I'd like to begin by saying that I'll do my best to achieve clarity, but that I'll still probably manage (unintentionally) to say a few things that won't seem perfectly transparent the first time that you read them -- especially in discussing some of the subject matter. So, please feel free to ask questions!! Also, some of these statements likely will seem a lot clearer after you have studied the corresponding topics in the course. Accordingly, it's recommended to re-read parts of the material at various times during the semester and/or after the end of the semester.
1. Statement of purpose. First of all, here are one or two things that the course isn't. Even though most students take this course as a requirement for secondary math teaching licensure, this isn't a teaching methods class. And we don't go through the material in an order that parallels what's done in a high school geometry course -- and often not even in ways that "maybe could be" presented to high school students learning geometry for the first time.
So, if it's neither of those things, then ¿what is it about? and ¿why is it relevant? A brief answer is that that learning the material presented in this course will help you to (i) extend your knowledge of geometry, and (ii) deepen your knowledge of the subject. Both of these goals are plausible and feasible for the following reasons:
So, in summary: we'll accomplish our goals by (i) learning
some things about "what happens next" after Euclidean geometry, and (ii)
solidifying our understanding of the conceptual underpinnings of the subject.
Having done this, you'll be several steps ahead of what you'll be teaching,
presumably in a strategically chosen direction. And, hopefully, this will strengthen
your ability to make good choices about how to present the material that you will
be teaching, and to adapt successfully to changes in textbooks and/or the officially
recommended version of the syllabus of the geometry course that you're going to teach.
2. Some history. The
parallel axiom in Euclidean geometry can be stated as follows:
Starting with the ancient Greeks and continuing up to around the year 1800,
there was some lingering uneasiness about this axiom. The other axioms of
Euclidean geometry could be viewed being self-evident properties of the
physical world, but it did not seem to be possible to physically verify
the parallel axiom. Indeed, such a verification would seem to involve
measurements on a scale that was so large as to be impractical.
Instead, a number of mathematicians tried to show that the parallel
axiom could be proved from the other axioms of Euclidean geometry. All
of these efforts failed. We now know the reason for this: the
other axioms of Euclidean geometry, while consistent with the Euclidean
parallel axiom, also are consistent with the hyperbolic parallel axiom,
which says that there are at least two lines through a given
external point and parallel to a given line. The fact that the hyperbolic
parallel postulate is consistent with the other axioms of plane geometry was
discovered independently by Nicolai Inanovich Lobachevsky and János
Bolyai, who published their findings in 1829 and 1832 respectively
[Cederberg]. Apparently, Karl Friedrich Gauss may
have discovered this fact some years previously but did not publish his
results, writing in 1829 that this was because he feared the "screams of
the dullards", so entrenched were the ideas of Euclid [Ryan].
The coordinate plane R2
studied in analytic geometry classes is the most familiar model of Euclidean
geometry. Indeed, it satisfies all of the axioms of Euclidean geometry,
including the parallel axiom. But the coordinate plane also provides the
materials for constructing a model of hyperbolic geometry.
In this course, we'll study the Poincaré half plane
model. In this model the points are the usual points of
the upper half plane (y-coordinate > 0), and there are two
kinds of lines:
Some work is needed to prove that the Poincaré half plane model
satisfies the axioms of hyperbolic geometry, including the hyperbolic
parallel postulate. This brief introduction is not the place to present
extensive details. But I definitely want to emphasize the remarkable
fact that the standard model of Euclidean geometry actually contains the
materials for constructing a model of hyperbolic geometry !!
For more detailed information about the Poincaré half plane model, see
[Stahl].
3. Back to the beginning.
Having established some of the context, it's time to say something more
specific about what we're actually going to do in this course. Since we'll
study the chapters of the text in order, we'll organize this account similarly.
Basics of vector geometry This is similar to
the vector geometry that's used in multivariable calculus, although some
of our notation may be new to you. We'll use linear algebra methods to prove
a few propositions whose content is basically geometric. For instance, the
Cauchy-Schwartz inequality (a linear algebra fact that we'll learn
about) will be used to prove the triangle inequality -- which basically
says that the length of one side of a triangle is less than the sum of the
lengths of the other two sides [or maybe equal, if the "triangle" is flat].
Congruence, isometries, and barycentric coordinates
By definition, an isometry is a transformation of the plane that
preserves distance, i.e., a transformation
T: R2 --> R2
such that the distance from T(P) to T(Q) is
equal to the distance from P to Q for every pair of
points P and Q. We'll say that two figures are congruent
(by definition) if there's an isometry which maps one of them to the other one.
Using a linear algebra concept -- the inner product, also called the
dot product -- we'll prove that isometries also preserve angular measure.
This leads to proofs of the side-angle-side (SAS) and angle-side-angle (ASA)
criteria for congruence of two angles. We'll also develop some matrix
formulas that define isometries.
Triangles and other plane figures We'll use vector
geometry methods to study geometric topics like centroid, incenter, and
circumcenter for triangles.
Classifying isometries We'll classify isometries
of the Euclidean plane according their geometric types [translations, rotations,
reflections, and glide reflections], and we'll study the
symmetries of planar figures, i.e., the set of
isometries which map a given figure to itself.
Conformal mappings By definition, a
conformal mapping is a transformation which preserves
angles. As noted above, this includes isometries. Transformations that
preserve similarity represent another type that's related to high
school geometry. Still another type of conformal mapping is the
circle inversion. Here's what this looks like in
the case of the standard unit circle. Thus, if
X = (a,b) in
R2, let
||X||2 =
a2 + b2
be the square of the usual norm of
X. We define the circle inversion
[in the unit circle centered at the origin] by mapping
X = (a,b)
to (1/||X||2)X = ( a/(a2 + b2) , b/(a2 + b2) ). Since points on the
unit circle satisfy ||X|| = 1, each point on the
unit circle is mapped to itself. Otherwise, the interior of the unit circle
[except for the origin ... ] is mapped to the exterior
of the unit circle, and the exterior is mapped to the interior.
The axioms of geometry A somewhat systematic
study of the axioms of geometry, along with models
of various subsets of the axioms. We won't have time to do an exhaustive study,
as that would force us to re-trace a lot of our steps. But we'll discuss some
interesting stuff such as the between-ness axioms. The need for
those axioms was overlooked by Euclid, and also by various commentators during
the 1800's, but was noticed shortly before 1900 by David Hilbert
[Hilbert]. (Also see [Moise] for
discussion of formulations of the axioms of geometry by Hilbert,
G. D. Birkhoff, and others.)
Introduction to hyperbolic geometry Lines, angles, and
area in the Poincaré half plane model. (For an approximate description
of this, see 2. above; for lots of details specific
to our approach, see [Stahl]. For a radically different
approach that includes a "real-world application" of hyperbolic geometry,
see [Ryan], pp. 150-156.) Other topics, including
Poincaré isometries and trigonometry, if time permits.
4. Text:
5. Other useful books
All of these books are available in the Mathematics Library
(310 Vincent Hall).
6. Prerequisites, etc.
7. For further information:
Please
send me an e-mail or call me
at the phone number listed below.
Back to the class homepage.
Prof. Joel Roberts
Office: 351 Vincent Hall
The fact that there are two kinds of lines does tend to introduce some
extra complexity into the proofs. But it will be manageable if we
remain mindful of the need to consider various cases.
Note: Reading that section of Ryan's book requires
a slightly more sophisticated viewpoint about bilinear forms than is provided
by many undergrad linear algebra courses. But I'll be happy to explain the
pertinent ideas to member of the class who is interested.
(Available from Alpha Print in Dinkytown -- 1407 4th St. SE)
For Fall 2005: specify bin #12
Springer-Verlag 1989.
W. H. Freeman 1980.
Chicago, The Open Court Publishing Company, 1902.
Addison-Wesley, 1963.
Cambridge University Press 1986.
Jones and Bartlett, 1993.
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts