Non-Euclidean Geometries



Non-Euclidean Geometries

Introduction

A Flexible Geometry

Some Finite Geometries

Three Point Geometry

Five Point Geometry

An Incidence Geometry

Taxicab Geometry

Spherical Geometry

Hyperbolic Geometry

Answer to exercises

Introduction

An axiomatic system is a formalized foundation for a subject. Axiomatic systems for geometry have the following structure:

Underlying foundations:

The language of logic,

The rules of proof,

The axiomatic system

Undefined terms,

Axioms,

Definition, and

Theorems.

Understanding a system requires modeling, exploration, concentration, and arguments. We’ll start with simple systems and we’ll build up to large complicated geometrical systems.

Undefined terms are a small set of nouns or adverbs that start the whole program off.

We take these as basics and build with them.

Axioms are true statements that lay down the rules of association for the undefined terms.

It is almost always possible to visualize the undefined terms and to build a model of the geometry using the axioms.

Definitions build on the axioms and the undefined terms. They bring richness to the system.

Theorems are truths about the system that require a proof.

We’ve seen one of the bigger geometries: Euclidean Geometry. All the other geometries are in a big set called “Non-Euclidean Geometries”.

We’ll look at some petite geometries. Some are so small, they’ll fit on one page!

And we’ll look at Taxicab Geometry – that’s a famous one. Then we’ll move into Spherical Geometry and finish with Hyperbolic Geometry. These are two big geometries and are as important today as Euclidean Geometry.

A Flexible Geometry **

Undefined terms: point, line, on

Axioms: A1 Every point is on exactly two distinct lines.

A2 Every line is on exactly three distinct points.

Models:

There are lots of very different models for this geometry.

Here are two:

One has a finite number of points and the other has an infinite number of points.

Model 1: 3 points, 2 lines

Points are dots and lines are S-curves. One line

is dotted so you can tell it from the second line. Nobody ever said “lines” have to be straight things, you know.

Note, too, that there are only 3 points so my lines are composed of some material that is NOT points. Some non-point stuff. Luckily they’re undefined terms so I don’t have to go into it.

Model 2: an infinite number of points and lines

This is an infinite lattice. Each line is slanted at 45° and has 3 points along it. It continues forever left and right

Your Model:

You will be asked to invent a model of this on your own in your homework

Ideas for

Definitions:

Biangle – each two-sided, double angled half of the first model…like a triangle but only two points. Do biangles exist in Euclidean geometry (ah, no…check the axioms…two lines meet in exactly ONE point in Euclidean geometry.)

Quadrangle – each diamond-shaped piece of the second model

Parallel lines – Parallel lines share no points.

The second model has them; the first doesn’t. How many lines are parallel to a given line through a given point NOT on that line in the second model? (two! This, too, is really different than Euclidean Geometry).

Collinear points – points that are on the same line.

Midpoints – are these different from endpoints in a way that you can explicate in a sentence for Model 2? Does it make sense to have a “distance” function in this geometry – maybe not…maybe this is something we’ll just leave alone.

What do you notice that cries out for a definition in your model? Making a suggested definition is a homework question.

Theorems: Consider the following questions and formulate some proposed theorems (called “conjectures” until they’re proved)

Flexible Geometry Exercise:

Are there a minimum number of points?

Is there a relationship between the number of points and the number of lines?

Why is this a Non-Euclidean Geometry?

**This geometry is introduced in Example 1, page 30 of

The Geometric Viewpoint: a Survey of Geometries by Thomas Q. Sibley;

1998; Addison-Wesley (ISBN 0-201-87450-4)

Some Finite Geometries

Next we have 2 geometries that are much more closely specified by their axioms than

“A Flexible Geometry”. In fact, there are a specific number of points in each one.

The following two examples of finite geometries each has a model with a different number of points, and neither has an alternative model with more or fewer points. The axioms are quite specific and controlling on this issue.

Note that the axioms are quite specific about which undefined terms are “incident” or bearing upon one another in all three geometries.

Then we will explore another type of geometry is called an Incidence Geometry.

The axioms for an Incidence Geometry are specific about a couple of things but do allow at least two distinctly different models.

The Three Point Geometry

Undefined terms: point, line, on

Axioms: A1 There are exactly three distinct points.

A2 Two distinct points are on exactly one line.

A3 Not all the points are on the same line.

A4 Each pair of distinct lines is on exactly one point.

Model:

Three Point Geometry exercise:

There’s only one basic model for this geometry. Sketch it here:

Possible Definitions:

Collinear Points – points that are on the same line are collinear.

Theorems:

Theorem 1: Each pair of distinct lines is on exactly one point.

Theorem 2: There are exactly 3 distinct lines in this geometry.

Proof of Theorem 1

Theorem 1: Each pair of distinct lines is on exactly one point.

Suppose there’s a pair of lines on more than one point. This cannot be because then the two lines have at least two distinct points on each of them and Axiom 2 states that

A2 Two distinct points are on exactly one line.

Thus our supposition cannot be and the theorem is true.

QED

This type of proof is called a proof by contradiction. It works like a conversation.

Someone asserts something and someone disagrees and contradicts them. The assertion is the theorem and the contradiction is the sentence that begins with “Suppose…”.

Then the first person points out why the supposition cannot possibly be true…which has the handy property that it proves the theorem.

The proper contradiction to an assertion that “exactly one” situation is true is to suppose that “more than one” is true.

The proof for Theorem 2 is a homework problem.

The Five Point Geometry

Undefined terms point, line, on

Axioms A1 There are exactly five points.

A2 Any two distinct points have exactly one line on both of them.

A3 Each line is on exactly two points.

Models

Points: {P1, P2, P3, P4, P5}

Lines: {P1P2, P1P3, P1P4, P1P5, P2P3, P2P4, P2P5,

P3P4, P3P5, P4P5}

Note that the lines crossover one another in the interior of the “polygon” but DO NOT intersect at points. There are only 5 points!

Possible Definitions

Triangle -- a closed figure formed by 3 lines. An example: P2P1P4 is a triangle.

How many triangles are there? (5)

Quadrilateral – a closed figure formed by 4 lines. An example: P2P5P4P3 is a quadrilateral. How many quadrilaterals are there? (5)

Distance – probably not meaningful since every point is connected to every other point.

Angle measurement – probably not meaningful – it’s another distance and this geometry is too small for distance to mean much.

Parallel lines – two lines are parallel if they share no points.

Note that line P1P2 is parallel to line P4P5.

5 Point Geometry Exercise:

How many pairs of parallel lines are there?

Other possible defnitions:

Collinear points

Interior

Plane

Theorem

Each point is on exactly 4 lines.

An Incidence Geometry

Undefined terms: point, line, on

Axioms: A1 There is exactly one line on any two distinct points.

A2 Each line has at least two distinct points on it.

A3 There are at least three points.

A4 Not all the points lie on the same line.

Models:

Two examples follow; there are others.

Definitions – We’ll look at the models and see what makes sense…

Parallel lines

The distance from point one to point two

Angle measurement

Intersecting lines

Triangles

Quadrilaterals

Between or interior

Concurrent lines

Theorems:

Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.

Theorem 2: Each point is on at least two lines.

Theorem 3: There is a triple of lines that do not share a common point.

An Incidence Geometry, continued

A six point model:

The ONLY points are the 6 dots that are labeled. Note that in the interior of the “polygon” there are NO intersections of lines at points.

Imagine the points are little Styrofoam balls and that the lines are pipe cleaners…where two pipe cleaners lay on top of each other there’s no intersection only a “crossover”. Only at the ends where the ends are stuck into the balls is there a point and an intersection.

The points are: A, B, C, D, E, and F.

The lines are:

AB BC CD

AC BD CE

AD BE CF

AE BF

AF

A1 There is exactly one line on any two distinct points. See the list

A2 Each line has at least two distinct points on it. See the “endpoints”.

A3 There are at least three points. There are 6 which is “at least 3”.

A4 Not all the points lie on the same line. See the list.

Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.

Theorem 2: Each point is on at least two lines.

Theorem 3: There is a triple of lines that do not share a common point.

Theorem 1: For example: lines BF and BE intersect only at B.

The “crossovers” in the interior are not intersections.

Theorem 2: Each point is on 5 lines which is “at least 2”.

Theorem 3: All you have to do with Theorem 3 is show one triple:

AB, AF, and ED do not share a common point.

Let’s look at the situation with respect to parallel lines.

We will use the definition that two lines parallel lines if they share no points.

In Euclidean Geometry, if you have a line and a point not on that line, there is exactly one line through the point that is parallel to the given line.

Let’s check this out:

Take line AC and point B. These are a line and a point not on that line.

Now look at lines BF and BD. Both of these lines are parallel to line AC.

(recall that the lines that overlap in the “interior of the pentagon” do NOT intersect at a point – there are only 5 points in this geometry).

So there are exactly TWO lines parallel to a given line that are through a point not on the given line. This is certainly non-euclidean!

An Incidence Geometry, continued

A Model with an infinite number of points and lines:

Points will be {(x, y) ( x2 + y2 < 1}, the interior of the Unit Circle,

and lines will be the set of all lines that intersect the interior of this circle.

So our model is a proper subset of the Euclidean Plane.

Model:

Note that the labeled points (except H) are NOT points in the geometry. A is on the circle not an interior point. It is convenient to use it, though.

H is a point in the circle’s interior and IS a point in the geometry.

We cannot list the number of lines – there are an infinite number of them.

Checking the axioms:

A1 There is exactly one line on any two distinct points.

This model is a subset of Euclidean geometry and the axiom holds.

A2 Each line has at least two distinct points on it.

Each line has an infinite number of points by Euclidean Axioms.

A3 There are at least three points.

The unit disc has an inifinite number of points.

A4 Not all the points lie on the same line.

True

Definitions:

Parallel lines: lines that share no points are parallel.

In Euclidean Geometry, there is exactly one line through a given point not on a given line that is parallel to the given line.

Interestingly, in this geometry there are more than two lines through a given point that are parallel to a given line.

Let’s look at lines GC and GB. They intersect at G…which is NOT a point in the geometry. So GC and GB are parallel. In fact, they are what is called asymptotically parallel. They really do share no points.

Now look at P1P2. It, too, is parallel to GC. Furthermore both P1P2 and GB pass through point H.

P1P2 is divergently parallel to GC.

Not only is the situation vis a vis parallel lines different, we even have flavors of parallel:

asymptotic and divergent. So we are truly non-euclidean here, folks.

Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.

Inherited from Euclidean Geometry.

Theorem 2: Each point is on at least two lines.

Each point is on an infinite number of lines.

Theorem 3: There is a triple of lines that do not share a common point.

FE, GC, and AD for example.

Incidence Geometry Exercise:

Find at least one way that this geometry is like Euclidean Geometry.

Find three ways that this geometry is different from Euclidean Geometry.

Answers to exercises

Three Point Geometry exercise:

[pic]

6 Point Geometry Exercise:

Note that line P1P2 is parallel to line P4P5.

How many pairs of parallel lines are there?

There are 30 pairs of parallel lines. Each line is parallel to three others.

Here are some examples:

P1P2 is parallel to P3P4, P3P5, and P4P5.

P1P5 is parallel to P2P3, P2P4, and P3P4.

P2P5 is parallel to P1P3, P1P4, and P3P4.

Flexible Geometry Exercise:

Are there a minimum number of points?

There are 3 points to the “minimum” model.

Is there a relationship between the number of points and the number of lines?

The ratio of points to lines is 3:2.

Why is this a Non-Euclidean Geometry?

The axioms are totally different from Euclidean Geometry.

Note that this geometry has several distinctly different models – and Euclidean Geometry has essentially only one model. (This is a new fact for you!)

Incidence Geometry Exercise:

Find at least one way that this geometry is like Euclidean Geometry.

The axiomatic structure, the framework, is the same format:

undefined terms

axioms

definitions

theorems

The points and lines are actually from Euclidean Geometry and are actually called points and line segments.

Find three ways that this geometry is different from Euclidean Geometry.

The axioms themselves are different.

There are at least two different models; Euclidean Geometry has only one model.

There are two types of parallel lines with infinite point model – this is totally

non-euclidean. The finite model has two lines parallel to a given line through a point not on the line; this, too, is different.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download