COURSE: MATH 5900-010 (Independent Study)

TIME: 2:15-3:45 TRF

PLACE: Gilbreath Hall in an empty classroom!

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: By appointment

PHONE: 439-6979 (308F), Math Office 439-4349

E-MAIL: gardnerr@etsu.edu WEBPAGE: www.etsu.edu/math/gardner/gardner.htm (see my webpage for a copy of this course syllabus and updates for the course).

INTRODUCTION: This part of your Independent Study class will be a one week introduction to non-Euclidean geometry with a review of Euclidean geometry. No text is required and notes will be posted online in PDF and PS formats for you to download. These are the notes we will cover in class, and all homework problems can be solved using only the information in the notes. We will cover three main topics: Euclidean geometry, hyperbolic geometry, and elliptic geometry. The notes for these topics are available, respectively, at:

• http://www.etsu.edu/math/gardner/noneuclidean/euclidean.pdf
  http://www.etsu.edu/math/gardner/noneuclidean/euclidean.ps
• http://www.etsu.edu/math/gardner/noneuclidean/hyperbolic.pdf
  http://www.etsu.edu/math/gardner/noneuclidean/hyperbolic.ps
• http://www.etsu.edu/math/gardner/noneuclidean/elliptic.pdf
  http://www.etsu.edu/math/gardner/noneuclidean/elliptic.ps

PRIMARY REFERENCES: The notes are based primarily on the following references:

1. An Introduction to Non-Euclidean Geometry, David Gans, Academic Press, 1973. (This is our main reference.)
2. The Thirteen Books of Euclid's Elements, Translated from the Text by Heiberg, Volume I (Introduction and Books I and II), Thomas Heath, Dover Publications, 1956.
3. Non-Euclidean Geometry, A Critical and Historical Study of its Development, Roberto Bonola, Dover Publications, 1955.
4. An Introduction to the History of Mathematics, 5th Edition, Howard Eves, Saunders College Publishing, 1983.

SECONDARY REFERENCES: Some brief use of the following may also by made:

1. A Survey of Classical and Modern Geometries, with Computer Activities, Arthur Baragar, Prentice Hall, 2001.
2. A Source Book in Mathematics, David Eugene Smith, Dover Publications, 1959.
3. Modern School Mathematics: Geometry, Ray Jurgensen, Alfred Donnelly, and Mary Dolcians, Houghton Mifflin, 1972. (This was my 10th grade high school geometry book.)

WARNING: We are interested in a survey of these areas and the level of rigor will not be up to the standard of most of our other graduate level classes. Also, I have quickly cobbled the notes together and the notation may be a bit inconsistent!

GRADES: Your grade for this part of the Independent Study (which will be about 20% of the course) will be based on your performance on assigned homework problems.

Johann Bolyai (1802-1860)

Karl Gauss (1777-1855)

Nicolai Lobachevsky (1793-1856)

Homework 1 (Euclidean Geometry)

1. Prove that nothing is lost by ending Postulate 5 with the word "meet." That is, assume the angles are as described and that the lines meet, then prove that the lines meet on the side on which the two angles are less than two right angles.
2. Prove that a line which meets one of two parallels also meets the other.
3. Prove that a line perpendicular to one of two parallels is also perpendicular to the other.
4. Prove that parallel lines are equidistant from one another (distance is measured by the length of a line segment perpendicular to both lines).
5. Prove that Playfair's Uniqueness Theorem and Postulates 1-28 imply the Parallel Postulate.

Homework 2 (Hyperbolic Geometry)

1. Prove that no two lines in hyperbolic geometry are equidistant from one another by showing that the distances from one line to another cannot have the same value in more than two places.
2. The set of points which are at the same distance from a given line and lie on the same side of it is called an equidistant curve, and the line is called the base line of the curve. Prove that in hyperbolic geometry no three points of an equidistant curve are collinear.
3. If the vertices of a triangle in hyperbolic geometry lie on an equidistant curve, show that the perpendicular bisectors of the sides are parallel, with a common perpendicular.
4. Show that a quadrilateral whose opposite sides are equal is a parallelogram, but that the converse is not true. (A parallelogram is defined to be a quadrilateral whose opposite sides lie on parallel lines.)
5. Show that there are no squares in hyperbolic geometry but that there are rhombuses with equal angles.
6. Show that two parallels with a common perpendicular are symmetrical to each other with respect to the point mentioned in Theorem H38 (this is used in the proof of Theorem H42).
7. Given an equilateral triangle, construct another with sides twice as long and prove that no angle of one triangle equals an angle of the other.

The following websites may be relevant to our study:

1. Euclid's Elements. This site includes statements of definitions, common notions, postulates, and propositions, but not proofs.
2. Hilberts Axioms of Euclidean geometry. A list of axioms to develope Euclidean geometry in a modern way. These are from Hilbert's The Foundations of Geometry. Another copy is available here.
3. The Poincare Disk, a website provided by Mathworld, including a few references and an animation showing different lines in this model of hyperbolic geometry.

Return to Bob Gardner's webpage.