Math 3379
Chapter 2 - Venema
Homework 2
Chapter 2
2.4
Problem 10
Problem 12
08 points
12 points
2.5
Problem 04
Problem 05
Problem 08
Problem 10
08 points
08 points
06 points
08 points
2.6
Problem 01
Problem 02
Problem 03
10 points
10 points
10 points
1
Chapter 2 Axiomatic Systems and Incidence Geometries
In more recent times, and after lots of work on the foundations of mathematics, we
use an axiomatic structure for many topics.
Axiomatic structure
Undefined terms
A starting point - usually few, less than 10
Axioms
consideration.
Statements that are true for the system under
Called Postulates by Euclid.
Independent – an independent axiom cannot be
proved OR disproved as a logical consequence of
the other axioms.
Consistent – no logical contradiction can be
derived from the axioms
Definitions
Technical terms defined by various manipulations
of the undefined terms and axioms or from notions
introduced by proofs.
Theorems
Statements about relationships in the system that
have been proved. Called propositions by Euclid.
Interpretations
and Models
A way to give meaning to the system, often visual,
models are often helpful. A system may have
more than one model. A system with exactly one
model, with all other models being isomorphic to
one another is called categorical.
2
Geometry #1 – A Five Point Geometry
not from the book
Undefined terms: point, line, contains
Axioms
1. There exist five points
2. Each line is a subset of those five points.
3. There exist at least two lines
4. Each line contains at least two points.
Let’s come up with at least 2 non-isomorphic models for this system.
3
Geometry #2 A Flexible Geometry **
not from the book
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,
so they are non-isomorphs.
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, it’s “line stuff”. Some non-point stuff.
Luckily they’re undefined terms so I don’t have to go into it.
4
Model 2: an infinite number of points and lines
This is an infinite lattice. Each line is has 3 points along
it. It continues forever left and right
Your Model:
Put it in the CW Chapter 2 #1 Turn in Page once you’ve got it figured out.
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
5
Parallel lines – Parallel lines share no points. See the CW questions!
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? Again, the CW.
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? **TURN IN CW Chapter 2 #1 right now.
**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)
6
INCIDENCE GEOMETRIES
Undefined terms:
page 16, Section 2.2
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Note that the axiom uses all 3 undefined terms and is defining a relationship
among them.
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
IA3
There exist three points that do not all lie on any one line.
Definitions:
Collinear:
Three points, A, B, and C, are said to be collinear if there exists one
line l such that all three of the points lie on that line.
Parallel:
Lines that share no points are said to be parallel.
7
Interpretations and models:
(Note: non-categorical!)
The Three-point Geometry
Label the points A, B, and C
Check the axioms.
What is exactly 1/3 of the way between B and C? In other
words, what are lines made of ?
Alternate, and isomorphic models:
Theorem 1: Each pair of distinct lines is on exactly one point.
Proof of Theorem 1
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 1
states that “two distinct points are on exactly one line”.
Thus our supposition cannot be and the theorem is true. QED
8
Theorem 2: There are exactly 3 distinct lines in this geometry.
Take a minute now and prove Theorem 2. You may work in groups or
individually. Turn in your proof in CW #2. Turn it in when I call time.
Last but not least:
How many parallel lines are there?
Could this be called non-Euclidean? Why?
9
The Four-point geometry
Undefined terms:
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Note that the axiom uses all 3 undefined terms and is defining a relationship
among them.
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
IA3
There exist three points that do not all lie on any one line.
Same axioms!
Model:
Note the “at least 2” in IA2!
A planar shape, a tetrahedron, or octant 1 in 3-space
C
B
A
D
10
What are some alternate views on this model?
List the points:
Interpret point to be the symbol. Or for the octant, use the usual Cartesian idea.
List the lines:
{A, B},
Interpret line to be a set of 2 symbols. Or for the octant, use the usual Cartesian
idea.
In the planar shape, what is in between A and B?
How many parallel lines are there? Expand from “lines that share no points” to the
Playfair statement: Given a line and a point not on that line, how many lines go
through the point and share no points with the given line.
11
Why is this a geometry?
Why is this a non-Euclidean Geometry?
12
The Five -point Geometry
Undefined terms:
not from the book
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Note that the axiom uses all 3 undefined terms and is defining a relationship
among them.
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
IA3
There exist three points that do not all lie on any one line.
Definitions:
Collinear:
Three points, A, B, and C, are said to be collinear if there exists one
line l such that all three of the points lie on that line.
Parallel:
Lines that share no points are said to be parallel.
13
Model:
P1
Points: {P1, P2, P3, P4, P5}
P2
P5
Lines: {P1P2, P1P3, P1P4, P1P5, P2P3, P2P4, P2P5,
P3
P4
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?
Quadrilateral – a closed figure formed by 4 lines. An example: P2P5P4P3 is a
quadrilateral. How many quadrilaterals are there?
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P1
P2
P5
P3
P4
Note that line P1P2 is parallel to line P4P5. So are P3P4 and P2P5…List them
ALL!
15
Five-point geometry Theorem 1:
Each point is on exactly 4 lines.
Let’s prove this right now – get in groups and get to work! First one done, get it
up on the board and we’ll tweak it together.
16
Another non-Euclidean!
The Seven-point geometry
page 18
Also known as Fano’s geometry.
(Gino Fano, published 1892)
A
F
B
G
C
D
E
{BDF} is a line! Nobody said “straight” in the axioms!
Where does {BDF} intersect {CBA}?
7 points and 7 lines…what’s the situation with respect to parallel lines?
17
Alternate axioms for Fano’s Geometry:
Axioms for Fano's Geometry
Undefined Terms. point, line, and incident.
Axiom 1. There exists at least one line.
Axiom 2. Every line has exactly three points incident to it.
Axiom 3. Not all points are incident to the same line.
Axiom 4. There is exactly one line incident with any two distinct points.
Axiom 5. There is at least one point incident with any two distinct lines.
A
F
B
G
C
D
E
Sometimes MORE THAN ONE list of axioms generates the SAME Geometry.
18
There are exactly 7 points in Fano’s Geometry. Count them in the model to make
sure. Now let’s get busy on CW Chapter 2 #3. Proving this, given 7 points,
exactly 7 lines…
Turn it in when I call time.
Enough with finite geometries – there’s an infinite number of them!
In fact, let’s talk about how many there are:
Is there a geometry with 17 points? 1927 points (why did I pick that number?)
N points?
19
A detour to a big well-known geometry:
Sphererical Geometry
The unit sphere is NOT a model for an incidence geometry but is very important in
the development of an understanding of modern geometry. We will spend a bit of
time on it.
Undefined terms:
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
The sphere fails to satisfy this axiom. WHY?
How can we change the axiom so it “works”?
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
True for the sphere.
IA3
There exist three points that do not all lie on any one line.
True for the sphere.
20
Interpretation and notation:
Point: an ordered triple (x, y, z) such that it satisfys x2  y 2  z 2  1 . In other words,
points are on the surface of the unit sphere.
Line: a great circle on the sphere’s surface. A Euclidean plane containing a great
circle includes the center of the sphere (0, 0, 0).
On:
is an element of a solution set
S2
will denote the unit sphere. It is embedded in 3 dimensional Euclidean
space.
Lines:
Between two points!
What are non-great circles and what makes them interesting?
What’s the situation vis a vis parallel lines in this model?
21
What is the minimal closed polygon in a sphere?
Let’s talk distance and angle measure
There are triangles, how do they compare to Euclidean triangles. Measure and sum
in small groups.
22
CW Chapter 2 #4
Comparing SG and EG, what’s the same, what’s different?
Let’s take a few minutes in small groups to make some lists
What’s the same as Euclidean Geometry?
What’s different from Euclidean Geometry?
Now fill out CW Chapter 2 #4 and turn it in.
23
Let’s look at the Cartesian plane:
Undefined terms:
page 19
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Note that the axiom uses all 3 undefined terms and is defining a relationship
among them.
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
IA3
There exist three points that do not all lie on any one line.
Interpretation and notation:
Point: any ordered pair (x, y)
Line: the collection of points whose coordinates satisfy a linear equation of the
form y = mx + b
On: A point is said to lie on a line if it’s coordinates satisfy the equation of that
line.
R2
will symbolize the Cartesian plane
Why is THIS symbol used?
First let’s use the definition of cross product from Modern Algebra…who knows
it?
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Why is this called the Cartesian plane and not just THE plane?
Argand Plane, among others
25
The Klein disk
page 20
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. “on” has the usual
Euclidean sense.
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.
A
B
H
P1
P2
C
D
G
F
E
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.
Is everybody clear on what is and is not in our space?
26
Checking the axioms:
Undefined terms:
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Inheriting…
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
Inheriting…
IA3
There exist three points that do not all lie on any one line.
Inheriting…
Definitions:
Collinear:
Three points, A, B, and C, are said to be collinear if there exists one
line l such that all three of the points lie on that line.
Parallel:
Lines that share no points are said to be parallel.
27
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.
A
B
H
P1
P2
C
D
G
F
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.
E
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.
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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.
Now for the usual:
CW Chapter 2 #5 Compare and contrast the Klein Disc to Spherical Geometry
Visit in groups and then we’ll turn in #5.
29
2.3
The Parallel Postulates in Incidence geometry
page 20
Definition:
Parallel lines:
Lines that are parallel do not intersect. i.e. they share no
points.
This works in 2D and we’ll be in 2D for this course, except for Spherical
Geometry. Now, in contrast to some high school geometry books: a
line cannot be parallel to itself in our course.
See the Official Definition, bottom of page 20
Definition 2.3.1
We find that there are several ways for a geometry to be configured with respect to
parallel lines.
One way is to have none, one way is the Euclidean way exactly one, and there’s a
third possibility, too, exemplified with the Klein Disc – more than
one.
Euclidean Parallel Postulate:
For every line l and for every point P that does not lie on l, there is exactly one line
m such that P lies on m and m is parallel to l.
Illustration
30
Which Incidence geometry models have this property?
Three-point geometry
Four-point geometry
Five-point geometry
Fano’s geometry
Klein disk
Spherical geometry
Elliptic Parallel Postulate
For every line l and for every point P that does not lie on l, there is no line m such
that P lies on m and m is parallel to l.
What about the Sphere? None. Which have we looked at that don’t have any
parallel line
Which Incidence geometry models have this property?
Three-point geometry
Four-point geometry
Five-point geometry
Fano’s geometry
Cartesian plane
Klein disk
Spherical geometry
31
Hyperbolic Parallel Postulate
For every line l and for every point P that does not lie on l, there are at least two
lines m and n such that P lies on m and n and both m and n are
parallel to l.
Which Incidence geometry models have this property?
Three-point geometry
Four-point geometry
Five-point geometry
Fano’s geometry
Cartesian plane
Klein disk
Spherical geometry
32
Independence of the Axiom concerning Parallelism:
We have just looked at an axiomatic system for Incidence geometries:
Undefined terms:
point, line, on
Axioms:
IA1
For every pair of distinct points P and Q, there exists exactly one line l such
that both P and Q lie on that line.
Note that the axiom uses all 3 undefined terms and is defining a relationship
among them.
IA2
For every line l there exist at least 2 distinct points P and Q such that both P
and Q lie on the line l.
IA3
There exist three points that do not all lie on any one line.
Definitions:
Collinear:
Three points, A, B, and C, are said to be collinear if there exists one
line l such that all three of the points lie on that line.
Parallel:
Lines that share no points are said to be parallel.
We have seen that all 3 of the incidence axioms are satisfied by a plethora of
models. Thus adding IA4, an axiom about parallelism will cause our outlines of
axiomatic systems to have subcategories. Because there are models of the first 3
axioms that have different situations with respect to parallel lines, we know that an
axiom about parallelism is INDEPENDENT from the first 3 axioms. It cannot be
logically derived from them.
33
Please read section 2.4 Axiomatic Systems and the Real World with an eye toward
your term paper. There are some valuable thoughts there.
34
Hyperbolic Geometry, an introduction
Similar to the Klein Disk, Hyperbolic geometry is in the unit circle with the circle
itself NOT in the space. The lines, however, are orthogonal circles.
35
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:
Geometry, second edition by Harold Jacobs. ISBN: 0-7167-1745-X (copyright
1987).
36
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.
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.
37
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) x 2  y2  r 2 }
[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!
38
fx =
1-x2
1.4
gx = - 1-x2
5
hx =
8
5
qx = -
 
 
- x-
8
5
4
- x-
5
4
2
+
2
+
1
1.2
4
1
1
4
0.8
0.6
0.4
0.2
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
Enrichment 1:
Lines in our space:
Poincaré Disk Model
This sketch depicts the hyperbolic plane H2 usin g the Poincaré disk model. In this model, a line through
tw o poin ts is def ined as the Euclidean arc passing through the poin ts and perpendic ular to the circle.
Use this document's custom tools to perf orm constructions on the hyperbolic plane, comparing your f in dings
to equivale nt constructions on the Euclid ean plane.
Disk Controls
B
D
A
C
P. Disk Center
blue circle...not part of our space
39
Here is a sketch from Sketchpad that shows hyperbolic line H AB 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 AB . Do you
see that the tangent lines are perpendicular?
Hyperbolic line CD 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.]
40
Enrichment 2:
Parallel Lines in Hyperbolic Geometry
F
Disk Controls
B
G
A
D
E
H
H AB is parallel to every other line showing in the disc.
Since H AB intersects H DF on the circle, these two have a type of parallelism
called “asymptotically parallel”.
H DH and H DE are “divergently parallel” to H AB .
41
So we have H AB and a point not on it: Point D and we have 3 lines parallel to
H AB 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.
Do CourseWork Chapter 2 #6 right now and turn it in.
42
Enrichment 3:
Vertical angles are congruent.
F
Disk Controls
mGDI = 57.5
B
I put
poin
A
ts I
D
and
J
J on
E
H
with
mEDJ = 57.5
the
Eucl
idea
n
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 doubleheaded arrow.
G
I
Are vertical angles congruent in Spherical geometry?
Euclidean Geometry?
The Klein Disk?
43
Enrichment 4:
Triangles and Exterior Angles
F
Disk Controls
L
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
H
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
44
Equality Theorem (the exterior angle measures the sum of the 2 remote interiors) is
a Euclidean theorem not a Hyperbolic Theorem.
Note that the defect of the triangle is 58.6. (The sum of the angles is 121.4) We
Do CW #7 right now.
Now we’ve completed Chapter 2 with some extra material. Do the homework
after you read Chapter 2.
45