Chapter 7: Hyperbolic Geometry

Slides:

Advertisements

Similar presentations

By: Victoria Leffelman.  Any geometry that is different from Euclidean geometry  Consistent system of definitions, assumptions, and proofs that describe.

Advertisements

§7.1 Quadrilaterals The student will learn:

Euclid BC Author of The Elements –13 books in all. –Standard textbook for Geometry for about 2000 years. Taught in Alexandria, Egypt.

Euclid’s Elements: The first 4 axioms

Ch. 1: Some History References Modern Geometries: Non-Euclidean, Projective, and Discrete 2 nd ed by Michael Henle Euclidean and Non-Euclidean Geometries:

Study Guide Timeline Euclid’s five axioms (300 BC) From Proclus (400AD) belief that the fifth axiom is derivable from the first four Saccheri (17 th century):

Math 260 Foundations of Geometry

Non-Euclidean Geometries

Spherical Geometry. The sole exception to this rule is one of the main characteristics of spherical geometry. Two points which are a maximal distance.

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.7 Non-Euclidean Geometry and Fractal Geometry.

History of Mathematics

Chapter 2: Euclid’s Proof of the Pythagorean Theorem

1 §23.1 Hyperbolic Models 1 Intro to Poincare's Disk Model. A Point – Any interior point of circle C (the ordinary points of H or h-points) Line – Any.

Chapter 9: Geometry.

A Short History of Geometry,

Non-Euclidean Geometry Br. Joel Baumeyer, FSC Christian Brothers University.

Area (geometry) the amount of space within a closed shape; the number of square units needed to cover a figure.

The Strange New Worlds: The Non-Euclidean Geometries Presented by: Melinda DeWald Kerry Barrett.

Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Hyperbolic Geometry Chapter 11. Hyperbolic Lines and Segments Poincaré disk model  Line = circular arc, meets fundamental circle orthogonally Note: 

1 Those Incredible Greeks Lecture Three. 2 Outline  Hellenic and Hellenistic periods  Greek numerals  The rise of “modern” mathematics – axiomatic.

Chapter 2 Greek Geometry The Deductive Method The Regular Polyhedra Ruler and Compass Construction Conic Sections Higher-degree curves Biographical Notes:

Warm-Up The graph shown represents a hyperbola. Draw the solid of revolution formed by rotating the hyperbola around the y -axis.

Geometry in Robotics Robotics 8.

Discrete Math Point, Line, Plane, Space Desert Drawing to One Vanishing Point Project.

INTRODUCTION TO Euclid’s geometry The origins of geometry.

Euclid’s Postulates 1.Two points determine one and only one straight line 2.A straight line extends indefinitely far in either direction 3. A circle may.

Parallels and Transversals Objective: Identify Angles formed by Two Lines and a Transversal.

A proof that can be proved in Euclidean geometry, yet not in Non-Euclidean geometry.

Euclid The famous mathematician Euclid is credited with being the first person to describe geometry.

1 §4.6 Area & Volume The student will learn about: area postulates, Cavalieri’s Principle, 1 and the areas of basic shapes.

Geometry and Measurement Chapter Nine Lines and Angles Section 9.1.

H YPERSHOT : F UN WITH H YPERBOLIC G EOMETRY Praneet Sahgal.

Euclid and the “elements”. Euclid (300 BC, 265 BC (?) ) was a Greek mathematician, often referred to as the "Father of Geometry”. Of course this is not.

Non-Euclidean Geometry Part I.

Definition: Rectangle A rectangle is a quadrilateral with four right angles.

The Non-Euclidean Geometries

Euclidean Geometry

Warm-Up x + 2 3x - 6 What is the value of x?. Geometry 3-3 Proving Lines Parallel.

Geometry Review Lines, Angles, Polygons. What am I?

Chapter 3 Parallel and Perpendicular Lines

Warm-Up The graph shown represents a hyperbola. Draw the solid of revolution formed by rotating the hyperbola around the y-axis.

Geometry on a Ball Spherical Geometry… a Non-Euclidean Geometry a Non-Euclidean Geometry.

The Parallel Postulate

Ch. 4: The Erlanger Programm References: Euclidean and Non-Euclidean Geometries: Development and History 4 th ed By Greenberg Modern Geometries: Non-Euclidean,

3.5 Parallel Lines and Triangles

MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180- degrees and apply this fact to find unknown measure of angles. Block 26.

The Elements Definition 10 When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is.

2 nd Half of 11.1 Basic Geometric Shapes By: Nicole Garcia & Denise Padilla.

The reason why Euclid was known as the father of geometry because, he was responsible for assembling all the world’s knowledge of flat planes and 3D geometry.

Geometry 2.2 And Now From a New Angle.

3.1 Identify Pairs of Lines and Angles 3.2 Use Parallel Lines and Transversals Objectives: 1.To differentiate between parallel, perpendicular, and skew.

T AXICAB G EOMETRY An exploration of Area and Perimeter Within city blocks.

Chapter 9: Geometry. 9.1: Points, Lines, Planes and Angles 9.2: Curves, Polygons and Circles 9.3: Triangles (Pythagoras’ Theorem) 9.4: Perimeter, Area.

MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180- degrees and apply this fact to find unknown measure of angles. Block 26.

HYPERBOLIC GEOMETRY Paul Klotzle Gabe Richmond.

3.1 – 3.2 Quiz Review Identify each of the following.

Euclid’s Postulates Two points determine one and only one straight line A straight line extends indefinitely far in either direction 3. A circle may be.

Euclid’s Definitions EUCLID’ S GEOMETRY

3.2- Angles formed by parallel lines and transversals

3-3: Proving Lines Parallel

9.7 Non-Euclidean Geometries

Proving Lines Parallel

ZERO AND INFINITY IN THE NON EUCLIDEAN GEOMETRY

Book 1, Proposition 29 By: Isabel Block and Alexander Clark

3.2- Angles formed by parallel lines and transversals

Euclid’s Geometry Definitions Postulates Common Notions

Chapter 2 Greek Geometry

3-2 Proving Lines Parallel

Presentation transcript:

Chapter 7: Hyperbolic Geometry References: Euclidean and Non-Euclidean Geometries: Development and History 4th ed By Greenberg Modern Geometries: Non-Euclidean, Projective and Discrete 2nd ed by Henle Roads to Geometry 2nd ed by Wallace and West Hyperbolic Geometry, by Cannon, Floyd, Kenyon, and Parry from Flavors of Geometry http://myweb.tiscali.co.uk/cslphilos/geometry.htm http://en.wikipedia.org/wiki/Tessellation http://www.math.umn.edu/~garrett/a02/H2.html http://www.geom.uiuc.edu/~crobles/hyperbolic/hypr/modl/

Euclid’s Postulates (Henle, pp. 7-8) A straight line may be drawn from a point to any other point. A finite straight line may be produced to any length. A circle may be described with any center and any radius. All right angles are equal. If a straight line meet two other straight lines so that as to make the interior angles on one side less than two right angles, the other straight lines meet on that side of the first line.

Euclid’s Fifth Postulate Attempts to deduce the fifth postulate from the other four. Nineteenth century: Carefully and completely work out the consequences of a denial of the fifth postulate. Alternate assumption: Given a line and a point not on it, there is more than one line going through the given point that is parallel to the given line.

People Involved F.K. Schweikart (1780-1859) F.A. Taurinus (1794-1874) C.F. Gauss (1777-1855) N.I. Lobachevskii (1793-1856) J. Bolyai (1802-1860)

Why Hyperbolic Geometry?

Circle Limit III by M. C. Escher (1959) from http://en. wikipedia http://www.math.umn.edu/~garrett/a02/H2.html

Disk Models Poincare Disk Klein-Beltrami Model

Upper Half Plane Model

Minkowski Model