Thursday, January 29

Puzzles and Polygons [1.2]
• Dissections, cut and paste methods of measurement.
• Cutting and reassembling polygons.

• Tangrams.
• Tangram Activities last class

• Tangoes: a commercial game based on tangrams
Tapestry Project from previous Math 103 students.

Cutting and reassembling polygons.

Convex:    Any two points in the figure have a line segment connecting them. If that line segment is always  inside the figure, then the figure is called "convex".

Making Dissection Puzzles:

• Dissections (Junkyard)
• Equidecomposable polygons (translation from Portugese)

•
Where we are going:
* Scissors congruence: A sc= B means figure A can be cut into pieces that can be reassembled to form figure B.
This is also described using the word "equidecomposable".  "A and B are equdecomposable to B."
•  SC =  is a reflexive, symmetric, and transitive relation. [like congruence and similarity in geometry and equality in arithmetic]
• Theorem I : A sc= B implies Area(A) = Area(B)
• Theorem II [The converse of Theoerm I!]: Area(A) = Area(B)  implies A sc= B !!
• The presidential puzzles: Washington, ..., Jefferson,...,Lincoln, ... ,Clinton, Bush II.

• Simple cases as evidence and a foundation for building toward the proof of Theorem II.:
• A triangle is SC to a rectangle. Activity: • A rectangle is SC to a square. radius = r =  (a+b)/2   #
• length of DC  =  a- r
=  r - b
• x2 + (CD)2 =  r2
• x2  =  r2 - (CD)2  [Revised! 2-3-04]
=r
2 - (r-b)2
= r2 - (r2 -2rb+ b2)
=  2rb - b
2
=    (2r-b)*b     [Now notice that from # 2r-b = a!]
= ab

• Two squares are SC to a single square.
• A polygon is SC to a square.
• Triangulation .: Any polygom can be decomposed into triangles!
• If two polygons have equal area, then they are SC to the same square!