Tuesday,  March 2
Review of prvious discussion on
Space
: How do we understand objects in space?

How can the Flatlander experience the sphere and space?
Recall assignment: Make a torus with 2 and 1 piece!

Cross sections: Look at the octahedron with cross sections : squares, rectangles, triangles and hexagons depending on how the octahedron passes through the plane.

Shadows: Recall our previous class activity when we considered how the octhedron might case shadows.

Fold downs- flattened figures: Consider how the cube can be assembled from folded down squares in two different configurations: a cross or a "zig-zag."


What does a folded down flattened torus look like?
A rectangle with opposite sides resulting from cutting the torus open making a cylinder and then cutting the cylinder along its length.

    A torus





New:
analogue...  point... line.... polygon.... polyhedron......

What is the difference between Euclidean Geometry, Projective Geometry, and Topology

Euclid: congruence, similar... measurements, scale
Questions: Is a triangle congruent/similar to another triangle?
Is an circle congruent /similar to an ellipse?

Is a triangle congruent/similar to a square?


Projective: We will discuss this in greater detail later in the course.Projections preserve lines, points of intersection and contact (tangency).
Questions: Is a triangle projectively related to another triangle?
Is an circle projectively related to an ellipse?
Is a triangle
projectively related to a square?



Next class we will see how this formula can help solve some problems on the plane and elsewhere.