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.

Making Dissection Puzzles:

• Dissections (Junkyard)
• visit Dissection Animations
• visit The Puzzling World of Polyhedral Dissections by Stewart T. Coffin.

 

 

• 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]
          =r² - (r-b)²
          = r² - (r² -2rb+ b²)
           =  2rb - b²
           =    (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!