Modern Geometry –

mth405

(3 credits)

This course explores geometry from heuristic, axiomatic, and computational angles. Students examine ancient results, Euclid, non-Euclidean geometry via the Poincaré disk, and transformational geometry.

This undergraduate-level course is 5 weeks. To enroll, speak with an Enrollment Advisor.

Topics and Objectives

Axiomatic Systems and Ancient and Neutral (Finite) Geometries
- Solve elementary problems from Western and Non-Western sources.
- Demonstrate properties of components of axiomatic systems.
- Prove elementary theorems in neutral geometry.
Euclidean Plane Geometry and Constructions
- Prove theorems in Euclidean Geometry.
- Perform basic constructions with straight-edge and compass.
Transformational Geometry
- Demonstrate properties of isometries: reflections, rotations, translations, and glide reflections.
- Analyze the compositions of linear transformations.
- Identify the invariants of matrix transformations.
- Represent geometric objects and linear transformations using matrices.
- Represent direct and indirect isometries using matrix methods.
Non-Euclidean Geometry
- Explain the significance of denying the fifth axiom of Euclid's Playfair's Postulate.
- Construct objects in the Poincaré Disk.
- Calculate distances, angle measures, and areas in the Poincaré Disk.
- Prove Simple theorems of hyperbolic geometry.
- Summarize contributions of pivotal geometers.
Symmetry and Fractal Geometry
- Demonstrate properties of types of finite plane symmetry groups.
- Prove elementary results involving finite plane symmetry groups.
- Identify characteristics of self-similar geometric objects.
- Calculate the similarity dimension of geometric objects.

The University of Phoenix reserves the right to modify courses.
While widely available, not all programs are available in all locations or in both online and on-campus formats. Please check with a University Enrollment Advisor.
Transferability of credit is at the discretion of the receiving institution. It is the student’s responsibility to confirm whether or not credits earned at University of Phoenix will be accepted by another institution of the student’s choice.

Modern Geometry –

mth405

Axiomatic Systems and Ancient and Neutral (Finite) Geometries

Euclidean Plane Geometry and Constructions

Transformational Geometry

Non-Euclidean Geometry

Symmetry and Fractal Geometry

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