Linear Geometric Constructions presentation.pdf · Compass Twice-Notched Straightedge Postulates...

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

Introduction

Mathematicians

Introduction

Mathematicians

Definitions and Rules for Basic Construcitons

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

ToolsCompass Straightedge

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

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

Basic constructions

Perpendicular Lines

Parallel Lines

Squareroots

Bisecting an angle

Basic constructions

Perpendicular Lines

Parallel Lines

Squareroots

Bisecting an angle

Basic constructions

Perpendicular Lines

Parallel Lines

Squareroots

Bisecting an angle

Basic constructions

Perpendicular Lines

Parallel Lines

Squareroots

Bisecting an angle

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.

Theorem

Given a constructible number a, we can construct√

Theorem

Given an angle BAC , we can bisect the angle

Theorem

Given an angle BAC , we can bisect the angle

Fields

What is a field?

Operations

AdditionSubtractionMultiplicationDivision

Properties

AssociativeCommunitiveDistributive

Identities

Additive identityMultiplicative identity

Inverses

Additive inverseMultiplicative inverse

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Fields

What is a field?

Operations

Properties

Identities

Inverses

Examples of Fields

Q, the rational numbers

R, the real numbers

C, the complex numbers

E, the constructible numbers

Examples of Fields

R, the real numbers

Examples of Fields

R, the real numbers

Examples of Fields

R, the real numbers

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

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?

Trisecting an angle

Why is this?

Trisecting an angle

Why is this?

Trisecting an angle

Why is this?

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

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

Pivot the twice notched straightedge around a point

Slide the twice notched straightedge along lines and circles

Examples of Neusis Constructions

Trisection of an angle

Cube roots

Examples of Neusis Constructions

Trisection of an angle

Cube roots

Neusis Constuction for trisection of an angle

Theorem

A triscetum construction using Neusis and given angle A’BC’

Theorem

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.

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.)

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?

Conclusion

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.

Linear Geometric Constructions presentation.pdf · Compass Twice-Notched Straightedge Postulates...

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