Created by: Mr. Young EUCLID If you have a line and a point, it is just obvious that there is only one line through that point that is parallel to the line. Non-Euclidean Geometry? How can you have Geometry without me? EUCLID I wonder if people realize that they have me to thank for having to write proofs. Most people don’t even know that regular geometry is Euclidean Geometry. You’re just a crazy old man. Created by: Mr. Young plaza.ufl.edu/youngdj/powerpoint/noneuclidean.ppt

Outline 1. What is Euclidean Geometry? 2. What is Non-Euclidean Geometry? 3. Spherical Geometry 4. Spherical Geometry: A Real World Application 5. Euclidean vs. Spherical Geometry 6. Other Geometries

1. What is Euclidean Geometry? Euclidean Geometry is the Geometry that is taught in High School Geometry Classes It is based primarily on a series of books called “The Elements” written by a Greek Mathematician named Euclid who lived from about 325-265 B.C. I wrote that book. It’s Elementary! The Elements

1. What is Euclidean Geometry? Euclidean Geometry deals with points, lines and planes and how they interact to make more complex figures. Euclid’s Postulates define how the points, lines, and planes interact with each other. Remember: A Postulate is statement that is assumed to be true.

1. What is Euclidean Geometry? Euclid’s First Four Postulates are as follows: 1. Through any two points there is exactly one line 2. Through any three points not on the same line there is exactly one plane 3. A line contains at least two points 4. All right angles are congruent

1. What is Euclidean Geometry? Euclid’s Fifth Postulate, called the Parallel Postulate seems obvious, but is the source of much debate. 5. Through a point not on a line, there is exactly one parallel to the given line. Isn’t it obvious? You can say that Euclidean Geometry is Geometry in which the parallel postulate holds.

2. What is Non-Euclidean Geometry? Non-Euclidean Geometry is any Geometry that uses a different set of postulates than Euclid used. Most of the time Non-Euclidean Geometry is Geometry in which the Parallel Postulate does not hold to be true.

2. What is Non-Euclidean Geometry? If the parallel postulate is not true that means that given a line and a point not on the line there is NOT exactly one line through the point which is parallel to the line. How is this possible? Remember that points, lines, and planes are undefined terms. Their meaning comes only from postulates. So if you change the postulates you can change the meaning of points, lines, and planes, and how they interact with each other. This is most easily seen by example…

3. Spherical Geometry The main difference between Spherical Geometry and Euclidean Geometry is that instead of describing a plane as a flat surface a plane is a sphere. That’s crazy! = A line is a great circle on the sphere. A great circle is any circle on a sphere that has the same center as the sphere. Points are exactly the same, just on a sphere.

Are Euclid’s Postulates true in Spherical Geometry? 1. Through any two points there is exactly one line TRUE Of course They’re true! 2. Through any three points not on the same line there is exactly one plane TRUE

Are Euclid’s Postulates true in Spherical Geometry? 3. A line contains at least two points TRUE 4. All right angles are congruent Duh! I wouldn’t Write them if they weren’t TRUE

Is the Parallel Postulate true in Spherical Geometry? 5. Given a line and a point not on that line how many lines can be drawn through the point that are parallel to the line? NONE, Therefore the Parallel Postulate is FALSE in Spherical Geometry What the…!? That’s just weird

Is the Parallel Postulate true in Spherical Geometry? 5. Given a line and a point not on that line how many lines can be drawn through the point that are parallel to the line? NONE, Therefore the Parallel Postulate is FALSE in Spherical Geometry Common Mistake: Except for the circle in the middle, these horizontal circles do not share a center with the sphere and are therefore can not be considered parallel lines, even though they appear to be parallel.

Other strange things happen in Spherical Geometry Lines always intersect at 2 points, not one. That’s crazy talk

Other strange things happen in Spherical Geometry In the diagram below B is between A and C, but... A is between B and C, and... Whatever. C is between A and B. A B C

Other strange things happen in Spherical Geometry The angles in a triangle don’t have to add to 180º In the diagram below ∆ABC has 3 right angles, which add to 270. A C B What are you Trying to say? That I’m wrong?

4. Spherical Geometry: A Real World Application If Spherical Geometry is so strange why do we even bother studying it? Because the Earth is a Sphere. No It’s Flat. Euclidean geometry can not be used to model the Earth because it is a sphere. Instead of the Cartesian coordinates used in Euclidean Geometry Longitude and Latitude are used as to define position of points on the Earth.

4. Spherical Geometry: A Real World Application Lines of Longitude are great Circles running between the North and South Poles. The “Center” Longitude is called the Prime Meridian Degrees West Degrees East Longitude is measured in degrees East or West from the prime meridian. Prime Meridian, 0°

4. Spherical Geometry: A Real World Application Lines of Latitude are parallel horizontal circles, but not great Circles The “Center” Latitude is called the equator Degrees North Equator, 0° Latitude is measured in degrees North or South from the equator Degrees South

4. Spherical Geometry: A Real World Application Any Location on the Earth can be found with its latitude and longitude. Clarks Summit, PA Lat. = 41.5° N Long. = 75.7° W Clarks Summit, PA CLARKS SUMMIT ROCKS!

4. Spherical Geometry: A Real World Application Latitude Longitude 0° 60° E 120° W 30° E 90° E 120° E 150° E 30° S 60° S 90° S

4. Spherical Geometry: A Real World Application This picture shows the angles that define the degrees for longitude and latitude OOOOH. Angles.

4. Spherical Geometry: A Real World Application Astronomers use a similar concept to define the position of stars and other objects in the sky. What’s your Sign baby?

Euclid vs. The Sphere 4. Euclidean vs. Spherical Geometry Which Geometry is right? Were you just watching that whole globe thing?I am right. Duh! I am obviously right Oh it’s on now. You better watch out old man.

4. Euclidean vs. Spherical Geometry Spherical Geometry must be used in some cases: Finding long distances for flights, driving or sailing. Predicting paths of weather Map making Did you hear that? Most people live their lives without you. Everyone needs me. See! You can’t live without me. Take that old man. But Euclidean Geometry works well in most cases Finding most distances or lengths Most everyday activities that require geometry like construction, drawing, etc.

Euclid vs. The Sphere 4. Euclidean vs. Spherical Geometry Which Geometry is right? Neither Geometry is the “right” Geometry, but since Euclidean Geometry works in most cases and is simplest, it is taught in schools. Good teachers teach spherical geometry anyway. I guess we have to agree to get along. But at least I’m taught in schools.

5. Other Geometries Spherical Geometry is just one Example of Non-Euclidean Geometry Any Geometry that starts with a different set of postulates is Non-Euclidean. Some other Geometries have practical applications and some are just theoretical

5. Other Geometries Hyperbolic Geometry is used to model space since Einstein’s theories imply that space is curved. Here is a model of a three dimensional hyperbolic curve, that Hyperbolic Geometry would be based on. Now I know You’re crazy.

Euclidean Geometry is not the only type of Geometry. Summary Euclidean Geometry is not the only type of Geometry. But it is the most common Spherical Geometry is one example of Non-Euclidean Geometry that has definite practical applications. I can accurately model the real world. Euclidean Geometry sufficiently describes the world that most of us deal with day to day, so it is the primary Geometry studied in School.

The End I’m sorry I resorted to violence. It’s just that I am so passionate about parallel lines. I’m sorry I called you an old man. You actually look pretty good for being almost 3,000. Let’s be friends. Want to learn more about Non-Euclidean geometry? Do a web search. Or better yet major in Math and take a course on it in college.