible numbers
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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 field extensions 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) r is constructible with only a straight edge and compass. p is constructible with only a straight edge and compass q
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass?
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle Taking the cube root of a constructible number
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Limitations of Basic Constructions
What constructions are we not able to do with simply a straightedge and compass? Trisecting an angle Taking the cube root of a constructible number Why is this?
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction?
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
Postulates for Neusis Constructions
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
Twice-Notched Straightedge
Postulates for Neusis Constructions Using a straightedge and compass we can:
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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))
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
Friday, July 8th , 2011.
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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:
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
Friday, July 8th , 2011.
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
Friday, July 8th , 2011.
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point Slide the twice notched straightedge along lines and circles
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
Friday, July 8th , 2011.
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Definitions and Rules for Neusis Construcitons What is a Neusis Construction? Tools Compass
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 point Slide the twice notched straightedge along lines and circles
Using the intersections of the slid and/or pivoted straightedge and those previously constructed lines and cirlces to constuct new points Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Examples of Neusis Constructions
Trisection of an angle
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Examples of Neusis Constructions
Trisection of an angle Cube roots
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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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 Linear Geometric Vickers () Constructions
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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 Linear Geometric Vickers () Constructions
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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 Linear Geometric Vickers () Constructions
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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 Linear Geometric Vickers () Constructions
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Neusis Constuction for cube roots
Theorem
√ Given a constructible length a, it is possible to find 3 a using a compass and twice-notched straightedge. √ Claim: We can calculate x = 2 3 a by using similar triangles and proportions. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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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 two notches and compass (i.e. α ∈ the constructible numbers with two notches.)
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Conclusion
Basic Constructions versus Neusis Constructions
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions Numbers we can construct
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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Conclusion
Basic Constructions versus Neusis Constructions Fields and field extensions Numbers we can construct Is there a real number not degree 2p 3q over Q that can be constructed using a twice notched straight edge?
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Acknowledgements
We would like to thank our graduate mentor Laura Rider for her help on the project and Professor Smolinsky teaching the class. We would also like to thank Professor Davidson for allowing us to participate in this program.
Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Linear Geometric Vickers () Constructions
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