- 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

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)

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=a^{2 }/c + b^{2 }/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.

- 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.
**A quadrilateral can be made from two triangles...**

so the sum of its interior angles is 2 * 180 = 360.**A pentagon can be made from 3 triangles... so the sum of its interior angles is 3* 180 = 540. If the pentagon has all angles congruent( of equal measurement) then each angle will be 540/5 = 108 degrees!**

**A hexagon can be made from 4 triangles... so the sum of its interior angles is 4* 180 = 720. If the hexagon has all angles congruent( of equal measurement) then each angle will be 720/6 = 120 degrees!**