## TOPICS AND THEMES

Axiomatic System of Reasoning * * *

- What is geometry and how has it developed in history, art and science?
- What is the historical literature for a system of geometry? We will examine Euclid’s Elements, Book I.
- Changing our axiomatic system for geometry: what happens without the fifth Euclidean postulate? Or what is taxicab geometry? (Non-Euclidean systems in contrast)
- Categorizing our knowledge of 2D objects allows us to classify cases of shapes by essential, remarkable characteristics.

### Connecting Representations of Shapes: old and new ways to draw with tools

- Use a variety of representations for geometric ideas (drawings, text, software, objects);
- Exploring dynamic computer sketches of geometric ideas helps support argumentation and explanation of our generalizations about 2D space objects and relations.

### Measures

- What does it mean to measure paths or regions of the plane, or regions of space? What is a metric space?
- How can we analyze geometric plane objects in the Cartesian Coordinate Plane?
- How do we apply geometry to measure the earth and in astronomy more broadly?
- How does geometry promote navigation and way-finding on the surface of Earth?

### Symmetry of Structure: Mappings that preserve structure

- What is symmetry (point/axis and line/plane symmetries)?
- What is an isometry (a structure preserving operation)?
- What types of projections allow us to portray 3D objects on a 2D mapping?
- What happens when we combine symmetric operations or other transformations?