Curves and Sectioning Angles Nicholas Molbert, Julie Fink, and Tia - PowerPoint PPT Presentation

Introduction Basic Constructions The Hyperbola The Problem of Trisection Curves and Sectioning Angles Nicholas Molbert, Julie Fink, and Tia Burden July 6, 2012 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Table of Contents Introduction 1 History Notation Basic Constructions 2 Propositions and Pictures The Hyperbola 3 General Definitions Bridging the Gap from General Definition to Γ The Curve Γ The Problem of Trisection 4 Construction of Trisection Proof of Trisection Investigation of Further Curves Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Three Greek problems of antiquity Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry: Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry: Hippocrates (460-380 BC) labels points and lines 1 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry: Hippocrates (460-380 BC) labels points and lines 1 Hippias (460-399 BC) and the quadratix 2 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry: Hippocrates (460-380 BC) labels points and lines 1 Hippias (460-399 BC) and the quadratix 2 Menaechmus (380-320 BC) and conic sections 3 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Pappus (early fourth century) used a hyperbola 2 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Pappus (early fourth century) used a hyperbola 2 Descartes (1596-1650) used the curve y = x 2 3 Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Pappus (early fourth century) used a hyperbola 2 Descartes (1596-1650) used the curve y = x 2 3 The link of Francios Viete (1540-1603) Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Pappus (early fourth century) used a hyperbola 2 Descartes (1596-1650) used the curve y = x 2 3 The link of Francios Viete (1540-1603) Wantzel’s proof of impossibility Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection History of the Problem of Trisection Trisections with extra tools: Archimedes (287-217 BC) used a marked straight edge 1 Nicomedes (280-210 BC) also used a marked straight edge 2 along with a conchoid The successful trisections: Apolonius (250-175 BC) used conic sections 1 Pappus (early fourth century) used a hyperbola 2 Descartes (1596-1650) used the curve y = x 2 3 The link of Francios Viete (1540-1603) Wantzel’s proof of impossibility From Greek constructions to abstract algebra Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection Notation ← → XY denotes the line through points X and Y . Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection Notation ← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection Notation ← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C ( X , Y ) denotes a circle with center X and radius XY . Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection Notation ← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C ( X , Y ) denotes a circle with center X and radius XY . XY denotes the magnitude of segment XY Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions History The Hyperbola Notation The Problem of Trisection Notation ← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C ( X , Y ) denotes a circle with center X and radius XY . XY denotes the magnitude of segment XY ∠ XYZ denotes the measure or name of an angle, depending on the context. Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles