MATH 412 Syllabus

Modern Geometries

Revised: November 2006

Course Description

Finite geometries, transformations, motions of the Euclidean plane and 3-space, similarity transformations, convexity, and other topics. Prerequisite: Junior standing or permission of department head.


1. Develop an understanding of the meaning and importance of the properties of consistency and independence of axiom systems.

2. Examine the concept of finite geometries, in particular those of Fano and Young.

3. Develop an understanding ofisometries and similarities in transformational geometry and the ability to treat them analytically.


Smart, James. Modern Geometries, Fourth Edition. Brooks Cole, 1993.


1. Krause, Eugene, Taxicab Geometry, Dover Publishers, 1986;


2. Kenney, Margaret J., Bezuszka, Stanley J., and Martin, Joan D., Informal Geometry Explorations, Dale Seymour Publications, 1992.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

  • Finite Geometry (3 weeks)
    Facts and relationships in several finite geometries;
  • Transformational Geometry (3-4 weeks)
    Mathematical transformation on given point sets and the Euclidean and non- Euclidean properties of these mappings; group and subgroup properties; transformations using vectors and matrices;
  • Convexity (2-3 weeks)
    Basic ideas of convexity (open, closed, bounded, interior, etc.) on given point sets in two and three space; convex hulls;
  • Non-Euclidean geometry (2-3 weeks)
    Fundamental properties of various non-Euclidean geometries (taxicab and classical--Riemann and Lobachewsky);taxi distance, taxi conic sections, taxi polygons, applied taxi problems, and metric space properties;
  • Ge ometric Explorations (2 weeks)
    Counting patterns, geometric constructions, congruence, symmetry, tiling, and related applications.
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