Geometry Presentations, Interim 2003

Course Information

Syllabus

PRESENTATION GUIDELINES

Guidelines:

It is essential to maintain class interest (class involvement is a plus when it can be accomplished without sacrificing intellectual content). To facilitate class understanding, each presentation should include the distribution of a written outline and/or summary of the content of the presentation. This handout should also include a bibliography of two or three of the most pertinent sources.

Presenters should make every attempt to present the topic in an instructive and professional manner using appropriate terminology and illustrations. Presentations will be graded on understanding and communication of content, depth of research, effectiveness of presentation, and evidence of active involvement by all members of the group.

Suggestions for Preparing Presentations:

• Assign research responsibilities to each member of the group.
• Define specific goals for the presentation (What do you want to convey to the audience?)
• Decide what materials are needed for the presentation.
• Outline and prepare a class handout.
• Assign presentation responsibilities to each member of the group.
• Meet with the instructor at least one day prior to the presentation to review your plans.
• Practice the presentation several times so you can avoid reading notes and check your timing.
• RELAX.

A bibliography and other resources for each topic will be distributed at the initial presentation meeting with the instructor. If you need supplies for the presentation, see the instructor at least one day in advance.

Topics:

Each group must choose one of the topics listed below. The listed subtopics give an indication of the types of ideas that should be included.

1. Early Geometry: Euclid and Before (Jan. 14):
• Methods and contributions of ancient civilizations - both western and non-western.
• Mathematics of ancient Greece (the role of geometry, Pythagoreans, 3 classical construction problems, axiomatic method)
• Euclid's work as a "culmination" of previous western geometry.
2. Development of non-Euclidean geometry (Jan. 14):
• Early questions about independence of fifth postulate
• Major contributions of Saccheri, Gauss, Bolyai, Lobachevsky and others
• Final answer to the question, reasons for the delay, etc.
• Effects of this answer on understanding of mathematical and philosophical "truth"
3. Symmetry in Culture, Art, Science, Nature, Music, etc. (Jan. 15):
• What is symmetry? How has the meaning of the term evolved?
• Significance in the areas mentioned.
• Specific examples in one or two of these areas.
• Its increasing importance in mathematics.
4. Mathematics of Paper Folding (Jan. 15):
• Construction of conic sections via paper folding
• Use to determine all possible Platonic solids
• Other polyhedra
• Demonstration of theorems from Euclidean geometry
5. Introduction to Intuitive Topology -- (Jan. 20):
• Intuitive description of topology and topological transformations
• Variants and invariants under topological transformations (including no. of sides, edges, genus)
• Presentation of several famous topological problems
6. Other Tilings (Jan. 22):
• Use of periodic tilings in creating Escher-type designs
• Aperiodic and Penrose tilings of the Euclidean plane
• Tilings of the hyperbolic plane
7. Development of Projective Geometry (Jan. 27):
• Origins in artistic problem, attempts at solutions
• Introduction of vanishing point, horizon line, observation point, 1,2 and 3 point perspective using DGS
• Perspective views of common objects
• Introductory overview of projective geometry
8. The Fourth Dimension (Jan. 27):
• Early literary references, occurrences in art
• Review and description of Abbot's Flatland
• Generation of a hypercube by a cube
• Applications of the fourth dimension in mathematics and physics
9. Fractals in Culture, Art, Science, Nature, Music, etc. (Jan. 28):
• Cultures in which fractals have played significant roles.
• Significance of fractals in other areas mentioned.
• Specific examples in one or two of these areas.
• Their use in movies.

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