Module 7

by Leigh Hollyer

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Table of Contents

Module 7

Undefined Terms:

Hyperbolic Postulates

A4 If two intersecting lines cross at adjacent congruent angles, then each of these is congruent to any other angle created the same way.

This is necessary because Euclid’s proof of SAS used the notion of moving parts of one triangle over another to achieve coincidence or superposition.

It is necessary to define “moving” objects and the act of defining movement opens the door to our next module: Transformational Geometry. For now, keep “moving” out of the picture.

Hyperbolic Axioms

We may also use any theorems we proved (or proved by anyone!) that don’t implicitly or explicitly use the existence of parallel lines.


Klein-Beltrami Model

This model is excellent for illustrating some startling hyperbolic facts about parallel lines.

In Euclidean geometry,

Why does mathematics have such a term? (ie Why do we care?)

Notice that two lines can be parallel and are NOT equidistant.

They could be parallel and share a boundary point ( this is called asymptotically parallel) since boundary points are not in the model.

So, we have kinds of parallelism* and we’ve lost the transitivity of parallel lines.

The distance formula for the Klein-Beltrami model

PPT Slide

Imagine what a triangle looks like in the Klein-Beltrami model. It looks normal, Euclidean…and we’ve already agreed the the sum of the interior angles is less than 180 degrees…so the image of a triangle is a problem.

A more visually intuitive model, the Poincare disk, comes with a more difficult visualization of what lines are, but the distance formula is slightly simpler.

Poincare Disk

Let’s look at a sketch of orthogonal and non-orthogonal circles on the overhead.

Back to the model:

The distance formula is

Two segments, AB and CD, are called Poincare-congruent if D(A,B) = D(C,D).

Upper Half-Plane Model

A line will be any portion of a circle whose center is on the x axis.

The algebraic formula for a line is

The Upper Half-Plane model is an unbounded model. It is a proper subset of the Euclidean plane.

The formula for distance is

Cosh(x) is a function that is more formally known as hyperbolic cosine. Arccosh(x) is the inverse to hyperbolic cosine. This function has uses on its own -- it describes curves that happen in nature and was known before its use here.

The curve of a power line suspended from two light poles has a shape defined by f(x) = a cosh(x). The shape is also known as a catenary. The hyperbolic cosine formula is

One of the original researchers who learned how to work with hyperbolic sines and cosines was Johann Lambert, whose extension of the Saccheri quadrilateral is called a Lambert quadrilateral.

The types of parallel lines still show very nicely in this model. In fact, this model is quite easy to set up in Sketchpad or on graph paper.

Minkowski Model

Points are points on the surface of half of a two-sheeted hyperboloid…this is a special shape in three dimensions.

The distance formula requires even more knowledge of mathematics -- and angle measure is a trigonometric function of a vector quantity. Calculus III is a minimum requirement to work on the Minkowski model.

I’ve included it to show you that, like with Spherical geometry, geometry can be defined on surfaces.

Notebook Problem


2. There are no triangles with the sum of the interior angles is 180 degrees.

4. There are no lines that are everywhere equidistant.*

6. No rectangles exist.

11. There are triangles that cannot be circumscribed.*

As one of my favorite authors* wrote:

Connections to Real Numbers

In Group V, Hillbert deliberately introduced the notion of length and tied in both rational and irrational numbers.

In some versions of the axioms of Euclidean Plane Geometry you will find a “Ruler Postulate” (SMSG Postulate 3):

3b. To every real number there corresponds exactly one point of the line.

Using the Klein-Beltrami model and the simplest line that there is: the diameter that is the x axis, I want to review that distance formula with pictures and numbers.

You will see some versions of the distance formula using the natural logarithm “ln(x)” instead of the base 10 logarithm “log(x)” … this creates some slight numerical changes but doesn’t change the ability of the function to map the whole real line into a bounded space.

I’ll work this example on the overhead.




An example


Are the area formulas in the non-Euclidean geometries really different?

“Area” in geometries

It is, of course, more than unsettling to realize that a collection of objects with area equal to zero has a non-zero real number representing the collective area…but that’s an issue for real “foundations of mathematics” buffs.

Theorem 2

So, as in Euclidean geometry, area is additive. Decomposition is the technique of choice for figuring out the areas of polygons…what do you suppose a regular pentagon looks like in the Poincare model?

How do you suppose that you would calculate the area of the pentagon? Remember that k is fixed, so triangles with the same defect have the same area no matter what they look like...

In Euclidean geometry, we used the definition of the area of a rectangle to get to the definition of the area of a triangle.

Theorem 3

Theorem 4

First I’ll outline the construction and then I’ll show you on the overhead. (note: this process is a picture, not a proof)

Next draw perpendiculars to this line from each vertex of the triangle.

I’ll be using the Klein-Beltrami model because it’s the easiest for these pictures. Please keep in mind the inaccuracy of the image -- the shapes will look Euclidean and they’re really, really not.

Before sketching the proof, it is necessary to realize that AAS is a congruence condition in Hyperbolic geometry. It was a situation peculiar to Spherical geometry that caused AAS to be unreliable: the existence of poles.

As all the cases are reasonably similar, I’ll only outline Case I:

By similar reasoning,

This makes ACRP a Saccheri quadrilateral. (Claim 1)

Now, PQ = PM + MQ and since

Now notice that ? MAP ? ? MBQ. This means that

For similar reasons,

m?MAP + m?BAC = m?MBQ + m?BAC

Adding both of these equations gives

The left hand side adds angles 1,2, 5, and 6 giving the sum of the summit angles (the angles that are acute, the angles where the defect occurs.)

So, we’ve shown that this construction creates an associated quadrilateral for any triangle, with base measurement twice that of the line segment joining the chosen midpoints and with the same defect as the triangle.

Do you think that this idea of having an associated Saccheri quadrilateral translates (within reason) to

Biangles revisited

Hyperbolic geometry has biangles, too. (And hyperbolic biangles have alternate names; it’s author-dependent.)


Saccheri Quadrilaterals

Saccheri quadrilaterals are called rectangles in Euclidean geometry.

The line segment joining the midpoints of the base and summit is perpendicular to both of them.

In Hyperbolic geometry, Saccheri quadrilaterals are called just that.

The summit of a Saccheri quadrilateral is longer than its base.

The base and summit are ultra parallel, as are the other two sides.

In Spherical geometry, a Saccheri quadrilateral is called that.

The line joining the midpoints of the base and summit is perpendicular to them both.

Neutral properties

Biperpendicular quadrilateral --

Geometry Specific

Called another name in Euclidean geometry (rectangle)


2. Two lines perpendicular to the same line are

3. Parallel lines are

4. Sum of interior angles of a triangle:

5. Area of a triangle is

6. A triangle of maximal area

7. Two triangles with congruent corresponding angles are

8. Biangles

9. The intersection of two distinct lines is

10. Straight lines are

11. “betweenness” makes sense

12. If a line intersects one of two parallel lines, it

13. The summit angles of a Saccheri quadrilateral are

14. Exhibits plane separation*

Anyone who chooses to do the Mobius Strip project will be working with plane separation issues. A line in Euclidean geometry separates the plane into two regions that share the line as a boundary…a Mobius strip can help model a way to connect 2 points without crossing the line.

Many theorems depend on having an interior and exterior to a shape. In a geometry that lacks this property, none of those theorems hold.

Working on Mobius strips emphasizes the difference between infinite and boundless, too.

Changing gears a bit

What are some equivalent statements to this?

The summit angles of every Saccheri quadrilateral are 90o (P/2 in radian measure).

This is a list of the reasons to look for in theorems when you are trying to use a familiar Euclidean one in a non-euclidean geomety. If these appear, you’re warned that the proof depends on Euclidean parallel lines.

So now we’ve looked at lists of properties that are the same or are different.

Neutral Geometry

Euclid did this, too…note that 29 of his propositions (theorems) were proved without the fifth postulate.


**a major caveat here:

We’re ok with Euclid’s work up to Book I, Proposition 28.

Our undefined terms:

Incidence Axioms* from SMSG

P7 Any three points line in at least one plane, and any three non-collinear points lie in exactly one plane.


Equivalence relation:

Given set elements: a, b, and c and a relationship, R, on S

and symmetric means

and transitive means

A relationship that has all three qualities is called an equivalence relation.

N Theorem1

N Theorem 2

N Theorem 3

N Theorem 4

N Theorem 5


Since L is a line in the plane so that it separates the plane into two sets.

Since L doesn’t intersect CB, then B and C are on the same side of L.

Theorem 6

Theorem 7

Have proofs of these in mind for the upcoming test.


Theorem 8


Let E be the midpoint of side BC.

?AFC ? ? BFE and ? ACF ? ? FBE


N Theorem 9

Logical language

The conditional

Conditional, cont.

N Theorem 6:

N Theorem 10

N Theorem 11

N Theorem 12

All of these lead to a

N Theorem 13

Proof: Consider DABC. We must show that AC + CB > AB.

Select the point on ray AC called S such that

This creates an isosceles triangle CBS by construction. Note that

Since C is in the interior of ?ABS

Using AC + CS = AS

N Theorem 14

N Theorem 15

One theorem found in many texts is:

An amazing aside

In Euclidean geometry, the length of the segments joining two midpoints of the sides of a triangle is 1/2 the length of the other side.

They fixed the observers head to avoid motion parallax and used three lights in a triangular pattern (symmetric, low intensity in a dark setting, along a eye-level plane). A fourth light is introduced and the observer as asked to place the light on the left, equidistant from the two endpoints and then a fifth light on the other side. This is locating those midpoints.

After some additional instruction and practice, the experiment began and the location of “equidistant” was recorded numerous times for seven subjects.