Wednesday, February 5, 2014
Class #5

Reminder to read Flatland.
Look over reading and assignments

Discussed briefly two types of symmetry for assignment due next week:

Reflection Symmetry across a vertical or horizontal line.
Rotational symmetry by 180 degrees.

Miscellaneous: Some topics we will study (subject to time for adequate discussions).

Symmetry.

Tiling.

The Sphere and the Torus. Surfaces

Proof and Truth in Geometry and Arithmetic.

Should we believe these dissection proofs? Closing discussion?

Discussion: The angles of the triangles in the figure are not complementary.
The triangles are not similar: the green/purple triangles have sides 3 by 8 while the triangle in the quadrilaterals have sides 2 by 5. Therefore they do not combine to have a right angle in the corners of the 5 by 13 figure.

Curry's Paradox: How Is It Possible?

Other visual tricks ...

so why should we believe Pythagoras? Because we demonstrated that in the argument for the activity we did, the angles did form a right angle because the triangles were all congruent.

Theme Question: Who first showed the earth was a sphere or shaped like a sphere?

More discussion on the The Sphere, The Plane, and the Torus.

Question 1. How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface?

Question 2. How can one distinguish the sphere from a torus based solely on experiences on the surface?

Dissection of the plane--- Tilings of the plane:
Tessellations

Activity: Tile plane with playing cards.