Other Geometries

There are MANY geometries other than the familiar Euclidean. We will look at several of them. Please pay attention to what is similar and what is different about each as we progress.

Finite Geometries Incidence Geometry Taxicab Geometry Spherical Geometry Hyperbolic Geometry

1 Introduction to Axiomatic Systems

In studying any geometry, it is important to note the axiomatic framework of the geometry and keep it in mind. Often students are so challenged by the details that they forget that there is a structure to geometry. Each geometry has a framework called its axiomatic system. An outline of a typical axiomatic system is below.

Any axiomatic system has four parts:

undefined terms axioms (also called postulates) definitions theorems

The undefined terms are a short list of nouns and relationships. These terms may be visualized but cannot be defined. Any attempt at a definition ends up circling around the terms and using one to define the other. These are the basic building blocks of the geometry. It is usually a good idea to have a mental image of the undefined terms – a visualization of the objects and how they relate.

Axioms (or postulates) are a list of rules that define the basic relationships among the undefined terms and make clear the fundamentals facts about a system. Axioms are always true for the system. No deviation from the facts they state is permitted in working with the system.

Definitions and theorems build on the axioms and undefined terms, clarifying relationships and auxiliary facts.

We will be using, with slight modification, the set of undefined terms and axioms developed by The School Mathematics Study Group during the 1960’s for this module. This list of axioms is not as brief as one that would be used by graduate students in a mathematics program nor as long as some of those systems in use in middle school textbooks. One definite advantage to the SMSG list is that it is public domain by design. We will be using the Cartesian coordinate plane as our visualization of the undefined terms of Euclidean geometry.

Once we have spent time learning the axioms, some definitions and a few theorems we will move to the second module on Euclidean Topics and look at geometric shapes and proofs that require using the axioms, definitions, and theorems in concert

2 In the following geometries:

In the following two examples of finite geometries, each has exactly one model 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.

In TCG, EG, SG there are only one model. HG has several 2D and a 3D model!

In Euclidean Geometry, there is exactly one line through a given point not on a given line that is parallel to the given line. In the following geometries, some have NO parallel lines, others have more than one through that point. It’s a variable kind of thing!

3 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 at least 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.

4 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.

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.

Note: we have no need of a distance formula or a way to measure angles with this simple little geometry. These things MUST be defined in the axioms for you to have them!

Given a line and a point not on that line, how many lines are there through that point and parallel to the given line?

5 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

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

Lines: {P1P2, P1P3, P1P4, P1P5, P2P3, P2P4, P2P5, P2 P5 P3P4, P3P5, P4P5}

Note that the lines crossover one another in the interior of the P3 P4 “polygon” but DO NOT intersect at points. There are only 5 points! LINE STUFF!

Possible Definitions

Triangle -- a closed figure formed by 3 lines. An example: P2P1P4 is a triangle. How many triangles are there?

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

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

Note that line P1P2 is parallel to line P4P5. So are P3P4 and P2P5…

Let’s look at a given line and a given point…

Other possible definitions:

Collinear points Interior Plane

Theorem

Each point is on exactly 4 lines.

7 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.

Note: no distance, no angle measure

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 Intersecting lines Triangles Quadrilaterals Between or interior Concurrent lines – share a point…more than 2 are “concurrent”

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.

Theorems must be totally true in EVERY MODEL!

8 An Incidence Geometry, continued

A six point model:

The ONLY points are the 6 dots that are labeled. Note

C that in the interior of the “polygon” there are NO A 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 D F “crossover”. Only at the ends where the ends are stuck into the balls is there a point and an intersection. E The points are: A, B, C, D, E, and F. The lines are:

AB BC CD DE EF AC BD CE DF 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, CF, and ED do not share a common point.

9 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, BE, 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 6 points in this geometry).

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

10 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 A interior point. It is convenient to use it, though. B H P1 P2 H is a point in the circle’s interior and IS a point in the geometry. C We cannot list the number of lines – there are an G D 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

11 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 A G…which is NOT a point in the geometry. So B H P1 GC and GB are parallel. In fact, they are what is P2 called asymptotically parallel. They really do share no points. C Now look at P1P2. It, too, is parallel to GC. D G Furthermore both P1P2 and GB pass through point H. F E 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.

12 The SMSG Axioms for Euclidean Geometry

Undefined Terms: point, line, and plane

We take as our beginning point the undefined terms:

point, line, and plane.

Most people visualize a point as a tiny, tiny dot. Lines are thought of as long, seamless concatenations of points and planes are composed of finely interwoven lines: smooth, endless and flat.

Think of undefined terms as the basic sounds in a language – the sounds that make up our language for the most part have no meaning in themselves but are combined to make words.

The grammar of our language and a good dictionary are what make the meaning of the sounds. This part of language corresponds to the axioms and definitions that you will find next in the module.

From there the facts, flights of fancy, and content-laden sentences are built – these are the theorems and definition in an axiomatic system.

The conventions of the Cartesian plane are well suited to assisting in visualizing Euclidean geometry. However there are some differences between a geometric approach to points on a line and an algebraic one, as we will see in the explanation of Axiom 3.

13 SMSG Postulates for Euclidean Geometry:

A1. Given any two distinct points there is exactly one line that contains them.

A2. The Distance Postulate:

To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points.

A3. The Ruler Postulate:

The points of a line can be placed in a correspondence with the real numbers such that A. To every point of the line there corresponds exactly one real number. B. To every real number there corresponds exactly one point of the line, and C. The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.

A4. The Ruler Placement Postulate:

Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.

A5. A. Every plane contains at least three non-collinear points.

B. Space contains at least four non-coplanar points.

A6. If two points line in a plane, then the line containing these points lies in the same plane.

A7. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

A8. If two planes intersect, then that intersection is a line.

A9. The Plane Separation Postulate: Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that

A. each of the sets is convex, and

B. if P is in one set and Q is in the other, then segment PQ intersects the line.

14 A10. The Space Separation Postulate:

The points of space that do not line in a given plane form two sets such that

A. each of the sets is convex, and

B. if P is in one set and Q is in the other, then the segment PQ intersects the plane.

A11. The Angle Measurement Postulate:

To every angle there corresponds a real number between 0 and 180.

A12. The Angle Construction Postulate:

Let AB be a ray on the edge of the half-plane H. For every r between 0 and 180 there is exactly one ray AP with P in H such that m  PAB = r.

A13. The Angle Addition Postulate:

If D is a point in the interior of  BAC, then m  BAC = m  BAD + m  DAC.

A14. The Supplement Postulate:

If two angles form a linear pair, then they are supplementary

A15. The SAS Postulate:

Given an one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

A16 The Parallel Postulate:

Through a given external point there is at most one line parallel to a given line.

A17. To every polygonal region there corresponds a unique positive number called its area.

A18. If two triangles are congruent, then the triangular regions have the same area.

A19. Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2.

A20. The area of a rectangle is the product of the length of its base and the length of its altitude.

15 A21. The volume of a rectangular parallelpiped is equal to the product of the length of its altitude and the area of its base.

A22. Cavalieri’s Principal:

Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane, the two intersections determine regions that have the same area, then the two solids have the same volume.

16 Taxicab Geometry uses the SAME undefined terms and axioms as EG up to SAS A15. is based on the Cartesian Plane. We will use the SAME points in the plane as we use for Euclidean Geometry and the same lines. We will use the same way of measuring angles, too. BUT we will use a DIFFERENT formula to measure distance.

Let’s first review the old distance on a line formula: in one dimension and in two dimensions:

Where Euclidean Geometry then uses the Pythagorean Theorem to get distance, we will simple ADD the two distances:

Do you see WHY it’s called “Taxicab”?

17 HANDOUT 6.1 A

Let’s find some Taxicab distances and some Euclidean distances and see what’s going on.

Find the distances between the following points:

Points Distance E Distance T (0, 0) to (0, 4) (1, 1) to (3, 4) (−1, 2) to ( −5, −2) (0, 0) to (2, 2) (0, 0) to (5, 0)

What do you notice?

Another interesting point to note is that the Euclidean distance is always less than or equal to the Taxicab distance (p. 211).

Let’s explore a triangle or two. BOTH geometries have triangles, but Taxicab Geometry has some surprises in theirs.

HANDOUT 6.1 B

18 Let’s study some triangles. Grab a piece of graph paper and get started:

Here’s one:

Find the vertices (1, 2), (3, 2) and (2, 3). Now let’s list the Euclidean characteristics.

Side lengths (Euclidean):

Angle measurements:

Now those angle measurement stay the same in TCG.

Side lengths (TCG):

Oops what did we find?

Here’s another: (5, 4), (5, 1) and (9, 1)

It’s a right triangle in both geometries, isn’t it?

Let’s measure the sides.

EG: TCG:

Ooops! What theorem is NOT true in TCG?

Here’s a third one:

19 Let’s move into Quadrant 2: Put on the base: (3, −5) and (3, −8). Let’s use TRIG to find the coordinates of the apex angle:

Recall: we can solve for the height with this!

Now, let’s see if it’s equilateral (EG)…and what about those angle measures, then?

Now what about in TCG?

Back to the section:

Now let’s review congruence in the Big picture:

For two figures to be congruent, corresponding sides and corresponding angles need to be congruent. Let’s note that the perimeter and the area of congruent triangles have the same measure.

Now, in reality, that is SEVEN things to check:

First that you have a reasonable correspondence, Then 3 angles and 3 sides.

But we all know the Euclidean shortcut: SAS.

Would it surprise you to know that the shortcut does NOT always work in TCG?

Let’s look at two situations: one where it does work and one where it doesn’t and see if we can figure out a way to predict when it’s safe to use the shortcut.

HANDOUT 6.1 B

20 Here’s a situation when it DOES work:

We’ll take a 3-4-7 right triangle (TCG; 3-4-5 EG, of course) and we’ll go [let’s translate that to English before we start]

Start with (5, 1), (9, 1) and (5, 4). Now transform it.

Ok. Are they congruent and does SAS work?

Now let’s take this pair:

(1, 2), (3, 2) and (2, 3) and let’s use the 2 – 90 degrees – 2 as our SAS

Compare this to (−3, 0), (−1, 0) and (−1, 2)

And we’ll use the 2 – 90 degrees – 2 as our SAS. Are THESE two congruent?

Now let’s get to the bottom of why the first one works and the second one doesn’t. Would a 45 degree rotation about the origin make the first triangle and the rotated triangle congruent in TCG? Would a translation of 6 cm along a line at 30 degrees to the x-axis work?

What was so special about that glide reflection I chose? Is it that it was a glide reflection or something else?

Now let’s continue looking at things that are the same and different in TCG and EG.

21 Circles! Both geometries have circles.

Let’s review Euclidean circles:

Definition: The set of points that are equidistant from a given center. The distance from the center to a point in the set is called the radius.

HANDOUT 6.1 C

Center:(0,0) Radius 4.

Grab some graph paper! Let’s graph a TCG circle with the above facts.

HANDOUT 6.1 D

Now let’s look at intersections and tangencies. Euclidean circles can share: No points One point (internal and external) Two points.

Taxicab circles can share No points, One point Two points An infinite number of points…let’s sketch some of these!

So, now there are some similarities and differences between TCG and EG. Let’s list a few:

The undefined terms and the first 14 axioms are the same. A15 is not true for TCG! Let’s make a list and have a discussion!

22 Points, the Cartesian Plane…these are the same

Distance formulas are different; angle measuring is the same

Triangles

Congruence

Circles

Now let’s do an exercise with a perpendicular bisector.

Definition: the line that is equidistant from the endpoints of a given segment

HANDOUT 6.1 E

Using the line segment (2, 3) and (6, 5). Find the midpoint of this line. Is the midpoint on the bisector?

Using the taxicab distance find 3 points above and 3 points below the line that are equidistant from the endpoints.

Now find 3 more above and below. Did something unusual happen?

What’s the same and what’s different here?

23 Let’s talk about trigonometry. If we use a 3 – 4 – 5 Euclidean right triangle we know things about the angles of that triangle. We measure the angles in TCG with the same Euclidean Protractor that we always use. Does trig hold up if we change the hypotenuse length to the TCG required “7”?

Would it surprise you to know that the trig we learned is Euclidean and that there’s an entirely different trig called Taxicab Trigonometry?

24 Spherical Geometry

Spherical Geometry 6.2

How, exactly, do you indicate a point on the surface of a sphere? Because it is 3D, we need three items of information – two angles and one length.

(rho) is the length of the vector from the center of the sphere to the point. The formula is

 theta) is the angle that the vector makes when projected into the x-y plane. It is measured from the positive x-axis just like we do for Euclidean trigonometry work

 Fe, as in Fe Fi Fo Fum) is measured from the positive z-axis down to the vector.

We will stick to radian measure for  and can use either degrees or rad for .

Let’s get a sketch* or two up: First let’s look at  then we’ll look at 

*leave them both up!

We’ll symbolize the x-y plane as a line and show the z-axis going up. There’s 

25 Then we’ll look down straight at the z-axis into the x-y plane and see theta. Do you see “r”

The radius? It’s still . And that’s a reason why the vector length is rho.

Now let’s look at some Euclidean trig in the first drawing. Do you see the right angle with it’s base in the x-y plane?

Note that we can use the motion principle and move the base of the angle “r” to the top to discuss the trig of fe.

Sin(fe) = opp/hyp = r/rho r = rho times sin(fe)

Cos(fe) = adj/hyp = z/rho z = rho times cos(fe)

Now fe is limited to and theta goes from 0 to 2pi.

26 We have a similar situation in the xy-plane for theta and trig functions:

x = r cos(t) y = r sin(t) tan(t) = y/x x cannot be zero

Now (x, y, z) coordinates can be transformed into (rho, theta, fe) in the following way:

ILLUSTRATION 6.2 A

If rho = 2, theta is 60 and fe is 30 degrees…sketch the point and show the Cartesian Coordinates.

Spherical Distance:

27 Let’s review:

Euclidean Distance

Taxicab Distance

Now: Distance in Spherical Geometry is

Now you’ll need a calculator for the cosine inverse part of things.

Let’s practice!

Rho is 3, (0,3,0) and (

What is the distance between these points in SG and where are they on the sphere of radius 3?

So all of these geometries have distance, given in the axioms, and each uses a different formula for it.

28 Latitude and Longitude:

Latitude – horizontal circles (only one, the equator, is a great circle!) and Longitude – these are all great circles through the North and South Pole.

The latitude degrees are the complement to fe! Take 90° – fe to get the degrees latitude. Now each degree is divided into 60 minutes and each minute is divided into 60 seconds…Babylonian based arithmetic here!

And you measure off the equator N or S

The book uses sigma, , for this angle measure.

29 Now let’s look at Spherical Geometry in a holistic sense.

There are triangles. Compare and contrast these triangles to Euclidean ones!

What about the “interior” and “exterior” triangles in Spherical?

How does this compare to the Cartesian Plane?

How many degrees to the interior angle sum? “excess”

How about the exterior of the triangle? Do we concern ourselves with the exterior in Euclidean geometry? Here, we do. We can discuss the angle sum of the exterior of a triangle!

The measure of the exterior of an angle is 360° minus the measure of the interior of an angle. (in radian measure 2pi!)

So the angle sum of the exterior of a triangle is adding up all the measures of the exterior angles (and in radian measure this is 6pi – the angle sum of the interior measures)

30 Example: can we have a spherical triangle with 75, 85, and 45 degrees?

75 + 85 + 45 = 205. What is the excess? What is the measure of each exterior angle?

How are the angle sums of the interior and exterior related?

Note that you can talk about the excess of the exterior, too. It’s the angle sum of the exterior minus pi!

What this the excess of the exterior of the 75-85-45 from above?

Do these triangles have area? Yes! And it is calculated with a formula, too. The Spherical area of a triangle formula is based on the angle measure NOT the side lengths! The area of a spherical geometry triangle is the triangle’s excess times rho-squared. The area of the EXTERIOR of a spherical triangle is the excess of the exterior times rho-squared.

Take the 75-85-45 triangle from above. What is it’s area? What is the area of the exterior?

31 Note that if you are using a sphere with radius one, and you add the area of the interior and the area of the exterior, you get 4pi which is the surface area of the unit sphere! P. 224

What about biangles?

Lunes or Biangles are a shape unique to Spherical geometry.

This is the simplest polygon on our surface. Build a lune on your model.

Keep in mind that the vertices are antipodal points**. Pairs that are 180° apart. Does Euclidean geometry have antipodal points?

It is important to focus on the geometric facts that we’ve been discussing all along:

Shapes (biangle), names for places and parts of figures (vertices), measurements of angles (rads) and of AREA:

Area of a lune = 2rho2

Where alpha is the lunar angle measure.

The summit angles are congruent in biangles

32 How about points? All the spherical points are actually points in Euclidean 3 space.

BUT SG has “Antipodal Points”**

BETWEENESS is different

Let’s sketch a great circle and talk about between. Now let’s sketch a Euclidean line and talk about between

How about circles? Do we have them?

Well, yes and no..we need to distinguish between Great Circles (lines) and circles…

And how about that two center thing?

33 Can we have isometries in Spherical Geometry? SURE! And we use matrices to accomplish them, too!

Let’s look at a cross section of the sphere bordered by a great circle. We can put on the circle and the center. Now if we sketch two radii (length rho, remember?)– with a central angle of theta and then connect the circle points, we have a CHORD, of length L.

Sketch this.

Now we can use the Law of Cosines to find the length of L (given the measure of theta), or to find cos (theta) if given the length L.

The formula for the Law of Cosines is

Let’s look at how you back solve for theta.

Example:

Given rho = 4 and theta = 45 degrees find L

Given rho = 2 and L = 3, find theta.

34 Now, let’s remember that using an orthogonal matrix makes an isometry in 3D. And an orthogonal matrix is special: .

A reflection across the yz-plane – picture this – is F (x, y, z) = F (− x, y, z) and the matrix is

Let’s try this with (2, 1, 0)

A rotation about the z axis by an angle theta – picture this! Uses this matrix

Example:

Filling in 90° for theta will move a point from the x axis to the y axis – Let’s check this with

(3, 0, 0)

Remember that cos 90 = 0 and sin 90 = 1 (on the unit circle, (0, 1) is 90 degrees!)

Note, too, that composition of isometries doesn’t work because matrices do NOT commute!

See the example on page 228.

Saccheri Quadrilaterals

36 This particular kind of quadrilateral appears in Euclidean, Spherical, and Hyperbolic geometries. It was discovered in the early 1700’s. Saccheri was a Jesuit priest and logician. He was a student of Ceva, another famous geometer.

Here’s the definition:

A bi-perpendicular quadrilateral with right angles at the base and congruent legs.

Let’s sketch this.

Now, in Euclidean Geometry the summit angles are right angles. In Spherical Geometry the summit angles are obtuse and in Hyperbolic Geometry (coming next week), the angles are acute.

Let’s discuss the lengths of the summit and base. EG? Let’s check on the model for SG and what do you suppose it is for HG.

Some theorems for the Saccheri Quadrilateral that are true in all three geometries:

37 The diagonals are congruent.

The summit angles are congruent.

38 Hyperbolic Geometry Enrichment and Workbook

39 Historical Background of Non-Euclidean Geometry

About 575 B.C. Pythagoras wrote his book on Geometry. Some of the material was known in other cultures centuries before he wrote it down, of course. His was the first axiomatic approach to organizing the material. Interestingly, he did as much work as he could before introducing the Parallel Postulate. Many people have interpreted this progression in his work as indicating a level of discomfort with the Parallel Postulate. It’s not really possible to know what he was really thinking. About 400 years after the birth of Christ, Proclus, a Greek philosopher and head of Plato’s Academy, wrote a “proof” that derived the Parallel Postulate from the first 4 Postulates. thereby setting the tone for research in Geometry for the next 1400 years. Johann Gauss, the great German mathematician, actually realized that there was another choice of axiom but didn’t choose to publish his work for fear of getting into the same trouble as other scholars had with the Catholic Church.

Around 1830, two young mathematicians published works on Hyperbolic Geometry – independently of one another. The world took no note of them. In 1868, the Italian Beltrami found the first model of Hyperbolic Geometry and in 1882, Henri Poincare developed the model we’ll study.

120 years later, Hyperbolic Geometry is finally making it into high school textbooks. My favorite is a text that St. Pius X used in the 90’s. If you ever get a chance to look at it – it’s just terrific. And it includes a section on Spherical Geometry as well:

Note that St. Pius isn’t a flagship diocese school – they actually have a very full section of “Algebra half”, the course for the kids not ready for Algebra I.

40 In the 1400 years of work on the axioms of Absolute Geometry there was always a special group of people who endeavored to prove that the Parallel Postulate was actually a theorem. In fact, Saccheri and Lambert who both came so close to realizing that there was an alternate geometry out there waiting to be discovered never got all the way past believing that the axiom was a theorem. Here is a list of statements that are equivalent to P1, the Euclidean Parallel Postulate:

 The area of a triangle can be made arbitrarily large.  The angle sum of all triangles is a constant.  The angle sum of any triangle is 180.  Rectangles exist.  A circle can be passed through any 3 noncollinear points.  Given an interior point of a angle, a line can be drawn through that point intersecting both sides of the angle.  Two parallel lines are everywhere equidistant.  The perpendicular distance from one of two parallel lines to the other is always bounded.

In the Hyperbolic Workbook section, I’ll ask you for illustrations from the Hyperbolic sketches in Sketchpad that show that each of these is NOT true in Hyperbolic Geometry.

So now let’s adopt

Axiom P-2 Hyperbolic Parallel Postulate

If L is any line and P any point not on L, there exists more than one line through P parallel to L.

41 Hyperbolic Geometry Workbook

The Poincare model for Hyperbolic Geometry is the following:

Points are normal Euclidean points in the Cartesian Plane that are included in a disc:

{(x, y)} [We usually pick r = 1.]

The points of the circle that encloses the disc are NOT points of Hyperbolic Geometry nor are any points exterior to the circle.

Lines are arcs of orthogonal circles to the given circle. A circle that is orthogonal to the given circle intersects it in two points and tangent lines to each circle at the point of intersection are perpendicular. Note that diameters of the disc are lines in this space even though they don’t look like arcs; each diameter is said to be an arc of a circle with a center at infinity.

Here are two orthogonal circles in a Sketchpad graph. Draw in the tangent lines to see this!

42 fx = 1-x2 1.4 gx = - 1-x2

5 5 2 1 1.2 hx = - x- + 8  4  4

5 5 2 1 1 qx = - - x- + 8  4  4 0.8

-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5

43 Enrichment 1: Lines in our space:

Poincaré Disk Model

This sketch depic ts the hyperbolic plane H2 us ing the Poincaré disk model. In this model, a line through tw o points is def ined as the Euc lidean arc pas sing through the points and perpendicular to the circ le. Us e this document's custom tools to perf orm c onstructions on the hyperbolic plane, comparing your f indings to equivalent constructions on the Euc lidean plane.

Dis k Controls

C P. Disk Center

blue circle...not part of our space

Here is a sketch from Sketchpad that shows hyperbolic line H which is part of the Euclidean circle intersecting the big blue circle (the given circle that is the space boundary). Sketch in the tangent lines to the blue circle and H. Do you see that the tangent lines are perpendicular?

Hyperbolic line is a diameter of the blue circle. It, too, is a line in our space. Draw in the tangents and you’ll see why.

To find this space in Sketchpad: open the Sketchpad Program files, select Samples/Sketches/Investigations/PoincareDisk. You have to use the sketching tools under the double headed arrow down the vertical left menu to construct lines and measure angles and such – you must not use the tools on the upper toolbar (those are Euclidean tools). [You might have to start with My Computer/local disk/Program files/Sketchpad, etc. – it depends on how the tech loaded Sketchpad in the first place – it IS worth finding, though.]

44 Enrichment 2: Parallel Lines in Hyperbolic Geometry

Dis k Controls F

H is parallel to every other line showing in the disc.

Since H intersects H on the circle, these two have a type of parallelism called “asymptotically parallel”.

H and H are “divergently parallel” to H .

So we have H and a point not on it: Point D and we have 3 lines parallel to H through D right there on the sketch. This illustrates our choice of parallel axiom. And we now have two types of parallelism: asymptotic and divergent.

45 Do Workbook Problem 1 right now

Let’s check some axioms, theorems, and definitions just to make sure they’re still true. You know they’re true, but seeing it is helpful.

46 Enrichment 3: Vertical angles are congruent.

Dis k Controls F mGDI = 57.5

B I put

mEDJ = 57.5

points I and J on with the Euclidean tool bar “points on arc” at the top of the page AND I measured these angles using the Hyperbolic Angle Measure from the tool bar on the left under the double-headed arrow.

47 Enrichment 4: Triangles and Exterior Angles

Dis k Controls F

mBAH = 35.3 B mAHB = 40.5 mABH = 35.6 A M mLBM = 144.4 m1+m2+m3 = 111.44

m1+m2 = 75.80

Yes, we have triangles. No, the sum of the interior angles is not equal to 180; it is LESS THAN 180 as promised. The difference between 180 and the sum of the interior angles of a given Hyperbolic triangle is called the DEFECT of the triangle. In Spherical geometry the difference between the sum of the interior angles of a spherical triangle and 180 is called the EXCESS of the triangle.

And, as promised in the Exterior Angle Inequality Theorem, the exterior angle (LBM in our example above) has a greater measure than either remote interior angle. Thus this theorem is true in both Euclidean and Hyperbolic geometry.

In fact, its measure is greater than their sum (very non-Euclidean here – remember ALL the facts after the Parallel Postulate in the Axioms section are Euclidean facts and NOT applicable here in Hyperbolic Geometry). The Euclidean Exterior Angle Equality Theorem (the exterior angle measures the sum of the 2 remote interiors) is a Euclidean theorem not a Hyperbolic Theorem.

48 Note that the defect of the triangle is 58.6. (The sum of the angles is 121.4) We use the lower case Greek letter delta for defect (  ).

Do Workbook Problem 2 right now.

Enrichment 5: Right triangles

Dis k Controls

C mCDB = 90.0 mDCB = 17.6 mCBD = 18.1

D B DC = 1.81 DB = 1.79 CB = 2.96 A Distance2+Distance2 = 6.48

Here’s a right triangle – the measures of the angles are shown. I built it using an H and the Hperpendicular bisector of that line (again: Hyperbolic tools on the left menu).

 = 54.3

Does the Pythagorean Theorem hold in Hyperbolic Geometry?

Demonstrably not – see the calculations in the sketch.

49 Enrichment 6: Distance

The distance (metric) axioms are in the Axioms section

Measure the following distance in Euclidean Geometry:

You use these Euclidean measurements to calculate the Hyperbolic distance with a formula: the Hdistance from A to B is

is called the “cross product”.

The absolute value of the natural log of the “cross product” is a very clever way to measure distances.

Let’s look at some consequences of this formula.

50 Poincaré Disk Model

Dis k Controls Euclidean distances MA = 0.86 in. BM = 2.07 in. AN = 1.91 in. BN = 0.69 in. Hyperbolic distances

MA = 35.48 MABN HDAB ln = 1.89 MB = 37.37  BMAN  NA = 36.38

NB = 33.50 EG AB = 1.25 in. A B

AB = 1.89 N M

Here’s a picture with Hyperbolic distances on the left and Euclidean distances on the right. The hyperbolic distances were measured using the hyperbolic distance tool on the left and the Euclidean distances were measured using the Euclidean distance tool on the top menu. On the right, the calculation for Hyperbolic distance is shown. Remember that the calculation uses Euclidean distances. The Hyperbolic tools do the calculation automatically for you.

In Euclidean Geometry the distance between points on a segment is fixed and independent of location in the plane. In Hyperbolic Geometry, however, you have an interesting stretching of calculated distances that depends on whether the points are close to the center of the disc or close to the edge of the disc. Points can be the SAME Euclidean distance apart and have different Hyperbolic distances depending on their location in the disc. This is a function of the distance formula.

Here’s an illustration with two points that are .09 apart in Euclidean geometry and located in two different spots in the disc. Note that the Hyperbolic distances are different and the points are further apart out near the edge of the disc.

51 Poincaré Disk Model

Disk Controls

Euc lidean Distance AB = 0.09 in.

Hyperbolic dis tance AB = 0.53 A B M N

Euclidean distance .09

Hyperbolic distance .52

Disk Controls

A B N Euc lidean Distance AB = 0.09 in.

Hyperbolic dis tance AB = 0.12

Euclidean distance .09

52 Hyperbolic distance .12

Notice that the Hyperbolic distance depends on WHERE you are in the disc. Points that are the same Euclidean distance apart in different locations on the disc are different Hyperbolic distances apart. Problems 3 and 4 in the Workbook Homework explore this fact.

53 Enrichment 7: Quadrilaterals

Dis k Controls

mIGH = 51.1 mJHG = 51.0 GH = 2.59

G H GI = 0.93 JH = 0.93

mGIJ = 90.0 mHJI = 90.0 I J IJ = 1.97

This is the only kind of quadrilateral that you can have in all 3 of the big geometries: a Saccheri Quadrilateral.

Review the definition and the properties.

54 Do Problem 5 in the Workbook now.Enrichment 8:

Each of the following is logically equivalent tot the Euclidean Parallel Postulate. Thus each of the following are NOT true in Hyperbolic Geometry. I’ve illustrated below how one equivalence is not true in HG. The others are Workbook Problem 5.

 The area of a triangle can be made arbitrarily large.  The angle sum of all triangles is a constant.  The angle sum of any triangle is 180.  Rectangles exist.  A circle can be passed through any 3 noncollinear points.  Given an interior point of a circle, a line can be drawn through that point intersecting both sides of the angle.  The perpendicular distance from one of two parallel lines to the other is always bounded.

Illustration:

 Two parallel lines are everywhere equidistant.

This sketch depicts the hyperbolic plane H2 us ing the Poincaré disk model. In this model, a line through tw o points is def ined as the Euc lidean arc pas sing through the points and perpendicular to the c irc le. Us e this document's custom tools to perform c onstructions on the hyperbolic plane, comparing y our findings to equivalent constructions on the Euclidean plane.

Dis k Controls

Lines LK and IJ are parallel. I built perpendiculars to LK and points at the intersection of the perpendiculars on IJ. In meas uring the dis tance -- the "perpendic ular" distanc e -- note that the lines are NOT equidistant. Equidistanc e parallel lines are a Euclidean phenominon. I

D J L F LI = 0.76 G K E

KJ = 1.27