Thursday, September 4
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 !!
-
Simple cases as evidence and a foundation for building toward the proof
of Theorem II.:
-
A triangle is SC to a rectangle.
-
A rectangle is SC to a square.
-
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!
-
Discussed the presidential puzzles: Washington, ..., Jefferson,...,Lincoln,
... ,Clinton, Bush II.