Geometry.pdf

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Geometry
Meighan I. Dillon
Mathematics Department
Southern Polytechnic State University
1100 S. Marietta Pkwy
Marietta, GA 30060
mdillon@spsu.edu
December 11, 2006
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Introduction
Geometry literally means
earth measure.
When geometry was developed by
the ancients of various cultures, it was probably for the purposes of surveying,
i.e., measuring the earth. In mathematics, geometry usually refers to a more
general study of curves and surfaces. Different branches of geometry— differ-
ential geometry, algebraic geometry, geometric analysis— are distinguished
in part by the objects of interest and in part by the different sets of tools
brought to bear. In differential geometry, for example, the objects are curves
and surfaces in complex space and the tools arise largely from differential
calculus. Algebraic geometry is the study of objects that can be described
using rational functions. The tools are typically, but not always, of an alge-
braic nature. Some of them are quite elaborate and rarely encountered by
undergraduates, and only encountered by graduate students who will spe-
cialize in algebraic geometry or in an area that relies on algebraic geometry,
for example computer science.
Mathematicians who study geometry usually do not talk much about
similar triangles or alternate interior angles or angles in a circle or most of
the other things you probably think of when you think “geometry.” A part of
Euclidean geometry that does come up in many branches of geometry studied
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today is
projective geometry.
Although this is not a course in projective
geometry, we will study certain projective planes here. In that sense, this
course forms a bridge from high school geometry to more advanced studies.
The high school geometry course you probably had was primarily a course
in
synthetic
Euclidean geometry. Synthetic geometry starts from first princi-
ples: definitions, and
axioms
or
postulates,
which are truths held to be self-
evident. The heart of the subject is logical synthesis applied to the axioms
in a rigorous way to uncover the facts (theorems) which relate the objects
of the geometry. This remains an attractive subject for students of any age
with any background because all assumptions are, ideally, clearly stated at
the beginning of the program. The objects are familiar and the relations
among them rich. No previous knowledge is necessary.
Planar Euclidean geometry is the geometry of straight edge and compass.
The straight edge is unmarked, so cannot measure length, and the compass
does not stay open once you lift it from the page. The objects and all rela-
tions among them must be produced using only a straight edge and compass
applied to a flat surface. One approach to studying Euclidean geometry is
constructive, and usually a portion of the high school course deals with ac-
tual constructions. You can think of the synthetic approach as supplying
detailed instructions for allowed constructions.
The book that brought synthetic geometry to us in the West was
The
Elements
of Euclid, a masterwork dating from around the year 300 B.C. Not
only has
The Elements
been studied more or less continuously since it was
written, it has remained the most important source for the subject since that
time. High school students usually study textbooks based on
The Elements,
not
The Elements
itself. These textbooks are, more or less, the products of
interpretations, corrections, and “improvements” to
The Elements—
many
of questionable merit — that have been incorporated into the subject since
the time of Euclid. One view of these sorts of textbooks is that they attempt
to provide the elusive “royal road to geometry,”
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fit for schoolchildren.[2], p.
vi.
Our course will start with
The Elements
itself, in a translation that dates
to the beginning of the last century. Using this as a departure, we explore
There is a legend that Ptolemy, the king of Egypt, asked Euclid, his tutor, if there
was not a shorter way to learn geometry. Must one go through
The Elements?
Euclid’s
response was that there was “no royal road to geometry.” Similar stories are attributed
to other mathematicians in response to complaints from other kings trying to learn math-
ematics. See [2], p. 1.
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some developments in Euclidean geometry since the glory days of Greece.
Since a great deal has happened over the last few centuries, what you see
here represents a very small sampling of topics taken from what is available
for study under the heading ‘geometry.’
Well before the twentieth century, mathematicians recognized that
The
Elements
had gaps and inaccuracies. There were many significant contribu-
tions to rectifying
The Elements
but David Hilbert’s
Grundlagen der Geome-
trie
(Foundations
of Geometry),
dating from the early 1900s, became the de-
finitive correction and essentially closed the matter once and for all. Hilbert
was able to control some of the difficulties with
The Elements
that had been
recognized but had remained unresolved since antiquity. One of the more
significant contributions of Hilbert was his recognition of the necessity of
leaving certain terms undefined in an axiomatic development. Aristotle, who
predated Euclid, thought a great deal about defining basic geometric objects
and noted that “ ‘the definitions’ of point, line, and surface ... all define the
prior by means of the posterior” [2], p. 155. In other words, for predeccesors
of Euclid through the dawn of the 20th century, it was common but uneasy
practice to work with definitions where, for instance, you might use lines to
define points and points to define lines. Moreover, it was well-known that this
was a logical subversion. Hilbert recognized that the solution to the problem
was to jettison some definitions altogether. One must start with a minimal
set of basic terms that remain undefined: their properties are then detailed
as postulates. For example, we do not define ‘point’, ‘line’, or ‘plane’ but we
understand the nature of these objects through the postulates, for example,
two distinct points determine a unique line, and a line together with a point
not on that line determine a unique plane. All other definitions, for example
‘angle’, are then carefully crafted using this basic set of undefined terms.
Our foray into synthetic geometry includes contributions from the Renais-
sance that amounted to the initiation of the study of the projective plane
through the addition of ideal points to the Euclidean plane. Our study of
Euclid and Hilbert culminates with the nine point circle.
If Euclidean planar geometry derives from straight-edge and compass con-
structions, affine geometry derives from straight-edge constructions alone.
This sounds poorer than Euclidean geometry but we come into a wealth of
ideas when we use linear algebra to develop a model for affine geometry.
This is an extension of the familiar ideas of coordinate geometry and leads
naturally to the three dimensional vector space model for a projective plane,
an important tool in use today.
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Projective geometry dates to the 16th century, thus predates many of the
fundamental ideas that we think of as typical of the modern era in mathemat-
ics, such as sets and functions. Nonetheless, projective geometry has a strong
flavor of modernity as it provides a setting in which curves and surfaces have
few or no exceptional configurations. This gives projective geometry tremen-
dous power which we exhibit with a brief study of plane algebraic curves.
Algebraic curves can be described parametrically with polynomials. Plane
curves, that is, those that lie in
R
2
or
C
2
, necessitate the introduction of cal-
culations and tools that are manageable but rich enough to give the reader
an appreciation of some of the ideas handled in modern algebraic geometry.
Our goal in this final section of the course is Bezout’s Theorem, which is a
description of the intersection of two curves in a plane.
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2.1
Synthetic Geometry
Background on Euclid and
The Elements
There are no contemporary accounts of Euclid’s life but there are some details
about Euclid that scholars have been able to cobble together indirectly and
about which there is little controversy.
It is generally held that Euclid founded a school of mathematics in Alexan-
dria, Egypt.
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In particular, Archimedes, 287-212 B.C., sometimes described
as the greatest mathematician who ever lived, studied in Alexandria at
Euclid’s school and cited Euclid’s work in his own writing. It is clear then
that Euclid predated Archimedes. On the other hand,
The Elements
has de-
tailed references to the works of Eudoxus and Theaetetus. To have learned
their work, Euclid would have to have gone to Plato’s Academy.[2], p. 2.
The Academy was established outside Athens around 387 B.C.
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There is
general agreement that Euclid was too young to have studied with Plato at
the Academy, and too old to have taught Archimedes in Alexandria. This
helps narrow down the dates for Euclid which are currently accepted as about
325-265 B.C. [6]
As a student at the Academy, Euclid would have been the product of
Alexandria is named for Alexander the Great, who, as a child, studied with Aristotle.
This is the origin of the word
academic. Academy
is actually the name of the place
where Plato set up his institution. The Academy remained in use until 526, another
astonishingly long-lived force in the intellectual world.
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a rich tradition of careful thought, schooled in the writings and teachings
of Plato, Aristotle, and the other Greek geometers. Aristotle, a student of
Plato himself, suggested in his own writings that his students had sources
which codified the principles of mathematics and science, including geometry,
that were accepted at that time. These sources presumably would have been
available to Euclid as well. In other words, Euclid’s was not the first geom-
etry text, even if we restrict attention to the West. This does not diminish
its greatness but it is important to maintain perspective. One commonly
accepted view of
The Elements
is that it pulled together what was known in
geometry at the time, with the goal of proving that the five platonic solids
are the only solutions to the problem of constructing a regular solid. There is
room for doubt there, though, as several books of
The Elements
have nothing
whatever to do with the construction of the platonic solids.[2], p. 2.
The Elements,
which is in thirteen books, starts with a set of 23 defini-
tions. We quote from [2], p. 153-154.
1. A
point
is that which has no part.
2. A
line
is a breadthless length.
3. The extremities of a line are points.
4. A
straight line
is a line which lies evenly with the points on itself.
5. A
surface
is that which has length and breadth only.
6. The extremities of a surface are lines.
7. A
plane
surface is a surface which lies evenly with the straight lines
on itself.
8. A plane
angle
is the inclination to one another of two lines in a plane
which meet one another and do not lie in a straight line.
9. When the lines containing the angle are straight, the angle is called
rectilineal.
10. When a straight line set up on a straight line makes the adjacent angles
equal to one another, each of the equal angles is
right,
and the straight
line standing on the other is called a
perpendicular
to that on which
is stands.
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