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Episode 303: Traffic--Explore the Math
Original Math Notes
Episode 303: Traffic
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 1:
Billions and billions of stars.

                 CHARLIE (O.S.)
          "God does not play dice with the
          universe."
          
The UNIVERSE dissolves into a PHOTO of EINSTEIN.

                 CHARLIE (O.S.) (CONT'D)
          Albert Einstein said this, in
          response to scientists who were
          postulating that our world, at its
          most fundamental level, is
          irreducibly random.

Lecture hall. Charlie projects SLIDES for an ungrad class.

                 CHARLIE
          Einstein wasn't alone in this
          belief.  Human beings in general
          have trouble with the concept of
          randomness.  We seek patterns, we
          perceive cause and effect.
          
A SLIDE of a COMMERCIAL AIRPORT with AIRLINERS.

                 CHARLIE (CONT'D)
          Let's say there are three airline
          crashes in two months. As happened
          in 1996.  Do you cancel your
          upcoming flight and drive instead?
          
A SLIDE of a HIGHWAY filled with traffic.

                 CHARLIE (CONT'D)
          What is essentially a random streak
          or "run," we perceive as a pattern. 

Leads in Coin Tossing

Consider a game in which a fair coin is tossed repeatedly. When the cumulative number of heads is greater than the cumulative number of tails, heads is in the lead. Tails moves to the lead when the cumulative number of tails is greater than the cumulative number of heads. Intuition might suggest that with a large number of coin tosses, heads and tails would spend roughly equal time in the lead.
Scene 15:
INT. DINING ROOM. EPPES HOUSE - DAY

                 MEGAN
          Charlie, in mathematics, can
          something be too random?
          
                 CHARLIE
          Randomness isn't quantifiable.
          Something can't be a little random -
          - or very random.  Or too random.

Random Number Generation

A random or pseudorandom number generator (RNG) is a computational or physical device designed to generate a random sequence of numbers. There are many different methods for generating random bits and testing their quality.
Scene 16:
INT. DINING ROOM. EPPES HOUSE - DAY

                 ALAN
          What if there's a pattern that you
          just haven't been able to detect?
          
                 CHARLIE
          That's Hidden Variable Theory. The
          idea that nothing in the world is
          ever really random -- because
          there's always some influence or
          force we can't detect.
          
                 ALAN
          And is that theory a good one?
          
                 CHARLIE
          That's what Einstein believed in
          1935, and it proved not to be true,
          at least in the way he believed it.
          Many scientists now believe that
          God does play dice with the
          universe.
          
                 ALAN
          But -- doesn't dice have rules?
          
                 MEGAN
          And a die only has six sides,
          right?
          
                 CHARLIE
          Okay, forget the dice, that
          metaphor's not so good.

Chaos Game

This Demonstration starts with the center C of a regular n-sided polygon and draws a new point at a fraction of the distance r between C and a random vertex. Repeating this process will, for certain initial conditions, produce fractal images. This procedure is known as the chaos game.
Scene 22:
INT. CALSCI. LARRY'S OFFICE -- DAY

Larry's working at his chalkboard. Charlie comes in.

                 CHARLIE
          Pythagorean theorem, Law of
          Cosines, metrics--
          
                 LARRY
          Equivalence principle.
          
                 CHARLIE
          Back to basics.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Scene 25:
INT. FBI BULLPEN -- DAY

Charlie pulls out the iPod again.

                 CHARLIE
          Remember I talked about the shuffle
          mode?
          
                 DAVID
          How people perceive patterns when
          there aren't any.
          
                 CHARLIE
          The shuffle mode on this music
          player is controlled by an
          algorithm.  That algorithm creates
          a random order for the songs to be
          played in.  But it's a psuedorandom
          order. You know why?
          
                 MEGAN
          No idea --
          
                 DAVID
          That's why we have you.
          
                 CHARLIE
          There's one thing the algorithm
          will never do ... repeat a song.
          It makes sure the same song is
          never repeated twice.

Using Rule 30 to Generate Pseudorandom Real Numbers

The rule 30 elementary cellular automaton is the engine behind Mathematica's generation of pseudorandom numbers. This Demonstration looks at an oversimplified version of how the central column in the evolution of this automaton can be used to generate pseudorandom real numbers between 0 and 1.
 
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