Links and Class Schedule for MATH 406 Geometric Structures
Publisher & Author Provided Web Resources:
Class Sessions:
Session #1 (January 26): Explore and Discuss Properties of the Medians of a Triangle
Session #2 (January 28): Validate Conjectures about the Medians of a Triangle
Session #3 (January 31): Continue Our Validation of Conjectures (Page 1 , Page 2)
Session #4 (February 2): Axiom System #1 (Page 1 , Page 2)
Session #5 (February 4): Axiom System #2 , Page 2
Axiom System #3 (Figure 1.2.3 in text)
Session #6 (February 7): Establishing Independence & Consistency for Axiom Systems
(The Role of Models; Axiom System #5)
Session #7 (February 9): Euclid's Definitions, Postulates, & Common Notions
Neutral Geometry (Section 1.3 in text)
Module for Hyperbolic Geometry (Several Models)
Session #8 (February 11): Characterization of the Real Number System, Hyperbolic Geometry
Session #9 (February 14): Synthetic Euclidean Geometry (System #4: Hilbert's Axioms)
Session #10 (February 16): Synthetic Euclidean Geometry Continued (System 4: Hemmer's Axioms)
Session #11 (February 18): Synthetic Euclidean Geometry Continued (System 4)
Session #12 (February 21): Synthetic Euclidean Geometry Continued (System 4)
Session #13 (February 23): Synthetic Euclidean Geometry Continued (System 4) Sample proofs.
Session #14 (February 25): Group Work, Questions, Discussion, Review, List of Synthetic Axioms
Session #15 (February 28): Questions, Discussion, Review
Session #16 (March 2): Student Presentations at Board in Response to Class Questions
Session #16 (March 4): Test #1 (Study Guide) (List of Axioms for Synthetic Geometry)
Session #17 (March 7): Parallelism & Non-Euclidean Geometries (Section 3.1); SMSG Axioms
Some Theorems in Basic Geometry
Session #18 (March 9): Poincare's Model of Hyperbolic Space (Section 3.2);
Supplementary Notes on Circular Inversion and Orthogonal Circles
Session #19 (March 11): Polygons in Hyperbolic Space (Section 3.3)
Session #20 (March 14): Congruence in Hyperbolic Space (Section 3.4)
Session #21 (March 16): Working in Hyperbolic Space (Continued)
Session #22 (Match 18): Some Notes on Hyperbolic Geometry (Page 1 , Page 2)
Session #23 (March 28): Analytic Model of the Euclidean Plane (Section 4.1)
Session #24 (March 30): Section 4.1 Continued; Introduction to Section 4.2 (Notes: Page 1 , Page 2)
Session #25 (April 1): Section 4.2 Linear Transformations, Isometries & Matrices (Continued)
Session #26 (April 4): Direct Isometries: Translations & Rotations
Session #27 (April 6): Test #2 (Study Guide: Covers Chapter 3 plus Some of Chapter 1)
Session #28 (April 8): Translations, Rotations, and Reflections (Sections 4.2 & 4.3) Class Notes
Session #29 (April 11) Indirect Isometries: Reflections (Section 4.4)
Session #30 (April 13): Composition of Transformations (Section 4.5)
Session #31 (April 15): Composition of Transformations (Section 4.5)
Session #32 (April 18): Invariants under Transformations
The Image of a Line (Theorem 4.2.4)
Session #33 (April 20): Invariants Continued & Session 33 Notes
Session #34 (April 22): Other Linear Transformations (Section 4.6)
Session #35 (April 25): Other Linear Transformations (Section 4.6)
Session #36 (April 27): Topics in Transformational Geometry
Session #37 (April 29): Topics in Transformational Geometry
Session #38 (May 2): Review
Session #39 (May 4): Sample Exercise on Invariant Points & Lines
Test #3 (Covers material in Chapter 4; Take home. Due May 9)
Session #40 (May 6): Spherical Geometry, Affine Geometry
Test #3 (Write Two definitions in Class)
Session #41 (May 9):
Session #42 (May 11):
Some Relevant Links:
M433 Modern Geometry: A Dynamic Approach, Gilbert An Excellent set of course notes with Sketchpad applications.
Axiomatic Systems for Geometry, G. Francis Compares four axiom systems for Euclidean geometry
Definitions of Some Geometric Terms
The Origins of Geometry, D. Royster Eucild in modern language and the Parallel Postulate
Neutral and Non-Euclidean Geometries, D. Royster, Hilbert's and Birkoff's Axioms for Neutral Geometry
Hilbert's Axioms for Euclidean Geometry
Birkoff's Axioms for Euclidean Geometry
Essay on Euclidean, Elliptic, and Hyperbolic Geometry
SMSG Axioms for Euclidean Geometry
Four Axiom Systems for Euclidean Geometry, R. Campbell at UMBC
Java Software for Constructions in Hyperbolic Geometry, Copyright©: Joel Castellanos
A Non-Euclidean Geometry Tutorial, Jacob Graves
Hyperbolic Geometry Illustrated with Cabrini Java
Web-Based Euclid's Elements, D. E. Joyce
A Escher Worksheet, Sarah J. Greenwald
Another Essay on Non-Euclidean Geometry , J J
O'Connor and E F Robertson
The Scandal of Geometry: An Essay on Euclid's Fifth Postulate, Circles and Hyperbolic Space
Modules for Non-Euclidean Geometry; http://www.towson.edu/~gsarhang/
Hyperbolic Circles, Describing circles in hyperbolic space
Review of Geometry: A List of Theorems
Review of Elementary Plane Euclidean Geometry, by Ken Monks (pages 25-30)
Some Ideas from High-School Geometry, James J. Madden
Essay on Circle Inversion, by Jon Challen UGA, Includes some GSP files
A Module on Hyperbolic Geometry with Applets, Published by Journal of OnLine Mathematics (JOLM)
Cabrini World 2001, An excellent introduction to Hyperbolic Geometry using CabriniJava figures by Tim Lister.
Geometric Transformations, Summary of matrix representations for geometric transformations by Bill Toll
Spherical Sketchpad A sketchpad type tool for spherical geometry.
High School Geometry Standards:
South Putnam HS District's (Indiana) Standards