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Mr. Young’s Geometry Classes, Spring 2005

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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?

If 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.

Your’e just a crazy old man.

EUCLID

Mr. Young’s Geometry Classes,

Spring 2005

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

I wrote that book.

It’s Elementary!

The

Elements

1. What is Euclidean Geometry?

- Euclidean Geometry is the Geometry that is taught in High School Geometry Classes

- It is based primarily on a book called “The Elements” written by a Greek Mathematician named Euclid who lived from about 325-265 B.C.

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

Isn’t it obvious?

1. What is Euclidean Geometry?

- Euclid’s Fifth Postulate, called the Parallel Postulate seems obvious, but is the source of much debate.

5. Given a line and a point not on that line, there is exactly one line through the point that is parallel to the line

- 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…

That’s crazy!

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.

=

- 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.

Of course

They’re true!

3. Spherical Geometry

- Are Euclid’s Postulates true in Spherical Geometry?

1. Through any two points there is exactly one line

TRUE

2. Through any three points not on the same line there is exactly one plane

TRUE

Duh! I wouldn’t

Write them if

they weren’t

3. Spherical Geometry

- Are Euclid’s Postulates true in Spherical Geometry?

3. A line contains at least two points

TRUE

4. All right angles are congruent

TRUE

What the…!?

That’s just weird

3. Spherical Geometry

- 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

3. Spherical Geometry

- 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.

That’s crazy talk

3. Spherical Geometry

- Other strange things happen in Spherical Geometry

- Lines always intersect at 2 points, not one.

Whatever.

B

C

A

3. Spherical Geometry

- 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...

C is between A and B.

B

What are you

Trying to say?

That I’m wrong?

A

C

3. Spherical Geometry

- 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.

No It’s Flat.

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.

- 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

NEWBERRY

ROCKS!

4. Spherical Geometry: A Real World Application

- Any Location on the Earth can be found with its, a latitude and longitude.

Newberry, FL

Lat. = 29.6° N

Long. = 82.6° W

Newberry, FL

4. Spherical Geometry: A Real World Application

- Latitude

- Longitude

90° N

60° N

30° N

180° E, 0 °

150° W

0°

60° E

90° W

60° W

30° W

30° E

90° E

120° W

120° E

150° E

30° S

60° S

90° S

OOOOH.

Angles.

4. Spherical Geometry: A Real World Application

- This picture shows the angles that define the degrees for longitude and latitude

What’s your

Sign baby?

4. Spherical Geometry: A Real World Application

- Astronomers use a similar concept to define the position of stars and other objects in the sky.

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

Euclid vs. The Sphere

Which Geometry is right?

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.

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

- But Euclidean Geometry works well in most cases
- Finding most distances or lengths
- Most everyday activities that require geometry like construction, drawing, etc.

I guess we have to agree to get along. But at least I’m taught in schools.

Good teachers teach spherical geometry anyway.

4. Euclidean vs. Spherical Geometry

Euclid vs. The Sphere

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.

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

Now I know

You’re crazy.

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.

But it is the most common

I can accurately model the real world.

Summary

- Euclidean Geometry is not the only type of Geometry.

- Spherical Geometry is one example of Non-Euclidean Geometry that has definite practical applications.

- Euclidean Geometry sufficiently describes the world that most of us deal with day to day, so it is the primary Geometry studied in School.

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.

The End

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.