Spring’13
Math 3379
Chapter 1
(2 lectures)
Homework 1
1.1
1.2
1.3
1.4
2
3
5 (use Google to help)
10
1 (a, c, e, j),
5 (do not say “yes.”…write a paragraph for each question
supporting your assertion!)
3 (a, f)
6
Upload the homework to the appropriate link in your CourseWare account under the tab
“Assignments” by the deadline. No late homework will be accepted. Illegible, upside down, and
horrible to read homework will not be awarded any points.
It is expected that you will read the book, more than once, on your own. In class I will highlight
certain points but these items will not be the only material you are responsible for. Additionally,
I will include material in the Notes that is NOT in the book and you are responsible for learning
that information, too.
1.1 Greek Geometry Before Euclid
Geo: the earth, Metry: to measure. Originally Earth-centered. Not so since 1850 or so.
Originated east of Greece in the Orient as an aid to business (dividing land among heirs,
bookkeeping, building tombs….)
Sumer (2100 BC)….Babylon (1600 BC) – 360 degrees, base 60 numbers…Egyptians added
many formulas and facts, vocabulary
Pythagoras (569-475 BC)
Not much notion of “number” per se…the PT was about areas! Evidently they
discovered irrational numbers…see page 4 for a plausible way they did this!
See p. 5 with doing algebra with areas.
Continuing the physical, manipulative style approach to algebra and math facts, let’s look at the
last illustration of the series on page 6, this is all about the “geometric mean”.
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Geometric mean
From Wikipedia, the free encyclopedia
In mathematics, the geometric mean is a type of mean or average, which indicates the central
tendency or typical value of a set of numbers.
A geometric mean is often used when comparing different items – finding a single "figure of
merit" for these items – when each item has multiple properties that have different numeric
ranges. For example, the geometric mean can give a meaningful "average" to compare two
companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0
to 100 for their financial viability. If an arithmetic mean was used instead of a geometric mean,
the financial viability is given more weight because its numeric range is larger- so a small
percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger
difference in the arithmetic mean than a large percentage change in environmental sustainability
(e.g. going from 2 to 5). The use of a geometric mean "normalizes" the ranges being averaged, so
that no range dominates the weighting, and a given percentage change in any of the properties
has the same effect on the geometric mean. So, a 20% change in environmental sustainability
from 4 to 4.8 has the same effect on the geometric mean as a 20% change in financial viability
from 60 to 72.
The geometric mean is similar to the arithmetic mean, except that the numbers are multiplied and
then the nth root (where n is the count of numbers in the set) of the resulting product is taken.
We will also need to review the AA Similarity Theorem:
In Euclidean Geometry whenever 2 pairs of corresponding angle of 2 triangles are congruent,
then the triangles are similar.
And a fact about inscribed chords of a certain type:
the triangle created in a semicircle by connecting segments from the diameter’s edges to the
point on the circle makes a right angle there.
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Page 6: x 2  ab
The geometric mean. Here’s the illustration from the book:
x
a
b
Connect the left hand chord and the right hand chord from the diameter endpoints to the point on
the circle. Call the two sides of the split right angle: A1 and A2, left to right. Note that the
remaining angles are 90 – A1 on the left side and 90 – A2 on the right side. (note that I’m being
sloppy here and naming the angle and it’s measure the same name – Bad Teacher!)
Note that A1 + A2 = 90 so that A2 = 90 – A1
This means that the left hand triangle is 90, A1 and 90 – A1
And the right hand triangle is 90, A2 = 90 – A1, and 90 – A2
So the triangles are similar by AA Similarity.
We can then invoke ratio and proportions to say that the ratios of the
medium side to the short side of each triangle is the same number:
a x

x b
Thus x 2  ab and when we take the square root of both sides we discard the negative solution
because we are discussing lengths – actual physical lengths. Hence the geometric mean applied
to a geometry situation!
We’ll revisit this proof later in the Chapter.
Note the time line on page 8. Check out the URLS on page 9
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1.2
Euclid and the Elements
Euclid 300 BC, department head at the University of Alexandria
Elements
First printed in 1482 (prior to that, copied by hand)
Table of Contents, page 12
Notably, a systematic presentation with proofs…the first
Definitions – about 2 dozen…
Postulates – 5 of these
Common notions – 5 of these
Propositions – 450+ of these
Logic: Aristotle (384 – 322 BC)
see page 13
Axiomatic Systems a similar approach refined over the last 250 years.
Undefined terms
Axioms
Definitions
Theorems
Definitions needed to discuss modern style axiomatic systems
Page 17
Consistent:
An axiomatic system is said to be consistent if neither the axioms nor the
propositions (theorems) of the system contradict one another.
Independent:
An axiom is said to be independent of the other axioms if it cannot be
derived from the other axioms.
Complete:
An axiomatic system is complete if it is possible for every properly posed
statement to be proved or disproved. Alternatively, it is not possible to
add a new independent axiom to the system.
Categorical:
All models of the axiomatic system are isomorphic.
4
Finite Geometries:
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:
An inclass project!
This is NOT a categorical geometry.
5
Ideas for Definitions:
Biangle
Triangle
Parallel lines
Collinear points
Alternate points
Explorations:
Are there a minimum number of points?
This might become a theorem.
** See pages 30 and 31, Sibley, for a very nice presentation of this in the larger context of types
of axiomatic systems.
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The Three Point Geometry
Undefined terms:
Axioms:
Model
point, line, on
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 are on exactly one point.
This IS a categorical system.
Theorems:
Theorem 1:
Each pair of distinct lines is on exactly one point.
Theorem 2:
There are exactly 3 distinct lines in this geometry.
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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.
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The Four Point Geometry
Undefined terms
Axioms
point, line, on
A1
There are exactly four points.
A2
Any two distinct points have exactly one line on both of them.
A3
Each line is on exactly two points.
Models – categorical: homomorphisms between all
Possible Definitions:
Parallel lines
Collinear points
Theorems
1. The four point geometry has exactly 6 lines.
2. Each point of the geometry has exactly 3 lines on it.
Part of this one is on page 17 in your textbook
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An Incidence Geometry or two
Undefined terms:
point, line, on
Axioms:
A1
A2
A3
A4
There is exactly one line on any two distinct points.
Each line has at least two distinct points on it.
There are at least three points.
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… two intersect…more than 2 are “concurrent”
Theorems:
Theorem 1:
Theorem 2:
Theorem 3:
If two distinct lines intersect, then the intersection is exactly one point.
Each point is on at least two lines.
There is a triple of lines that do not share a common point.
Theorems must be totally true in EVERY MODEL!
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A six point model:
B
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.
C
A
D
F
E
AB
AC
AD
AE
AF
BC
BD
BE
BF
A1
A2
A3
A4
There is exactly one line on any two distinct points. See the list
Each line has at least two distinct points on it. See the “endpoints”.
There are at least three points. There are 6 which is “at least 3”.
Not all the points lie on the same line. See the list.
Theorem 1:
Theorem 2:
Theorem 3:
CD
CE
CF
DE
DF
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:
EF
If two distinct lines intersect, then the intersection is exactly one point.
Each point is on at least two lines.
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.
Let’s look at the situation with respect to parallel lines.
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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.
B
C
A
D
F
E
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!
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An Incidence Geometry, continued
The Klein Disc:
Points will be {(𝑥, 𝑦)|𝑥 2 + 𝑦 2 < 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.
A
B
H
P1
H is a point in the circle’s interior and IS a point
in the geometry.
P2
C
D
G
F
We cannot list the number of lines – there are an
infinite number of them.
E
Checking the axioms:
A1
A2
A3
A4
There is exactly one line on any two distinct points.
This model is a subset of Euclidean geometry and the axiom holds.
Each line has at least two distinct points on it.
Each line has an infinite number of points by Euclidean Axioms.
There are at least three points.
The unit disc has an inifinite number of points.
Not all the points lie on the same line.
True
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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.
A
B
H
P1
P2
C
D
G
F
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.
Theorem 1:
Theorem 2:
Theorem 3:
If two distinct lines intersect, then the intersection is exactly one point.
Inherited from Euclidean Geometry.
Each point is on at least two lines.
Each point is on an infinite number of lines.
There is a triple of lines that do not share a common point.
FE, GC, and AD for example.
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The SMSG Axioms for Euclidean Geometry – A categorical system
A1.
Given any two distinct points there is exactly one line that contains them.
Comments:
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.
Comment
A3.
C.
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
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.
B.
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.
Every plane contains at least three non-collinear points.
Space contains at least four non-coplanar points.
Comments
15
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.
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.
Comments
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.
16
A16
The Parallel Postulate: Through a given external point there is at most one line parallel
to a given line.
Comments
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.
Comment
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.
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1.3
Neutral Geometry
History Early 20th century effort…Encompasses the Big Three that we’ll study
Note that these axioms do not take a stand on the business of parallel lines – for good reason:
Euclidean
Hyperbolic
Spherical
The Axiomatic Structure for Neutral Geometry
Undefined terms:
point, line, plane, space
Axioms:
A1
To each pair of points (A, B) is associated a unique real number, denoted AB, with least
upper bound, 
A2
For all points A and B, AB  0, with equality only when A = B.
A3
For all points AB = BA.
A4
Given any four distinct collinear points A, B, C, and D such that
A – B – C, then either D – A – B, A – D – B, B – D – C, or B – C – D.
A5
Each two points A and B lie on a line, and if AB <  , that line is unique.
A6
Each three noncollinear points determine a plane.
A7
If points A and B lie in a plane and AB < , then the line determined by A and B lies in
that plane.
A8
If two planes meet, their intersection is a line.
A9
Space consists of at least four noncoplanar points, and contains three noncollinear points.
Each plane is a set of points of which at least three are noncollinear, and each line is a set
of at least two distinct points.
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A10 (Ruler Postulate): Given line L and two points P and Q on L, the points of L can be
placed into one – to – one correspondence with the real numbers x such that
 x  (called coordinates) in such a manner that
1. points P and Q have coordinates 0 and k > 0, respectively
2. if A and B on the line have coordinates a and B, then AB is
 a  b if


2  a  b

A11
a  b 
if
a  b 
(Plane Separation Postulate): Let L be any line lying in any plane P. The set of all points
in P not on L consists of the union of two subsets H1 and H2 of P such that
1. H1 and H2 are convex sets
2. H1 and H2 have no points in common
3. if A lies in H1 and B lies in H2 such that AB < ,
line L intersects segment AB .
A12
Each angle  ABC is associated with a unique real number between 0 and 180 denoted m
 ABC. No angle can have measure 0 or 180.
A13
(Angle Addition Postulate): If D lies in the interior of  ABC, then
19
m  ABD + m DBC = m  ABC.
A14
(Protractor Postulate): The set of rays AX lying in a plane and on one side of a given
line AB , including ray AB , may be placed into one – to – one correspondence with the
real numbers x such that 0  x  180 (called coordinates) in such a manner that
1. ray AB has coordinate 0
2. if rays AC and AD have coordinates c and d, then
m  CAD = c  d .
A15
(Linear Pair Axiom): A linear pair of angles is a supplementary pair.
A16
(SAS Postulate): If two sides and the included angle of one triangle are congruent,
respectively, to two sides and the included angle of another, the triangles are congruent.
20
1.5
Famous Open Problems in Geometry
We will focus on pages 34 – 37: Constructible Numbers, the topic of paper 1.
Discussion of Paper 1
Rubrics, writing center appointment confirmation are to be attached to the paper in ONE pdf file.
TWC will be sending a document soon with your consultant assignment.
Constructible Numbers:
A number a is constructible if one may construct a line segment with length a in a finite
number of steps from a segment of unit length using ONLY a straightedge and compass.
Note that neither instrument has length or degree markings…they are without a way to tell
visually how big or small a mark is.
So let’s talk about a = 5.
Theorem 1.4.1
If a and b are constructible real numbers then so are
ab
a b
ab
ab
a
Note the following start to the theorem and the discussions:
SINCE both a and b are constructible….
21
Proving the illustrations are on your own up to
Figure 1.4.14
a
1
a
Since we KNOW we can make similar triangles: Use x on the upright:
1/x as x/a. giving x 2  a
Ah, yes, the geometric mean showing back up in constructible numbers!
There’s a reasonable discussion of the matter in the book. You’ll want to supplement with some
careful research. Note: Mathematical Dictionaries exist in the library on this campus!
Example 1.4.1 TYPE in the last line 4 cos CUBED alpha
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