Tuesday, January 25

• More Discussion (brief) on the The Sphere  and the Torus.

• How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface?
How can one distinguish the sphere from a torus based solely on experiences on the surface?

Use shadows?: look at shadows at the same time of day? This is a "local"  feature of the surface.
Observe "curvature"? This is also a local property.

Circumnavigate (Global)?:  Go West -> return from the East, then  go North-> return from the south.
On a sphere: there will always be 2 points of intersection of the curves determined by the two routes.
On a torus: There will be only one point of intersection of the two routes.
plane: Go West ->  you keep going... there seems to be no return???
Other issues: What about strange gravity? Finding an edge? How do you know when you start?

• Measurement and the Pythagorean Theorem (PT)

• a2 + b2 = c2

Show video on PT- (Video Number 950) Put on reserve in library!
Background: Similar triangles
Area of triangles = 1/2 bh
Area of parallelogram= bh
Scaling: a linear scale change of r gives area change of factor r^2.
3 questions: running, moat, wind power...
Proof of the PT: Similar right triangles: c= a2 /c + b2 /c .
applications and other proofs.
Prop. 47 of Euclid.
Dissection Proof.
Prop 31 Book  VI  Similar shapes.
Simple proof of PT using similar triangles of the triangle.
Use in 3 dimensional space.

Puzzles and Polygons
• Measuring angles, lengths and areas.
• Squares, rectangles  : 90 degree/ right angle
• parallelograms: opposite angles are congruent, sum of consecutive angles =180 degrees
• triangles  : add to 180 degrees- straight angle [Illustrated physically and with wingeometry]
• Dissections, cut and paste methods of measurement.
• Cutting and reassembling polygons.
• The "Square Me" Puzzle
• The triangle, quadrilateral, pentagon, and hexagon.
• More on measurements of angles and areas of polygons.