- Measurement and the Pythagorean Theorem (PT) [1.1]
a2 + b2 = c2
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?
From Wikipedia, the free encyclopedia.
The halting problem is a decision problem which can be
informally stated as follows:
Given a description of an algorithm and a description of its initial arguments, determine whether the algorithm, when executed with these arguments, ever halts (the alternative is that it runs forever without halting).
Alan Turing proved in 1936 that there is no general method or algorithm which can solve the halting problem for all possible inputs.
The importance of the halting problem lies in the fact
that it was the first problem to be proved undecidable. Subsequently, many
other such problems have been described; the typical method of proving
a problem to be undecidable is to reduce it to the halting problem.
Show video on PT- 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.
Making Dissection Puzzles:
* 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."
o SC=
is a reflexive, symmetric, and transitive relation. [like congruence and
similarity in geometry and equality in arithmetic]
o Discussion
Question: What kind of motions are used in reassembling pieces in a puzzle?
o Theorem I
: A sc= B implies Area(A) = Area(B)
o Theorem II
[The converse of Theoerm I!]: Area(A) = Area(B) implies A sc= B !!
o Simple cases
as evidencs and a foundation for building toward the proof of Theorem II.:
+ Parallelograms with common base between the same paralle lines are S.C..
+ Parallelograms with common parallels and same area are S.C..
+ Parallalograms with same area are S.C..
+ Triangles and parallelograms with same area are S.C.
+ Adding parallelograms.
+ Adding triangles.
Two issues that are consequences of the puzzle discussion: