What is Geometry What are some key concepts

What is Geometry? • What are some key concepts? • How is geometry used? • What are some adjectives that describe geometry? (ex fun, creative, boring, …) • Where does geometry show up in the classroom? • How does geometry connect with other areas of math or other subjects?

What is Straight? • “Straight is that of which the middle is in front of both extremities” Plato • “A straight line is a line that lies symmetrically with the points on itself” Euclid

When do you call a line straight? • Try to build a notion of straightness based on your experiences rather than accept a certain number of assumptions • Though it is difficult to formalize, straightness is a natural human concept • How do you check in a practical way that something is straight? Ex: hanging a painting on the wall

Symmetries of a Line • Reflection symmetry • Half-turn symmetry

Isometries and Symmetries • An isometry is a transformation that preserves distances • A symmetry of a figure is an isometry of a region of space that takes the figure onto itself • What other symmetries does a line have?

Straightness on a Sphere • Paths that are • All geodesics on spheres instrinsically straight on are great circles a sphere (or other surface) are called geodesics • At every point and in every direction on a smooth surface there is a unique geodesic going from that point in that direction

Why do we care? • When do we use “straight” lines? • What do we do with straight lines?