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Linear Geometric Constructions Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers Friday, July 8 th , 2011. Friday, July 8 th , 2011. Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric

SLIDE 1

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

SLIDE 2

Introduction

What is a Geometric Construction?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 2 / 24

SLIDE 3

Introduction

What is a Geometric Construction? Types of Geometric Constructions

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 2 / 24

SLIDE 4

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

SLIDE 5

Definitions and Rules for Basic Construcitons

Tools

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 6

Definitions and Rules for Basic Construcitons

Tools Compass Straightedge

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 7

Definitions and Rules for Basic Construcitons

Tools Compass Straightedge Postulates for Basic Constructions

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 8

Definitions and Rules for Basic Construcitons

Tools Compass Straightedge Postulates for Basic Constructions

Assume we can construct two points (the origin and (1,0))

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 9

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

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 10

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

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 3 / 24

SLIDE 11

Basic constructions

Perpendicular Lines

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 4 / 24

SLIDE 12

Basic constructions

Perpendicular Lines Parallel Lines

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 4 / 24

SLIDE 13

Basic constructions

Perpendicular Lines Parallel Lines Squareroots

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 4 / 24

SLIDE 14

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

SLIDE 15

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.

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 5 / 24

SLIDE 16

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.

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 6 / 24

SLIDE 17

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.

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 7 / 24

SLIDE 18

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

SLIDE 19

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

SLIDE 20

Examples of Basic Constructions

Theorem

Given an angle BAC, we can bisect the angle

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SLIDE 21

Fields

What is a field?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 22

Fields

What is a field? Operations

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 23

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 24

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Properties

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 25

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Properties

Associative Communitive Distributive

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 26

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Properties

Associative Communitive Distributive

Identities

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 27

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Properties

Associative Communitive Distributive

Identities

Additive identity Multiplicative identity

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 28

Fields

What is a field? Operations

Addition Subtraction Multiplication Division

Properties

Associative Communitive Distributive

Identities

Additive identity Multiplicative identity

Inverses

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 29

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 11 / 24

SLIDE 30

Examples of Fields

Q, the rational numbers

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 12 / 24

SLIDE 31

Examples of Fields

Q, the rational numbers R, the real numbers

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 12 / 24

SLIDE 32

Examples of Fields

Q, the rational numbers R, the real numbers C, the complex numbers

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 12 / 24

SLIDE 33

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

SLIDE 34

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) is constructible with only a straight edge and compass. p q is 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

SLIDE 35

Limitations of Basic Constructions

What constructions are we not able to do with simply a straightedge and compass?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 14 / 24

SLIDE 36

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 14 / 24

SLIDE 37

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

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 14 / 24

SLIDE 38

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?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 14 / 24

SLIDE 39

Definitions and Rules for Neusis Construcitons

What is a Neusis Construction?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 40

Definitions and Rules for Neusis Construcitons

What is a Neusis Construction? Tools

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 41

Definitions and Rules for Neusis Construcitons

What is a Neusis Construction? Tools Compass Twice-Notched Straightedge

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 42

Definitions and Rules for Neusis Construcitons

What is a Neusis Construction? Tools Compass Twice-Notched Straightedge Postulates for Neusis Constructions

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 43

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:

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 44

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 45

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 46

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 47

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 48

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 49

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 15 / 24

SLIDE 50

Examples of Neusis Constructions

Trisection of an angle

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 16 / 24

SLIDE 51

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

SLIDE 52

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

SLIDE 53

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

SLIDE 54

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

SLIDE 55

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

SLIDE 56

Neusis Constuction for cube roots

Theorem

Given a constructible length a, it is possible to find

√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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 21 / 24

SLIDE 57

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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SLIDE 58

Conclusion

Basic Constructions versus Neusis Constructions

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SLIDE 59

Conclusion

Basic Constructions versus Neusis Constructions Fields and field extensions

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 23 / 24

SLIDE 60

Conclusion

Basic Constructions versus Neusis Constructions Fields and field extensions Numbers we can construct

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 23 / 24

SLIDE 61

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 be constructed using a twice notched straight edge?

Holli Tatum, Andrew Chapple, Minesha Estell, and Maxalan Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 23 / 24

SLIDE 62

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 Vickers () Linear Geometric Constructions Friday, July 8th, 2011. 24 / 24