Original Math Notes
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Episode 418: When Worlds Collide--Explore the Math
Original Math Notes
Episode 418: When Worlds Collide
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 9:
                  FRALEY
          How do computers relate to a
          terrorist cell?
          
                  CHARLIE
          Okay, think of, say, a rowing team.
          
ENTER AUDIENCE VISION:
An eight-man scull with a coxswain skims along.
                  
                  CHARLIE (V.O.) (cont'd)
          Imagine one rower secretly working
          against the others.
          
CLOSE ON ONE ROWER, moving his oar counter to the others.

                  CHARLIE (V.O.) (cont'd)
          What the math will do is look at
          the water in the boat's wake --
          
ON THE WAKE -- graphics show oars making complex swirls
behind the boat.  Math expressions overlay the swirls.

                  CHARLIE (V.O.) (cont'd)
          -- and tell us which oarsman isn't
          working with the team.
          
The POV fixes on a set of patterns that doesn't match the
others, and follows it back to the one oarsman.   

Wave Interference

This Demonstration shows interference patterns made by waves from two sources.
Scene 19:
CRIME SCENE PHOTOS of Rafiq and Jamali on the big screen:
dead and brutally beaten. Blueprints of the schools.
Fraley, Don and Megan.

                  CHARLIE
          My analysis has been unable to
          identify components that would
          correlate to the expected system.

                  FRALEY
          What are you saying?

Charlie takes a breath as he gets ready to lower the boom.

                  CHARLIE
          The PRF does not fit the profile
          for any known terror group.

                  FRALEY
          According to your computer logic
          analysis.

Skolemization

The process of removing all the existential quantifiers from a formula is known as Skolemization. The result is a formula in Skolem normal form that is equivalent in computational complexity to the original. This Demonstration shows the rewriting process of a Skolemization step by step for all general cases up to three quantifier alternations and seven variables or constants.
Scene 19:
                  CHARLIE
          Phil's work is in crop development,
          designed to create new strains of
          food crops resistant to drought and
          insects.  Both big problems in the
          farming regions of Pakistan.

                  FRALEY
          You're a mathematician.  This is
          biology.

The Tree of Life

The biodiversity of Earth has captured the interest of naturalists for many hundreds of years. With modern-day molecular and genetic techniques, it is possible to better understand the relationships between organisms, which are often shown in a phylogenetic tree.
Scene 49:
                  CHARLIE
          I think we're looking at "Angels
          and Devils."

                  FRALEY
          Excuse me?

                  CHARLIE
          M.C. Escher's print of angels and
          devils, an example of the figure-ground
          effect in perceptual organization.
          
Enter Audience Vision:

ESCHER'S CIRCLE LIMIT IV. An interlocking black and white pattern of
BLACK DEVILS --

The Banach-Tarski Paradox

This animation shows a constructive version of the Banach-Tarski paradox discovered by Jan Mycielski and Stan Wagon. The three colors define congruent sets in the hyperbolic plane H, and from the initial viewpoint the sets appear congruent to our Euclidean eyes.
 
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