Wednesday, February 5, 2014

Class #5

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.

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.